Math 2 Flashcards
Simplify the fraction:
4 14 5
—- x —- x —-
21 13 8
Prime numbers and cancellation (numerators with denominators) is a convenient approach.
2 x 2 x 2 x 7 x 5 5
———————– = —-
3 x 7 x 13 x 2 x 2 x 2 39
Simplify the Fraction
6x
—-
70
35
To evaluate 6.75 x 10^3:
Move the decimal to the right 3 places.
6,750.
To evaluate 72.12 x 10^-4:
Move the decimal to the left 4 places.
0.007212
To evaluate 54.197 / 10^2:
Because we are dividing by 10^2, we move the decimal to the left 2 places.
Convert 70% as a Fraction and Decimal:
Fraction:
7
—
10
Decimal
0.7
Convert 100% as Fraction and Decimal:
Fraction:
1
Decimal
1.0
Sometimes it is convenient to rewrite a fraction in order to separate a variable.
For example, three quarters of “T”.
3T
—-
4
How could be the other form of the fraction?
3
— T
4
How many zeros are there in a Billion?
9.
17 Billion = 17,000,000,000
In the question, “How many one-fourths are in 3/5 of 25/2”, where does the variable goes?
x 3 25
—- = — X —
4 5 2
x = 30
Multiplier of 7.5% increase?
1.075
Multiplier of a reduction of 8.8%?
1 - 0.088 = 0.912
In a percent difference problem, when involved decimals (those squared get smaller), I have to perform the subtraction in order of appearance of the quantities.
After, I have to divide by the basis of the comparison, which is what follows the word “than”, in the problem statements.
In a percent difference problem, when involved decimals (which squared get smaller), I have to perform the subtraction in order of appearance of the quantities.
After, I have to divide by the basis of the comparison, which is what follows the word “than”, in the problem statements.
Can I reduce terms to simplify the math when multiplying fractions?
Yes.
3/8 x 12/5 x 5/2 reduces to:
3/8 x 6/1 x 1/1
It is cross cancelation. The numerator of one fraction can get reduced by the denominator of other fraction.
Can I reduce terms to simplify the math when multiplying fractions?
Yes.
3/8 x 12/5 x 5/2 reduces to:
3/8 x 6/1 x 1/1
It is cross cancelation. The numerator of one fraction can get reduced by the denominator of other fraction.
x - 4
x + 4
-16 and -4 turns positive +4
(2x) (3x) =
6x^2
Never forget to square the variables too.
2x^2 + 11 - 6 = 0
I can factor out the 2, and divide 11/2 = 5.5
In this exercises I can also work with decimals.
“DISTANCE” FORMULA
(Rate) (Time)
Remember the fraction D/RT, and isolate the variable you need.
“RATE” FORMULA
Time
Remember the fraction D/RT, and isolate the variable you need.
“TIME” FORMULA
Rate
Remember the fraction D/RT, and isolate the variable you need.
On a Distance / Rate / Time problems, remember:
- If the *cars are moving apart and I have to find the average speed of each *car, I have to pay attention to the “Total Distance Apart from each other” number (given in the problem).
- Use the proper formula.
- Then set an equation in which I add up the Distance 1 + Distance 2 = “Total Distance Apart from each other” number (given in the problem).
Pythagorean Theorem:
If I know two sides I can get the Hypotenuse. Get it with sides 6 and 7:
(6)^2 + (7)^2 = Square Root of 85
(6)^2 = 36
+
(7)^2 = 49
——
85. -> Then, I just add the Square Root symbol.
SLOPE (m) FORMULA
Second x - First x
COORDINATE GEOMETRY
How can I get the “y - intercept”?
In the Slope Formula (m): y = mx + b, “b” is the “y - intercept”.
- Take the “x” and “y” coordinates of the same point. Ex. (3,2)
- Plug them into the Slop Formula, into the “y” and “x” variables.
- Plug in the Slope for “m”.
- Solve for “b”.
k(2) + 3k(1) = 17
2k + 3k = 5k = 17 —-> k = 17/5
I arranged the “2” to properly perform the addition.
k(2) = 2k
SLOPE is
RUN
(-3x)^2 =
x = 2
36.
(-6)^2
Incorrect to square each the -3 and the x. The multiplication inside the parenthesis is first, then I can square.
SIMPLIFY
x - (3 - x)
2x - 3
x - 3 + x = 2x - 3
If a negative sign is right to the parenthesis I have to multiply if by the signs inside the parenthesis. That’s why I got the “+x”, resulting in the “2x”.
Distributive Law
A (B + C) =
AB + AC It works with addition and subtraction. A (B - C) = AB - AC It does not work with multiplication: 3(xy) does NOT equal (3x) (3y) ---> It equals 3xy
(x^a) (x^b) =
x^a+b
Is (a + b)^2 = a^2 + b^2 ?
No.
It is absolutely illegal to “distribute” an exponent across addition and subtraction.
Instead we use the square of a sum.
Explain this algebraic pattern:
a^2 - b^2
This pattern is known as the Difference of Two Squares.
a^2 - b^2 = (a + b) (a - b)
Elaborate the expression:
9x^2 - 16
See the pattern. It is a Difference of Two Squares.
3x - 4) (3x + 4
Difference of Two Squares
x^7 - 4x^5 =
x^5 (x^2 - 4) = x^5 (x + 2) (x - 2)
Remember that The Difference of Two Squares cannot be that obvious.
y^2 = x^2 + k
Then, “k” = y^2 - x^2 = (x + y) (x - y)
When Algebraically asked to find the side of a geometric figure, such as a Square, I might get two results of “x” (which is the side). One positive and one negative. So I have to pick…
The positive number. I have to discard the negative value.
The edge of a square must be positive.
In the “Which Quantity is Greater A or B”, GRE exercises, what are the options?
A. Quantity A is greater.
B. Quantity B is greater.
C. The two quantities are equal.
D. The relationship cannot be determined from the information given.
INEQUALITIES
When finding possible values of “x”, and I have been given two expressions, one way to get them is:
To align the expressions and add them up.
One variable must cancel out.
I have to solve for the desired variable.
Turn the sign if I’m dividing by a negative number.
INEQUALITY
|2x - 4| < 6
When solving for an absolute value of “x”, I have to:
Divide the inequality into two parts: 2x-4 < 6 2x-4 > -6 Solve each one. Finally, you will get a range in which "x" will be located. Express it in one range. -1 < "x" < 5
GEOMETRY
Acute Angles?
Ángulos Agudos
Miden menos de 90 grados.
GEOMETRY
Obtuse Angles
Ángulos Obtusos
Miden más de 90 grados.
GEOMETRY
3-D Formula to get the Volume of a Cube:
S^3
*S = Side
GEOMETRY
3-D Formula to get the Surface Area of a Cube:
6s^2
What does “This year the GRE All Stars had 3 times as many wins, and one-half as many losses as they had last year”, means?
2014 2015
Wins X 3X
Losses Y Y/2
Is 2sb = 2s * 2b ?
No.
2s * 2b = 4sb
Usually, how does a Function looks like in a graph? How can I identify the graph?
For example: g (x) = |x-1| -1
The function is an absolute value, which typically has a V-shape.
I can identify the correct graph by trying x = 0, which yields g (0) = 0, the origin.
Then, try x = 1, which yields g (1) = -1. And the point (0,-1).
I can try with some other numbers and see where they fall on a graph.
-3(-2) + 6/3 - (-5) =
13
6 + 2 + 5 = 13
-5^2
-25
Make sure to read this as -(5^2); NOT, (-5)^2 = 25
3 x 99 - 2 x 99 - 1 x 99 =
0
99 (3 - 2 - 1) =
99 (0) = 0
Can I solve a couple of equal equations to get the value of two variables?
No.
There would be the same equations.
I can only solve two equations for two variables if the equations are different.
GEOMETRY
Punto donde dos líneas se encuentran:
Vértice - Vertex
GEOMETRY
Acute Angles - Ángulos Agudos miden menos de:
90 grados
GEOMETRY
Ángulos Rectos - Right Angles, miden exactamente:
90 grados
GEOMETRY
Straight Angles - Ángulos Llanos, miden exactamente:
180 grados
GEOMETRY
Define “Complementary Angles”
Ángulos Complementarios.
Cuando la medida de dos ánguulos suma 90 grados
GEOMETRY
Define “Supplementary Angles”
Ángulos Suplementarios
Cuando la medida de dos ángulos suma 180 grados
GEOMETRY
Define “Perpendicular Lines”
Dos líneas que se encuentran en “ángulo recto” son perpendiculares.
Si te dicen que dos líneas son perpendiculares, piensa 90 grados.
GEOMETRY
Define “Polygons”
Un Polígono es una figura de dos dimensiones con tres o más lados rectos; llamados de acuerdo al número de lados que tengan.
3 - Triangle
4 - Quadrilateral
5 - Pentagon
GEOMETRY
¿Cómo calculo la suma de los ángulos interiores de un “Polygon”?
Suma de los ángulos interiores en un Polígono de “n” lados = (n-2)180
GEOMETRY
¿Cuál es la suma de los ángulos exteriores de cualquier Polígono?
360 grados.
GEOMETRY
En un Triángulo, la medida del ángulo exterior es igual a:
La suma de los ángulos remotos interiores.
Si tengo el Triángulo A,B,C, el ángulo fuera y hacia la derecha de “C” es A + B
GEOMETRY
Define “Isosceles Triangles”
- Los Triángulos Isósceles tienen dos lados iguales y dos ángulos iguales.
- Los dos ángulos iguales están opuestos a los dos lados iguales.
- Si sé que es un Triángulo Isósceles, con saber cuánto mide uno de sus ángulos, puedo calcular la medida de los otros dos.
GEOMETRY
Si un Triángulo tiene dos lados que miden lo mismo (y por ende, dos ángulos que miden lo mismo), sé que se trata de:
Un Triángulo Isósceles.
GEOMETRY
Define “Equilateral Triangles”
- Tienen tres lados iguales y tres ángulos iguales.
2. Como 180 grados entre 3 = 60, cada uno de sus tres ángulos interiores mide 60 grados.
GEOMETRY
Define “Right Triangles”
- Es cualquier triángulo con un ángulo de 90 grados.
- El lado opuesto al ángulo recto: Hipotenusa
- Tienen su propia regla: Pythagorean Theorem ->
a^2 + b^2 = c^2
GEOMETRY
Define “Special Right Triangles”
Dos Triángulos Rectos que con frecuencia aparecen en el exámen:
30-60-90 y 45-45-90
GEOMETRY
Características del “Special Right Triangle”, 30-60-90.
Leg : Leg : Hypotenuse
x x * Square Root of Three 2x
Hidden in “Equilateral Triangles”
GEOMETRY
Define “Equilateral Triangles”
- Tienen tres lados iguales y tres ángulos iguales.
2. Como 180 grados entre 3 = 60, cada uno de sus tres ángulos interiores mide 60 grados.
GEOMETRY
Características del “Special Right Triangle”, 30-60-90.
Leg : Leg : Hypotenuse
x x * Square Root of Three 2x
Hidden in “Equilateral Triangles”
GEOMETRY
Define “Special Right Triangle”, 45-45-90
También llamado: “Isosceles Right Triangle”.
Leg : Leg : Hypotenuse
x x x * Square Root of 2
Hidden in Squares
GEOMETRY
Define “Congruent Triangles”
Triángulos que tienen misma forma y tamaño. Son idénticos, pero tienen diferente orientación en el espacio.
GEOMETRY
Define “Similar Triangles”
Dos Triángulos son Similares si sus tres ángulos son idénticos.
Tienes la misma forma pero no necesariamente el mismo tamaño.
Probable pregunta GRE: Determinar cuál es el grado en el que un Triángulo es mayor que otro.
La relación (Ratio), de cualquier par de lados es la misma.
a b c
– = – = –
X Y Z
GEOMETRY
Define “Quadrilaterals”
- Polígono de 4 lados.
- Sus 4 ángulos internos suman 360 grados.
- Cinco tipos principales: Square, Rectangle, Parallelogram, Trapezoid, Rhombus.
GEOMETRY
Define “Parallelograms”
Un Paralelogramo es un Cuadrilátero.
Area = b x h
GEOMETRY
Define “Diagonals of a Rectangle”
La diagonal de un Rectángulo forma Triángulos Rectos que incluyen la diagonal y dos lados del rectángulo.
Si conoces dos de estos valores, puedes calcular el tercero con el Teorema de Pitágoras.
GEOMETRY
Define “Diagonals of a Square”
Una diagonal divide al cuadrado en dos “Special Right Triangles”, 45-45-90
Dos diagonales, dividen al Cuadrado en cuatro triángulos 45-45-90
GEOMETRY
Circumference of a Circle formula:
2 * PI * r
O bien:
c = PI * d
GEOMETRY
Podemos calcular la longitud del Arco de un Círculo, si conocemos:
- Radio
- La medida del ángulo central (ángulo cuyo vértice es el centro del Círculo), del inscrito (ángulo cuyo vértice se encuentra sobre la circunferencia del círculo), que forman el Arco.
Formula: Arc Length = n/360 x 2PIr
“n” es la medida en grados del Arco.
GEOMETRY
Define “Area of a Sector”
Un “Sector” es una “rebanada de pizza”.
n/360 * PI r^2
GEOMETRY
Define “Concentric Circles”
Dos o más círculos con el mismo centro
GEOMETRY
What does it mean: “Circle A is perfectly inscribed in a Square; and the Square is perfectly inscribed within Circle B?
- Circulo A, dentro del Cuadrado.
2. Cuadrado dentro del Círculo B.
WORD PROBLEMS
What it means: 8 more than “x”
x + 8
WORD PROBLEMS
What it means: “x less than 2”
2 - x
WORD PROBLEMS
What it means: “2 less 16”
2 - 16
WORD PROBLEMS
What it means: “y can take any value up to 4”
y <_ 4 Four or less
WORD PROBLEMS
What it means: “y has a value of at least 4”
y _> 4 4 or more
WORD PROBLEMS
What it means: “The quotient of a and b”
a/b
WORD PROBLEMS
What it means: “One-Third the difference of a and b”
(1/3) (a-b)
WORD PROBLEMS
What it means: “a seven times itself”
a^7
WORD PROBLEMS
What it means: “A is half the size of B”
A = 1/2B
WORD PROBLEMS
What it means: “A is 5 less than B”
A = B - 5
WORD PROBLEMS
What it means: “Twice as many Girls (G) as Boys (B)”
G = 2B
WORD PROBLEMS
What it means: “P is x per cent of Q”
P = x/100 Q
Or:
P/Q = x/100
WORD PROBLEMS
What it means: “The Tens digit of a 2-digit number is trice the Units digit”
El valor del número es:
(3x * 10) + (x * 1) = 30x + x = 31x
*10 porque es “decenas” *1 porque son Unidades
Si se invirtieran los dos números, el nuevo número sería 13x
How to find the minimum value of the Function f(x) = x^2 + 4x - 5
- Factor the quadratic = (x+5) (x-1)
- x = -5, 1
- A quadratic reaches its extreme value halfway between these solutions, that is: 1+(-5) / 2 = -2
- Substitute for “x”: f(-2) = 4 + 4(-2) - 5 = -9
Correct answer: -9
Which is the most effective way to combine two inequalities?
To line up the inequality symbols and add both sides. This can only be done if the inequality symbols face in the same direction.
You can “flip” one of the inequalities by multiplying for -1
CONJUNTOS Y DIAGRAMAS DE VENN
Formula para relacionar “Dos Grupos”
G1 + G2 - Ambos + Ninguno = Total
0!
1
How many different anagrams are possible for the word ATLANTA?
3! 2!
*Tenemos múltiples repeticiones: tres A’s y dos T’s
COMBINATIONS
En las combinaciones…
El orden no importa.
COMBINATIONS
Combination Formula
nCr = n! / r! (n - r)!
n = # Total de objetos r = # de objetos que selecciono
PERMUTATIONS
En las permutaciones…
El orden SÍ importa.
Por lo general los problemas incluyen el “wording”: “ranking”, “order”, “arrange”.
PERMUTATION FORMULA
nPr = n! / (n - r)!
CONJUNTOS Y DIAGRAMAS DE VENN
Formula para relacionar “Dos Grupos”
G1 + G2 - Ambos + Ninguno = Total
0!
1
How many different anagrams are possible for the word ATLANTA?
3! 2!
*Tenemos múltiples repeticiones: tres A’s y dos T’s
STATISTICS
Lo primero que hay que hacer cuando me piden “Median”
Ordenar los números en orden ascendente o descendente.
Median: es el número medio de un conjunto ordenado.
STATISTICS
Mode
El número que aparece con más frecuencia en el conjunto de números.
Puede haber más de una Moda, o no puede haber Moda.
STATISTICS
What is the sum of all even integers from 650 to 750, inclusive?
Sum of terms: (Mean) (Number of terms)
Mean (or Average) = Sum of the Terms / Number of Terms
Mean = Median -> el número de enmedio es 700.
50 + 1 = 51 (700) = 35,700
FUNCTIONS
Qué tengo que contestar cuando me preguntan: “For which of the following values of x is f(x) defined? Indicate all such values”.
Debo analizar los términos que me den, cada uno. De tal forma que a partir de la naturaleza y características de cada uno de ellos, pueda empezar a eliminar opciones de respuesta hasta quedarme solo con las correctas.
A partir del término 5 / x+2, puedo saber que “x” no puede ser -2, porque las fracciones con denominador “0” son indeterminadas.
Set A is the Set of all integers “x” satisfying the inequality 4 < |x| < 9
What are those?
What is the absolute value of the smallest integer in Set A?
Positive: 5,6,7,8. Four positive integers satisfy the inequality.
Negative: -5, -6, -7, -8. Four negative integers satisfy the inequality.
8 integers in Set A.
The smallest integer is -8. Its absolute value is 8.
If the information in the answer choices supports the argument above, the the answer choices…
Must be PREMISES, and the Conclusion is found in the Argument.
If the information in the body of the question supports the answer choices below, the Argument’s Conclusion…
The Argument’s Conclusion must be found in the answer choices (Inference question).
If we are looking at an Inference Question, what is our first line of defense to get the correct answer?
The “NO new information” filter. The answer choices must be potential conclusions.
And Conclusions must be based entirely on the information based in the Premises. They basically restate information already given.
(s^7) (t^7) =
(st)^7
(w^5)^-3
w^15
(x^-5) (y^5)
= (x^15)(y^-6)
x^15
= ————
y^6
When solving a System of Equations for “x” and “y”, like this one:
15x - 18 - 2y = -3x + y
10x + 7y + 20 = 4x + 2
Remember:
18x - 3y = 18 (This one can make it smaller by factoring by 3).
6x + 7y = -18 (I multiply it by -1, so I can get rid of the new -6x)
That the first step is arrange the variables in one side, and the numbers alone on the other side.
And to align them up so I can get rid of one variable to solve for the other.
Another way to write and see this fraction:
3x
—-
5
3
—- (x)
5
If the Ratio of “2x” to “5y” is 3 to 4, what is the Ratio of “x” to “y”
15/8
Procedimiento = Cross Multiplication
8x = 15y
x/y = 15/8
DISTANCE - RATE - TIME
“Two cars started from the same point and traveled on a straight course in opposite directions for exactly 2 hrs., at which time they were 208 miles apart. If one car traveled, on average, 8 m/h faster than the other car, what was the average speed for each car for the 2-hr trip?
Separate and organize the information given for each car:
1rst Car Data:
Rate = x m/h ; Time = 2 hrs ; Distance = 2x miles
2nd Car Data:
Rate = x + 8 m/h ; Time = 2 hrs ; Distance = 2(x + 8) miles
Equation:
Distance + Distance = 208 miles
2x + 2(x+8) = 208 —> And I’m ready to solve for “x”
COORDINATE GEOMETRY
If we are asked to solve the equation of a line for “y”, then we automatically put the equation into:
Slope-Intercept form.
y = mx + b
COORDINATE GEOMETRY
What is the Slope of the line with the equation
3x + 5y = 8?
We have to solve for "y", to put this into Slope-Intercept form: y = mx + b 3x + 5y = 8 ---> 5y = -3x + 8 "y" = 3/5x + 8/5 The Slope is "m" = - 3/5
COORDINATE GEOMETRY
Horizontal lines have a Slope of zero, because they are all “run” with no “rise”.
So, if a horizontal line has a y-intercept of 4, then “m” and “b” =
“m” = 0
“b” = 4
They belong to the Slope Intercept form: y = mx + b
COORDINATE GEOMETRY
What is the Slope of a Vertical Line?
Undefined, because the slope fraction is always:
(Something)/Zero
We cannot divide by zero.
COORDINATE GEOMETRY
When trying to know the quadrants that contain a point of certain line represented by an equation, such as: x-y = 18, remember:
Work it out by rewriting the line in the Slop-Intercept form:
y = x-18
Then, set “x” and “y” to be “zero” and solve accordingly.
I must end with two points (2 numbers for each point).
When solving for “x”, “y” is zero.
When solving for “y”, “x” is zero.
COORDINATE GEOMETRY
In order to find “the equation” of a line I have to consider:
y2 - y1
——— The result goes into the formula: y = mx + b
x2 - x1 where “+ b” is the point in which the line crosses the
y-axis.
COORDINATE GEOMETRY
To find the coordinates of the point of intersection of two lines defined by some equation, I have to consider:
To work with the two equations given as a system of equations (by adding them up). The goal is to get both variables (x,y).
There is no need to graph the two lines to find the point of intersection.
COORDINATE GEOMETRY
When trying to know which “Slope of the Line” is greater, I will be given Equation A, and Equation B.
What is the best approach?
To put each equation into Slope-Intercept Form (y = mx + b), and see which has the greater value for “m”.
FRACTIONS
What is greater 2/-5 or 5/-2
2/-5
If I want to know which fraction is bigger, I have to let the negative sign to the number it originally belongs (not to the whole fraction as usual).
Then cross multiplication can work correctly.
COORDINATE GEOMETRY
The distance between points, can be measured by:
I will be given two points for answer “A” and answer “B”.
Each two will construct a Right Triangle, then I can plug the proper values into the Pythagorean Theorem and solve for the Hypotenuse.
GEOMETRY
What can I know if I’m told that “a” and “b” are complementary angles?
They sum to 90 degrees.
GEOMETRY
When solving a Right Triangle problem, and if I’ve been given the Hypotenuse (c^2) and just one Leg (a^2), so I can solve for the second missing Leg, how do I work with it?
Example: 10, ?, 26
I have to work with the Pythagorean Theorem. a^2 + b^2 = c^2 10 + ? = 26^2 100 + ?^2 = 676 ?^2 = 576 ?= 24 <- Correct Answer
GEOMETRY
There is a cube with inside edges of 4, what is the diagonal of the cube?
4 Times Square Root of 3
1. Think about two Right Triangles: one located at the base of the cube, and the second one that has the actual “diagonal” required by the problem (its Hypotenuse).
2. Get the first Hypotenuse of the base Right Triangle:
4^2 + 4^2 = c^2
16 + 16 = Square root of 32
3. Get the second Hypotenuse of the second Right Triangle (which is the actual diagonal required):
4^2 + Square Root of 32, squared = Square Root of 48
Simplification: 4 Times Square Root of 3
GEOMETRY
When working with Equilateral Triangles (3 equal sides, 3 equal angles), remember:
- Two 30 - 60 - 90 “Special Right Triangles” are hidden. Those help me to get the area of the original Equilateral Triangle.
- The Ratio of the “Special Right Triangle” is:
Leg (1) Leg (2) Hypotenuse
x - Square Root of 3 - 2x
GEOMETRY
Any time you see a Right Triangle and one of the sides has a length of Square Root of 3, or a multiple of it…
I should check to see if it is a 30 - 60 - 90 degrees Triangle.
GEOMETRY
When working with a Special Right Triangle: 30 - 60 - 90, how can I know which Vertex belong to 30 and 60 degrees?
30 degrees is opposite the short Leg.
60 degrees is opposite the long Leg.
GEOMETRY
Usually, there are problems with a “fence” (valla), and “yard”story.
What they usually ask is:
To focus in the perimeter and the area.
Usually I have to work with the sides of a figure, such as: rectangle, or a square. The sides.
If I read: “one 40-foot side of the yard”, they are giving me one side of the figure (40).
GEOMETRY
To find the Surface Area of a Rectangular Solid, (“… how much will it cost to cover the surface of the tank?”), I have to:
Sum the individual areas of all six faces. Top and Bottom, Side 1, and Side 2.
Usually, the problem will give three quantities to work with.
GEOMETRY
The volume of a rectangular solid equals =
(Length) * (Width) * (Height)
GEOMETRY
To find the surface area of a Rectangular Solid =
Sum the individual areas of all six faces.
Surface Area: 2(lw + lh + w*h)
GEOMETRY
What is the volume of a Cube?
S^3
GEOMETRY
What is the surface area of a Cube?
6s^2
RATIO AND PROPORTIONS
When solving a “What is the Ratio” problem, remember:
I must respect the order in which the problem requires the ratio. For example: “What is the ratio of the cube’s surface area to its volume?”.
Answer: C’SA
——–
VOL
GEOMETRY
What does it mean that: Square ‘efgh’, is INSCRIBED within square ‘ABCD’?
That square ‘efgh’ is perfectly inside a bigger square: ‘ABCD’.
GEOMETRY
What is a Parallelogram?
It is a flat shape with opposite sides parallel and equal in length.
GEOMETRY
Formula to get the Circumference of a Circle?
The Circumference of the Circle is “PI”d.
d = Diameter
Alternatively:
2’PI’r
GEOMETRY
When solving a problem where the Surface Area of a Cylinder is involved, and I have to find the Diameter or the Radius of the Cylinder, remember:
- Cylinder Surface Area formula: 2(‘PI’r^2) + 2’PI’rh
2. Very likely to get two results for r = (r+#) (r-#) I have to use the positive number.
GEOMETRY
Formula to get the Volume of a Cylinder:
‘PI’r^2h
GEOMETRY
How to get the Area of a Sector =
Angle of a Sector/380 times () the Area of the Circle -PIr^2-
GEOMETRY
When working with solid shapes in which story they must be painted or covered for some substance, remember:
You are working with Surface Area.
If the story asks you to find the number of “paint buckets”, and by math you get a decimal as the answer, such as 10.2, remember stores do not sell fractional buckets, so you will need to purchase 11 buckets.
GEOMETRY
When working with Circles, and you are asked to find the length of an Arc, remember:
- You will be given an Angle, either Central (bigger number) or Inscribed (Smaller number). One is half the other, so with one of them you can get the other.
- The Central Angle must be divided by 360, of the…
- The Circumference of the Circle = PI*d
GEOMETRY
Square Root of 1?
1
GEOMETRY
The Sector of a Circle is:
The pizza slice.
GEOMETRY
When finding a Circle Arc Length, remember:
Knowing the Center Angle and the Radius, I set up a proportion, cross multiply and solve the Arc Length:
Central Angle Arc Length
—————- = ——————
360 2’PI’ r
NUMBER PROPERTIES
Seven people enter a race. There are 4 types of medals given as prizes for completing the race. The winner gets a platinum medal, the runner- up gets a gold medal, the next two racers each get a silver medal, and the last 3 racers all get bronze medals. What is the number of different ways the medals can be awarded?
Anagram Grid: 1 2 3 4 5 6 7 P G S S B B B Translate into Factorials: 7! ----- 1! 1! 2! 3! The numerator of the fraction is the factorial of the largest number in the top row, in this case 7!.
COMBINATORICS
“Or”, means:
“And”, means:
- OR means add.
2. AND means multiply.
COMBINATORICS
A local card club will send 3 representatives to the national conference. If the local club has 8 members, how many different groups of representatives could the club send?
Anagram Grid 1 2 3 4 5 6 7 8 Y Y Y N N N N N Then, set a Factorial Fraction 8! ----- 3! 5! There are 3 representatives chosen; represent them with Y. Use N to represent the 5 members of the group who are not chosen.
GEOMETRY
When working with Square Roots, remember:
To check if the number inside the root can be simplified.
For example: Square Root of 98.
It can be simplified to 7 times Square Root of 2.
Use Prime Factorization to determine it:
98 = 2x7x7
GEOMETRY
The sum of the measures of the interior angles of a polygon with “n” sides is:
(n-2) * 180
For example: a Hexagon has 6 sides. Therefore, substitute 6 for “n” into the formula an calculate.
(6-2) * 180 = 720
Therefore, the sum of the measures is 720.
GEOMETRY
Equilateral Triangles have three interior angles of:
60 degrees.
VARIABLES AND NUMERICAL PROVE
Which are good numbers to try in a problem?
1, 2, 0, -1, -2, 1/2
WORD PROBLEMS
Velocidad es:
La Distancia dividida entre el Tiempo.
WORD PROBLEMS
Speed Formula
Time
S = D/T
WORD PROBLEMS
Distance =
(Speed) (Time)
WORD PROBLEMS
Time =
Speed
WORD PROBLEMS
When to objects approach or leave to each other from different directions, along the same route, the speed they are approaching is:
The sum of their individual speeds.
Car 1 travels at 40 mph; Car 2 travels at 50 mph, the distance between the Cars is reduced at a rate Speed of 40 + 50 = 90 mph
WORD PROBLEMS
Cuando un objeto que se mueve a una velocidad más rápida se acerca por detrás a uno que se mueve (por la misma ruta), más lentamente, la velocidad a la que el rápido se acerca al lento es:
La Diferencia entre sus velocidades individuales.
Si un camión viaja a 25 mph y el otro a 35 mph, la Distancia entre ellos se reduce a una tasa de 35 - 25 = 10 mph.
Una vez que el rápido rebasa al lento, la Distancia entre ellos crece a una tasa de 10 mph.
WORD PROBLEMS
Paso 1 para resolver un problema de Distance, Speed, Time:
Asigna Variables.
WORD PROBLEMS
How many minutes is 1/4 of an Hour?
15 minutes
1/4 * 60 (minutes) = 15 minutes
WORD PROBLEMS
Si una Década se expresa matemáticamente como 10/10, ¿cómo se expresarían dos Décadas?
20/10
WORD PROBLEMS
How many minutes is 1/4 of an Hour?
15 minutes
1/4 * 60 (minutes) = 15 minutes
WORD PROBLEMS
Cuál es la fórmula de crecimiento exponencial?
A, “dos” = A, “uno” x f^t/T
t = Punto en el tiempo que busco.
T= El tiempo que le toma a A, “uno” crecer (o decrecer) f veces sí misma.
Ex. A certain bacteria triple in number every 20 min. How long will it take a million such bacteria to reach 27 million in number?
27 = 1 x 3^t/20
3^3 = 3^t/20 —–> 3 = t/20
t = 60 minutes
WORD PROBLEMS
El primer paso para resolver un problema:
Estandarizar las unidades.
1 km = 1000 mts
2 km = 2(1000) mts
1 hr = 60 min
WORD PROBLEMS
El primer paso para resolver un problema:
Estandarizar las unidades.
1 km = 1000 mts
2 km = 2(1000) mts
1 hr = 60 min
GEOMETRY
To find the length of an Arc of a Circle…
First, I need to calculate the Circumference of the Circle: 2’PI’* radio.
Divido los grados del Arco “entre” 360 grados. Si el resultado es fracción, por ejemplo 1/6, entonces multiplico el resto contrario de 5/6 “por” la Circunferencia.
GEOMETRY
When asked to get the Area of a Parallelogram, remember:
Inside, it might has two congruent (identical) Right Triangles.
So you can get the Parallelogram’s Area by getting one Right Triangle’s Area, and multiply it by 2 (because there are two equal Right Triangles in total).
GEOMETRY
When asked to solve a problem between two Rectangular figures, remember:
One of them might be a Square.
It is not necessary to work with to Rectangles.
GEOMETRY
¿Diferencia entre el Diámetro de un Círculo y la Cuerda (Chord) de un Círculo?
El Diámetro es la Cuerda (Chord) más larga que se puede dibujar en un Círculo. Va de lado a lado.
Por su parte, la Cuerda (Chord) puede ser más pequeña que el Diámetro.
GEOMETRY
The Slope is defined as…
Change in “Y” / Change in “X”
GEOMETRY
En el ‘Coordinate Plane’ me pueden preguntar por los posibles valores de ‘X’, con los que una ecuación como: y=x^3-x^2-6x, toca el x-axis?
Para resolver estas preguntas es nevesario establecer a “y” como cero.
Posteriormente factorizar una ‘x’ y trabajar por las dos raíces.
0 = x (x+2) (x-3)
Hence, y = 0 when x = 0, -2, or 3.
Uno o varios de estos números estarán en las respuesta correcta.
GEOMETRY
¿Qué es un Obtuse Angle?
Un ángulo con una medida mayor a los 90 grados.
Forma parte de un Triángulo, que vive en el Plano.
Para reconocer un ángulo obtuso en el ‘Coordinate Plane’, es necesario primero tener dos puntos. El tercer punto forma el Triángulo. Si éste esta más arriba o más abajo de los dos puntos, entonces el Triángulo tendrá un ángulo obtuso.
GEOMETRY
Cuando de un Cuadrado sólo sé que su diagonal es ‘x’, debo recordar que:
- La diagonal divide al Cuadrado en dos Triángulos Isóceles (45-45-90).
- El ratio de un lado del cuadrado y la diagonal es: 1 : Raíz Cuadrada de 2.
- Puedo encontrar la medida de un lado del Cuadrado, dividiendo
x / Raíz Cuadrada de 2 - Raíz Cuadrada de 2 es aproximadamente 1.4, o bien, 7/5
- x / 7/5 ———> 5x / 7
GEOMETRY
¿Cuál es la suma de los ángulos de un Cuadrilátero?
360 grados.
GEOMETRY
¿Cómo calcular dos ángulos restantes en un Cuadrilátero inscrito en un Círculo?
El problema me ha dado ya dos ángulos. Por ejemplo uno de 60 grados y el segundo de 110. Si les sumamos el ángulo opuesto de cada uno de ellos el resultado es 180 grados.
Así, el ángulo opuesto al ángulo de 60 grados, es 120, porque
120 + 60 = 180
El ángulo opuesto a 110 es 70 porque 110 + 70 = 180
GEOMETRY
Cuanto me enfrente a Círculo A y Círculo B, ambos inscritos en
Círculo C, recuerda:
Si todos comparten la misma línea de Diaámetro, no sumo los “PI” de cada uno de ellos. El “PI” se factoriza, de tal dorma que
Diámetro Uno + Diámetro Dos = Diámetro Tres
GEOMETRY
What is the conversion relation from Feet to Inches?
1 Feet = 12 Inches
SQUARE ROOT OF 3?
1.7
WORD PROBLEMS
The words ‘is’, or ‘are’ in the text correspond to:
The Equal sign: =
WORD PROBLEMS
Translate mathematically, “A is 50 more than B”
A = B + 50
WORD PROBLEMS
Translate: “A is 50% of B”
A = 0.5B
Or —> This is a good opportunity to multiply both sides by 2:
2A = B
WORD PROBLEMS
Translate: “A is 50% greater than B”:
A = 1.5B
WORD PROBLEMS
Translate: Right now, Steve’s age is half of Tom’s age.
S = T / 2
But more conveniently, we avoid fractions:
T = 2S <—– This is way more convenient to work with.
WORD PROBLEMS
Set the equation: “In eight years, twice Tom’s age will be five more than three times Steve’s age”.
2 (T + 8) = 3 (S + 8) + 5
Important: I have to choose variables to represent the ages NOW, and use ADDITION and SUBTRACTION to create expressions for ages at other times.
In this case, “+ 8”.
WORD PROBLEMS
What does this variables stand for?
D = RT
D = RT, is the formula for solving Motion Questions.
D = Distance
R = Rate or Speed
T = Time
*I can easily solve for R or T.
WORD PROBLEMS
Meters to Centimeters equivalence:
1 m =
100 cm
WORD PROBLEMS
Hour to Seconds, equivalence.
1 HR =
3600 Sec
WORD PROBLEMS
“A car moving at 72 km/hr moves how many meters in one second?”.
(1km = 1000 m)
- 1 hr = 3600 s
- Rewrite the proportion with the problem units:
72 km/hr = 72,000 m / 3600 s
*To change 72 km to m, I just multiply 72 km * 1000 m = 72,000 m
Correct Answer: 20 m/s
WORD PROBLEMS
Average Velocity / Speed Formula =
Total Distance / Total Time
WORD PROBLEMS
“Bob drove 120 miles at 60 mph, then another 120 at 40 mph. What was his AVERAGE SPEED for the TOTAL trip?”
48 mph —> The formula: Total Distance / Total Time
- First Leg, D =120 miles.
- Get the Time of First Leg, D. T = D/R —> T = 120 m / 60 mph = 2hr
- Second Leg, D = 120 miles.
- Get the Time of Second Leg, D. T = D/R —> 120 m / 40 mph = 3 hr
- Total Distance / Total Time = 120 m + 120 m / 2 hr + 3 hr = 48 mph
WORD PROBLEMS
In Average Speed problems, sometimes I don’t get to know the Distance or Speed values, so I can add them up into the Average Speed formula: Total Distance / Total Time.
What do I do?
Still, work with the variable you don’t have.
If “Emma drove from A to B at a constant 60 mph speed. Then returned on the same route at a constant speed of 20 mph”.
T-One = D / R = D / 60
T-Two = D / R = D / 20
Total Time = T-One + T-Two = D/15
Average Speed = 2D / [D/15] = 30 mph
GEOMETRY
Set the equation of “Martha and Paul started traveling from A to B at the same time. Martha traveled at a constant speed of 60 mph, and Paul at a constant speed of 40 mph. When Martha arrived at B, Paul was still 50 miles away. What is the distance between A and B?”.
D = RT —> D = 60T D - 50 = 40T
60T - 50 = 40T
20T = 50
T = 2.5 hr
D = 60*(2.5) = 150 miles
WORD PROBLEMS
When a Word Problem involves multiple travelers, multiple trips, or a trip with multiple legs, remember:
That each traveler, each trip, and Leg, deserves its own D = RT equation.
WORD PROBLEMS
What is the Work Equation?
A = RT A = Amount of work done. (Products manufactured, houses painted, pizzas made, etc.) R = The work rate. (Number of products per minute, houses per day, pizzas per hour, etc). T = Time
WORD PROBLEMS
“When Amelia and Brad detail a car together, 1 car takes 3 hrs. When Amelia details a car alone, 1 car takes 4 hrs. How long does it take Brad, working alone, to detail one car?”.
Rates = Cars / Hours
1/4 + 1/x = 1/3 —> 1/x = 1/3 - 1/4 —> 1/12
Brad = 1/12
WORD PROBLEMS
“How much HCL and how much water must we use to create 5 Liters of a 30% HCL solution?”.
The amount of HCL would be 30% as a decimal times the Whole:
0.30*5 = 1.5 Liters of pure HCL.
This means that the rest of the solution would require the following amount of water: 5 Liters - 1.5 Liters of pure HCL = 3.5 Liters of water.
WORD PROBLEMS
“Suppose we start with 5 liters of a 30% HCI solution. How much water must we add to create a 20% solution?”.
Amount of HCL = 0.30 * 5 = 1.5 liters
In the new solution:
Amount of HCI = 0.20 * x = 1.5 liters (1.5 liters of HCL. It does not change because all we are adding is water).
x = 7.5 liters total, which is the amount of the new solution.
Added water = 7.5 - 5 = 2.5 liters of extra water required to bring from 30% to a 20% solution.
WORD PROBLEM
“Suppose we start with 8 liters of a 60% HPO solution. We add 4 liters of x% HPO solution, and the result is 12 liters of a 50% HPO solution. What is x?”.
- Get how much HPO I begin with, and how much I end with.
Begin: 0.60 * 8 = 4.8 liters
End: 0.50 * 12 = 6 liters
Therefore, HPO (also called “solute”: sustancia disoluta), added: 6 - 4.8 = 1.2 L
Finally, I have to figure out 1.2 is what per cent of 4 = 30%
x = 30%
SKILL:
-3x = -7, what is “x”?
x = 7/3, the negative signs cancel out.
WORD PROBLEMS
“We start with unlimited supplies of a 20% HSO solution, and of a 50% HSO solution. We combine “x” liters of the first with “y” liters of the second to produce 7 liters of a 40% HSO solution. What does “x” equal?”.
Because there are two different solutions, I have to set up simultaneous equations. One will always be the “total” equation.
x + y = 7 liters.
The other equation will always be about the amount of solute:
.20x + .50y = 2.8
SKILL:
How can I make this equation easier to solve?
0.2x + 0.5x = 2.8
By getting rid of the decimals by multiplying everything by 10:
2x + 5x = 28
GEOMETRY
How to calculate any Arc Length?
x/360 * 2 ‘PI’ r
GEOMETRY
How to calculate any Sector?
X/360 * ‘PI’ r^2
PRIME NUMBERS BETWEEN 2 AND 7, INCLUSIVE?
2,3,5,7
PRIME NUMBERS BETWEEN 11 AND 19, INCLUSIVE?
11,13,17,19
PRIME NUMBERS BETWEEN 23 AND 29, INCLUSIVE?
23, 29
PRIME NUMBERS BETWEEN 31 AND 37, INCLUSIVE?
31, 37
PRIME NUMBERS BETWEEN 41 AND 47, INCLUSIVE?
41, 43, 47
PRIME NUMBERS BETWEEN 53 AND 59, INCLUSIVE?
53, 59
PRIME NUMBERS BETWEEN 61 AND 67, INCLUSIVE?
61, 67
PRIME NUMBERS BETWEEN 71 AND 79, INCLUSIVE?
71, 73, 79
PRIME NUMBERS BETWEEN 83 AND 89, INCLUSIVE?
83, 89
PRIME NUMBER BEFORE 100 AND IN THE 90’S TEN?
97
FIRST PRIME NUMBER AFTER 100?
101
WORD PROBLEMS: MATRIX METHOD FOR GROUPING AND SET
“In a company of 300 employees, 120 are females. A total 200 employees have advanced degrees, and the rest have college degree only. If 80 employees are males with college degree only, how many are females with advanced degrees?”.
Females Males Total
College Only 20 80 100
Advanced Degree 100 100 200
Total 120 180 300
* This method simplify problems in which each member is placed into two different kinds of categories.
WORD PROBLEMS Sometimes in 3 circled Venn Diagram problems, I might find as on portion of the problem something like this: "... If 220 students are in either Italian class or Baseball...". What does this mean?
“Italian class or Baseball”, means that all the regions in the circle “Italian Class”, and all the regions in the circle “Baseball” add up to equal 220.
WORD PROBLEMS
“Find the 41st term of the sequence: 14, 23, 32, 41, 50, 59…”
- a-One = 14
- Difference between terms: 9
- Formula: a-“n” = a-One + d*(n-1)
- Apply the Formula: a-Forty First = 14 + 9(n-1) = 14 + 9(40)
- 41st term is 374
WORD PROBLEMS
Skill: when working with RECURSIVE SEQUENCES (for example Each term ‘sub’ n-1), remember:
Each term is defined in terms of one or two previous terms.
There is no way to jump immediately to the value of term. Instead, I have to find each and every term from the start up to the desired term.
WORD PROBLEM
“How many multiples of 8 are there from 200 to 640, inclusive?”
- Divide 200 / 8 = 25
- Divide 640 / 8 = 80
- 80 - 25 + 1 (I add 1) = 56
- 56 multiples of 8 starting at 200 and going up all the way to 640.
WORD PROBLEMS
“What is the sum of all the multiples of 20 from 160 to 840, inclusive?”
- 160 / 20 = 8 (the eight multiple of 20)
- 840 / 20 = 42 (the 42nd multiple of 20)
- Inclusive Counting step: 42 - 8 + 1 = 35
- 35 / 2 = 17.5
- Answer the question: 17.5 * (160 + 840) = 17,500
COMMON WORD PROBLEM PHRASES
“The Profit is the Revenue minus the Cost”
P = R - C
COMMON WORD PROBLEM PHRASES
“n persons have x beads (perlas) each. “The total number of beads is”
nx
COMMON WORD PROBLEM PHRASES
“Five dollars every two weeks”
5 dollars / 2 weeks
—> 2.5 dollars a week
COMMON WORD PROBLEM PHRASES
“There are twice as many computers as there are printers”
C = 2P
COMMON WORD PROBLEM PHRASES
“Container A is three times as big as Container B”
A = 3B
COMMON WORD PROBLEM PHRASES
“One half of the students are learning French”
F = 1/2S
COMMON WORD PROBLEM PHRASES
“The are 10 more Grapes than Apples, and 1/4 as many Apples as Pears”
G = A + 10, and A = 1/4 P
COMMON WORD PROBLEM PHRASES
“Three friends sit down to eat 14 slices of pizza. If two of the friends eat the same number of slices, and the Third eats two more slices than each of the other two, how many slices are eaten by the Third friend?”.
F + F + T = 14 slices of pizza.
T = F + 2
F + F + F + 2 = 14
F = 4
Therefore, T = 6
*Sustituir T por un amigo individual.WORD PROBLEMS
PROFIT =
Revenue - Cost
WORD PROBLEMS
“Arnaldo earns $11 for each ticket that he sells, and a bonus of $2 per ticket for each ticket he sells over 100. If Arnaldo was paid $2,400, how many tickets did he sell?”.
Let x = the total number of tickets sold.
Therefore, (x - 100) = the number of tickets sold over 100.
11x + 2 (x - 100) = 2,400
x = 200
SKILL:
TOTAL REVENUE
Ingresos Totales
WORD PROBLEMS
Breaking even occurs when…
When total revenue equals total cost.
WORD PROBLEMS
“Every week, Renee is paid $40 per hour for the first 40 hours she works, and 80 dollars per hour for each hour she works after the first 40 hours. If she earned $2000 last week, how many hours did she work?”.
Let h = number of hours Renee worked. 40(40) + (h-40) (80) = 2000 -assuming h = 40. 1,600 + 80h - 3,200 = 2000 80h - 1,600 = 2000 80h = 3,600 h = 45
WORD PROBLEMS
En Problemas de Edad recuerda que ocasionalmente tienes que pensar en “escenarios extrmos”.
Jane started baby-sitting when she was 18. Whenever she baby-sat for a child, that child was no more than half her age at the time. Jane is currently 32, and she stopped baby-sitting 10 years ago. What is the current age of the oldest person for whom Jane could have baby-sat?”.
23
Tengo la edad actual de Jane: 32. Debo pensar en escenarios extremos.
A los 18 pudo cuidad a un niño de máximo 9. Hoy este niño tendría 23 años.
El otro extremos es que a los 22 ella pudo cuidar a un niño de máximo 11 años, por lo que 10 años después éste tiene 21 años.
23 > 21
WORD PROBLEMS
Cuando el problema hable sobre un “tank” al que se pretende llenar con agua, y cuyas medidas me son proporcionadas: “If the tank is 6 ft. long, 4 ft. wide, and 8 ft. deep, how many hours will it take to fill up the tank?”.
La capacidad del tanque se obtiene multiplicando:
6 ft. long x 4 ft. wide x 8 ft. deep = 192
After this step, to solve the problem I could use the formula RT = W
In this case W would be substituted for 192 cubic feet.
WORD PROBLEMS
SKILL: En problemas de “crecimiento exponencial” recuerda que un método que funciona para llegar a la respuesta correcta es:
Enlistar los valores correspondientes a cada crecimiento.
Hacer listas ayuda a encontrar la respuesta “aproximada” correcta.
SKILL: Recuerda que si divido 18/7, en ocasiones no hay porqué llegar a los decimales.
Se puede quedar en una “mixed fraction” :
2 4/7
WORD PROBLEMS
In “working together” problems recuerda:
Add the individual Rates.
Then, use the RT = W equation to find the total work done.
WORD PROBLEMS
Remember: The Average Rate is equal to…
The total work done divided by the time in which the work was done.
WORD PROBLEMS
When facing a “kiss” problem: when to objects are approaching to each other I work with a RTD Chart.
“A train leaves Kyoto for Tokyo traveling 240 m/h at 12 noon. Ten minutes later, a second train leaves Tokyo for Kyoto traveling 160 m/h. If Tokyo and Kyoto are 300 miles apart, at what time will the trains pass each other”.
R x T = D
Train: K to T 240 m/h t + 1/6 = 240t + 40
Train: T to K 160 m/h t = 160t
Total: = 300
Now, sum the Distances:
400t = 260 —> t = 13/20 —>
multiply everything by 3 = 39/60
Answer: 39 minutes
WORD PROBLEMS
A way to solve an Average Speed problem.
By using a Multiple RTD chart.
R x T = D
A to B
B to A
A to B
I have to get all the Individual Times, and the Total Distance.
If necessary I can chose SMART numbers for the Distance (a common multiple for the Rates in the problem).
WORD PROBLEMS
SKILL: Para responder problemas de mangueras (hose) que llenan albercas con agua (y problemas de trabajo), recuerda:
Me puedo apoyar en un chart R x T = W
Abajo de “R” pongo la suma de los Rates de las mangueras.
Si una manguera llena una alberca en 6 horas, el Rate es 1/6 por hora.
En “W” podría ir, por ejemplo, el Work que el problema me pida hacer.
STATISTICS
What is the Mean?
The sum of the numbers divided by the number of terms.
STATISTICS
What is the Median?
It is the Middle number.
STATISTICS "The class Mean score on a test was 60, and the Standard Deviation was 15. If Elena's score was within 2 Standard Deviations of the Mean, what is the lowest score she could have received?".
30
Elena’s score was within 2 standard deviations of the mean. Since the standard deviation is 15, her score is no more than 30 points from the Mean. The lowest possible score she could have received, then, is 60-30 = 30
STATISTICS
What is the set of the first five positive even integers?
2,4,6,8,10
No zero, because it is even but not positive nor negative.
COMBINATORICS
In how many different ways can the letters in the word “LEVEL” be arranged?
30 There are two repeated "E" and two repeated "L" in the word "LEVEL". To find the anagrams for this word, set up a fraction in which the numerator is the factorial of the number of letters and the denominator is the factorial for the number of each repeated letter. 5! ----- = 30 2! 2!
COMBINATORICS
“Amy and Adam are making boxes of truffles to give out. They have an unlimited supply of 5 different types of truffles. If each box holds 2 truffles of different types, how many different boxes can they make?”.
10 In every combination, two types of truffles will be in the box, and three type of truffles will not. Therefore, this problem is a question about the number of anagrams that can be made from the "word" YYNNN: 5! ------ = 10 2! 3!
COMBINATORICS
“A men’s basketball league assigns every player a two-digit number for the back of his jersey. If the league uses only the digits 1-5, what is the maximum number of players that can join the league such that no player has a number with a repeated digit (e.g. 22), and no two players have the same number?”.
20 In this problem, the order of the numbers matters. Each number can be either the tens digit, the units digit, or not a digit in the number. Therefore, this problem is a question about the number of anagrams that can be made from the "word", TUNNN. 5! -------- = 20 1! 1! 3!
COMBINATORICS
“A pod of 6 dolphins always swims single file, with 3 females at the front and 3 males in the rear. In how many different arrangements can the dolphins swim?”.
36
There are 3! ways in which the 3 females can swim. There are 3! ways in which the 3 males can swim.
Therefore, there are 3! x 3! = 6 x 6 = 36
COMBINATORICS
“Mario’s pizza has two choices of crust. The restaurant also has a choice of 5 toppings. Finally, Mario’s offers every pizza in extra-cheese as well as regular. If Linda’s team decides to order a pizza with four toppings, how many different choices do the teammates have at Mario’s Pizza?”.
20
Consider the toppings first. Model the toppings with the “word” YYYYN, in which 4 of the toppings are on the pizza and one is not. The number of anagrams for this “word” is:
5!
—— = 5
1! 4!
If each of the pizzas can also be offered in 2 choices of crust, there are 5 x 2 = 10 pizzas. The same logic applies for extra-cheese and regular: 10 x 2 = 20
COMBINATORICS
“Country X has 4-digit postal code assigned to each town, such that the first digit is non-zero, and none of the digits is repeated. What is rhe number of possible postal codes in Country X?”.
4,536
The first slot can be filled by any one of the digits from 1 to 9, since 0 is disallowed. The second digit has no restriction involving zero; however, the digit that was used in the first slot may not be used. Thus, the second slot also has nine possibilities. The third and fourth slots may not use previously used digits, so they may be filled with 8 and 7 different digits, respectively.
Total number of possible postal codes: 9 x 9 x 8 x 7 = 4,536
COMBINATORICS
“8 athletes compete in a race in which a Gold, a Silver, and a Bronze medal will be awarded to the top three finishers, in that order. What is the number of ways in which the medals can be awarded?”.
336
Anagram Grid: in the first row we can use the numbers 1 to 8 to uniquely designate each athlete. In the second row, G, S, and B designate the three medals; while the athletes who get no medal can each be associated with an “N”.
Athlete: 1 2 3 4 5 6 7 8
Medal: G S B N N N N N
Because 5 of the letters are repeated, the answer is give by
8!/5! = 8 x 7 x 6 = 336
COMBINATORICS
“Lothar has 6 stamps from Irapuato and 4 stamps of Cancún in his collection. He will give two stamps of each type to his friend Peggy. What is the number of ways Lothar can give four stamps (two of each type) to Peggy?”.
90 This is a successive "pick a group" problems. First, Lothar picks 2 out of 6 Irapuato stamps, and then 2 out of 4 Cancún stamps. Each selection may computed according to the general formula: Pool! Pool! --------- * ----------- = 90 In! Out! In! Out!
COMBINATORICS
And =
Or =
And = Multiply Or = Add
PROBABILITY
How many ways there are in which 2 dice can be thrown?
(6 x 6) = 36
COMBINATORICS
A small group drawn from a large group is a:
It is a Combination.
And in Combinations we don’t care about the order of selection. All we care is the final arrangement that we get.
COMBINATORICS
What does the notation ‘nCr’, means?
This is often read as “n Choose r”.
If we have a group of n different individuals, and we randomly select r of these individuals, then the number of combination is denoted as:
nCr
For example: 8C3 would be the number of different 3-Person combinations we could select from a pool of eight.
COMBINATORICS
nC1 =
n.
For example: 7C1 = 7 = The number of different ways I can choose one person from a pool of 7.
COMBINATORICS
We have a pool of 10 different items, and we want to select a set of 4. How many different sets of four could we pick?
I have to think about ‘10C4’.
First, Fundamental Counting Principle (FCP): N = 10987, and because the order of the 4 selected doesn’t matter, I divide by 4!
10987
10C4 = ———— = 210
4321
Just compute1098*7 would be correct if order mattered.
COMBINATORICS
“An amusement park has 12 different major rides. A coupon gives its holder access to any 3 of these rides for free. How many sets of three rides are possible?”.
220.
12C3 = 12 * 11 * 10
Then, divide by 3! = 6
COMBINATORICS
Name a case in which order matters in a problem.
Different people in different roles, such in a leadership teams.
One guideline is to consider the resulting arrangement and whether switching items around in this result would matter.
COMBINATORICS
“A librarian has a set of ten books, including four different books about Abraham Lincoln. The librarian wants to put the ten books on a shelf with the four Lincoln books next to each other, somewhere on the shelf. How many different arrangements of the ten books are possible?”.
(4!) (7!)
Since the four books must be together, think of them as one BIG book. There are six other books, and one BIG Lincoln book. In how many ways can we put these seven books in order on a shelf? Of course, 7!
After, we still can put the four Lincoln books in any order. There are 4! orders for the Lincoln books.
Total arrangements = N = (4!) (7!)
COMBINATORICS
(Permutation problem example)
“From a group of 10 committee members, three will be randomly selected: one as chair, one as treasurer, and one as secretary. These three will form the “board” of the committee. How many different possible “boards” can be formed?”.
The Fundamental Counting Principle is a much more efficient way to solve the problem.
10 9 8
—– —– —– = 1098 = 720 possible boards.
C T S
COMBINATORICS
Permutations VS. Combinations VS. FCP
1) The FCP is more basic, flexible, and more widely applicable than Permutations and Combinations.
2) If picking with no repeats, ORDER DOESN’T MATTER, then use Combinations.
3) If picking with no repeats, and different orders of the final selection are meaningfully different (i.e. ORDER MATTERS), then it’s a Permutation: use FCP.
PROBABILITY
If two events, say A and B are “mutually exclusive” then the Probability of A or B -P (A or B)-, equals =
The Probability of A + The Probability of B
P (A) + P (B)
PROBABILITY
What happens if events A and B are not mutually exclusive?
This means there is some “overlap” region counted twice; some occasions when both A and B can happen together.
Therefore, we need to subtract it once from the sum of the two regions.
Therefore, the Generalized OR Rule is:
P(A or B) = P(A) + P(B) - P(A and B)
PROBABILITY
Which are the two “OR” rules?
1) The simple “OR” rule (for mutually exclusive events only).
P(A or B) = P(A) + P(B)
2) The Generalized “OR” rule (always works, for any events).
P(A or B) = P(A) + P(B) - P(A and B)
PROBABILITY
“In Game M, the probability of outcome A is 0.6, the probability of outcome B is 0.7, and the probability of A or B is 0.9. What is the probability of A and B happening at the same time?”.
0.4
Generalized OR rule:
P(A or B) = P(A) + P(B) - P(A and B)
0.9 = 0.6 + 0.7 - P(A and B)
P(A and B) = 0.6 + 0.7 - 0.9 = 0.4PROBABILITY
If events A and B are independent, then what rule can I use?
The simplified AND rule:
P(A and B) = P(A) * P(B)
PROBABILITY
What is the notation for conditional probability?
P (A | B)
-For events NOT independent-
It means: “Assuming that, we know that event B is true, then given this condition, what is the probability that A happens?”
Or more briefly, “What is the probability of A, given B?”.
PROBABILITY
What is the generalized AND rule:
P (A and B) = P(A) * P(B|A)
Or,
P(A and B) = P(B) * P(A|B)
*Used when A and B are NOT independent.
SKILL: Antes de multiplicar una fracción:
Puedo simplificarla de forma “cruzada”.
5/12 * 4/11
= 5/3 * 1/11 = 5/33
PROBABILITY
When I see the words “AT LEAST”, use the shortcut:
The Complement Rule: P(not A) = 1 - P(A)
The probability of not getting a six in one dice roll is (5/6).
The probability of not getting a single six in eight rolls is (5/6)^8.
Answer: P(at least one six) = 1-(5/6)^8
SKILL
What are the answer options of the Quantitative Comparison Questions?
A) Quantity A is greater.
B) Quantity B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given.
Simplify, 2^29 + 2^29 =
2 (2^29) = 2^30
*It works just with number 2
PERMUTATION PROBLEM
“Five stand-by passengers are waiting for three open seats on an airplane flight. In how many different ways can three passengers be arranged in these seats?”.
60
I can just compute: 5x4x3= 60
Or, use the formula Pool! / Out!, where Out! Is the number of items NOT to be chosen: 5! / 2! = 60
*This is a problem in which order matters.
PROBABILITY
When two fair dice are rolled, what is the probability that at least one of the numbers will be even?”.
Pay attention to “at least” problems. In such cases is easier to use the “1-x” shortcut (the probability of the event NOT happening).
The Probability of each number coming up odd is 3/6 = 1/2. Since the two dice are independent, the probability of both coming up is
1/2*1/2 = 1/4, so the probability of getting at least one even number is
1 - 1/4 = 3/4.
In the proportion:
Vinegar Oil Water
14 21 15
At least what percent of the dressing will be vinegar?
28% of the mixture.
Vinegar Oil Water
14 21 15 = 50
Therefore, Vinegar makes up 14/50 = 7/25 = 0.28 * 100 = 28%
“A child must choose from among 5 balloons, each of a different color. What is the number of combinations of three different balloons he can choose?”.
10
The number of ways that a child can choose a set of 3 balloons out of 5 is given by “5 Choose 3”: 5! / 2! * 3! = 10 different possible sets.
*We are effectively dividing the 5 balloons into a “Yes” pool and a “No” pool.
WORD PROBLEMS
“Joe will pick 3 friends to join him on a road trip. Among his friends are 4 musicians and 3 poets. In how many different ways can Joe select his 3 traveling companions so that he has at least one musician and at least one poet among them?”.
30
1. Tengo que considerar cada caso posible: Joe tendría que seleccionar ya sea dos musicos y un poeta, o bien, dos poetas y un musico.
2. Realizar la división de factoriales por cada caso posible del problema. El número de maneras Joe puede seleccionar 2 músicos de entre 4 opciones, es: 4! / 2! * 2! = 24 / 6 = 4.
3. Posteriormente, se debe calcular el número de maneras para seleccionar 1 poeta de entre 3 poetas: 3.
Entonces, 6 x 3 = 18
WORD PROBLEMS
“Towns X and Y are 220 miles apart along a road. Car A, traveling at 20 miles per hour, leaves from Town X towards Town Y at the same time as Car B, traveling at 35 miles per hour, leaves Town Y towards Town X. How many miles will Car B have traveled when the two cars pass by each other?”.
140
RTD Chart
Car R (Miles / Hour) T (Hours) D (Miles)
A 20 t 20t
B 35 t 35t
The Times for the two cars are equal, because they start simultaneously. The sum of the distances traveled by the two cars must equal 220 miles. We can solve for “t” from 20t + 35t = 55t = 220, yielding “t” = 4 hours. The Distance traveled by Car B is then found as follows:
R x t = 35 x 4 = 140 miles.
WORD PROBLEMS
“The average weight of men in a meeting room is 170 pounds, and the average weight of the woman is 130 pounds. If more than 60% of the people in the room are men, which of the following could be the average weight of all the people on the room?”.
Select all choices that apply:
a) 144 b) 148 c) 150 d) 152 e) 154 f) 156 g) 158 h) 168
F, G, H
This problem can be solved as a Weighted Average Problem. The overall average weight will be between 130 and 170 pounds, but closer to 170 pounds because there are more men than women in the room. We can use the limiting percentage of 60 to establish the lower bound on the overall average weight.
The “weights” to use in the Weighted Average Formula are 60% or 3/5 for the men, and 40% or 2/5 for the woman. The lower bound on the average weight is thus: [3/5 * 170] + [2/5 * 130] = 102 + 52 = 154 pounds. 154 is NOT possible, because the percentage of men is greater than 60; however all the values higher than 154 are possible.
WORD PROBLEMS
“Two Journalists have 8 hours in which to copy-edit a total of 100 articles. If Journalist A copy-edits at a steady rate of 3.5 articles per hour, how many articles per hour must Journalist B copy-edit in order to complete the assignment on time?”.
9
Formula: W = RT —> 3.5 articles per hour x 8 hours = 28 articles. Journalist B must therefore copy-edit 100-28 = 72 articles.
To find Journalist B’s rate, we simply plug 72 for Work into the Work formula and solve for B’s rate. R x 8 hours = 72, so R = 9 articles per hour.
WORD PROBLEMS
“The ratio of Boys to Girls in a certain school is greater than 1. When 2 Boys leave and 3 Girls are added to the school, the ratio still favors Boys. What is the least number of Boys that could have been originally enrolled in the school, assuming there was originally AT LEAST one Girls?”.
7
Because the ratio of Boys to Girls is greater than 1, we know there are more Boys than Girls in the school: B > G.
When 2 Boys leave and 3 Girls are added the ratio is still greater than 1, so we can say: B - 2 / G + 3 > 1.
Solving: B - 2 > G + 3 —> B > G + 5
Thus, we choose G to yield B > 6
Therefore, the least number of Boys that could have been originally enrolled in the school is 7.
WORD PROBLEMS
“If 40% of the attendees at an event are over 50 years old, and another 20% are under 20 years old. What percent of the attendees are between 20 and 50 years old?”.
40%
Se establece el 100% y a partir de ahí empiezo a restar:
100% - 40% - 20% = 40%
PROBABILITY
“The probability of rainfall in City X on any given day is 30%. The probability of rainfall on any given day is independent of whether it rains on any other day.
A) What is the probability of rainfall in City X on AT LEAST one day out of two days.
B) What is the probability of no rainfall in City X on either of those two days”.
A) 0.51 or 51%
The probability of rainfall on at least one of the two days is equal to:
P(Rain 1st Day) + P(Rain 2nd Day) - P(Rain Both Days), since rainfall on each day is independent of what happened on other days. This probability equals 0.3 + 0.3 - (0.3 x 0.3) = 0.6 - 0.09 = 0.51 or 51%
Alternatively, we can use the “1 - x” trick here: the probability of rainfall on at least one of the two days is 1 - the probability of rain on neither day = 1 - 0.7 x 0.7 = 1 - 0.49 = 0.51 = 51%
B) The probability of no rainfall on either day, follows from the work above: 0.7 x 0.7 = 0.49 or 49%
B)
SKILL
¿Cuántos Centavos hay en $1 Dólar?
100 Centavos
WORD PROBLEMS
“Jennifer can purchase 11-Cent and 21-Cent stamps. If she intends to spend EXACTLY $2.84 on these stamps, which of the following is a possible number of 11-Cent stamps she can purchase?”.
A) 0 B) 1 C) 2 D) 3 E) 4
B = 1
La opción es probar cada opción que me dan.
Como tengo $2.84 dólares para gastar y “compro” 1 estama de 11 Centavos, resto $2.84 - 0.11 (porque son Centavos) =
$2.84 - $0.11 = $2.73.
Posteriormente, divido $2.73 / $0.21 = 13. Como 13 es la úniva división que resulta en entero, esa es la correcta.
“Each employee of a certain company is in either Department X or Department Y, and there are more than twice as many employees in Department X as in Department Y. The Average (Arithmetic Mean) salary is $25,000 for the employees in Department X and $35,000 for the employees in Department Y. Which of the following amounts could be the average salary for all the employees of the company”.
A) $26,000 B) 28,000 C) 29,000 D) 30,000
A, B
Una estrategia es irme al extremo con la información que me han dado: “… there are more than twice as many employees in Department X as in Department Y”.
-Mínimo es el doble-, entonces multiplico $25,000 por 2 y $35,000 por 1:
50,000 + 35,000
——————– = 28,333.
2 + 1 = 3
Entonces, las respuesta correcta tiene que ser menor a 28,333: letras A y B.
WEIGHTED AVERAGES
“On a ferry, there are 50 cars and 10 trucks. The cars have an average mass of 1,200 KG and the trucks have an average mass of 3,000 KG.
What is the Average mass of all 60 vehicles on the ferry?”.
1,500 Primero debo conseguir la suma de los KG tanto de los carros como de los Trucks: Carros: 50*1,200 = 60,000 KG Trucks: 10*3,000 = 30,000 KG Posteriormente, obtengo el promedio de ambos números, sumándolos y dividiéndolos entre el número total de autos: 90,000 -------- = 1,500 60
WEIGHTED AVERAGE
“In a certain company, 70% of employees are marketers who make an average of $40,000; 20% are programmers who make an average of $80,000; and 10% are managers, who make an average of $120,000. What is the average salary of all employees at this company?”.
$56,000
7 1 1
— $40,000 + — $80,000 + — $120,000 =
10 5 10
28,000 + 16,000 + 12,000 = 56,000
SKILL:
What is 46.7 rounded to the nearest integer?
47
SKILL:
Rectangle R has a perimeter of 80. The square S has a perimeter of 20.
The perimeter of S is what fraction of the perimeter of R?
20 1
— = —
80 4
WORD PROBLEMS
“Working alone at its constant Rate, Machine A produces “k” liters of a chemical in 10 minutes. Working alone at its constant Rate, machine B produces “k” liters of the chemical in 15 minutes. How many minutes does it take machines A and B, working simultaneously at their respective constant Rates, to produce “k” liters of the chemical?”.
6 minutes.
When the machines work simultaneously, the Rate at which the chemical is produced is the sum of these two Rates.
K K 15K + 10K 25K K
——— + ——– = —————- = ———— = —
10 min 15 min 150 min 150 min 6
To compute the Time r equired to produce “k” liters at the Rate K/6, divide the amount “k” by the Rate k/6 to get k/1 / k/6 = 6
SKILL
1.0869565 rounded to the nearest 0.1%?
108.7%
1.0869565 x 100 = 108.69565
Rounded = 108.7%
SKILL
What is 10.49 rounded to the nearest tenth?
- 5
- We identify the number placed in the “tenths” position: 4
- We look to the number located to the right of 4, in this case 9
- Is 9 greater or equal to 5? If yes, the 4 adds 1 to become 5.
- Result: 10.5
SKILL
What is the multiplier of a 17.6% increase?
1.176
SKILL
Round to the nearest 0.01 the number: 57.939624
57.94
SKILL
1 Mile equals how many Feet?
1 Mile = 5,280 Feet
“The average (arithmetic mean) of 100 measurements is 23, and the average of 50 additional measurements is 27. What is the average of the 150 measurements?”.
4.333 The Average of all 150 measurements would be the sum of all values divided by 150. 100 measurements x 23 = 2,300 50 additional measurements x 27 = 1,350 2,300 + 1,350 ----------------- = 24.333 150
“One of the roots of the equation x^2 + kx - 6 = 0, is 3; and k is a constant.
The question is: What quantity is bigger: k or -1?”.
Both quantities are equal.
If I have been given one of the roots of a quadratic equation, in this case 3, it means that when “x” is 3, the equation is correctly satisfied.
So what I have to do is to solve for “k” and compare its value with -1:
3^2 + k(3) - 6 = 0
“k” = -1
So both quantities are equal.
What is the Probability of an outcome?
Total outcomes
SKILL
If the problem says the word “approximately”, remember:
El resultado que obtenga de mis cálculos, NO necesariamente tiene que aparecer como una opción de respuesta. Si este es el caso, la respuesta correcta es el número que más se acerque a la respuesta que yo obtuve.
If the Diameter of Circle “C” is 3 times the Diameter of Circle “D”, then the area of Circle “C” is how many times the area of Circle “D”?
9
Circle “C” Circle “D”
3d d
Because Diameter and radius are directly related, if the radius of Circle “D” is “r”, then the radius of Circle “C” is 3r.
The area of Circle “C”: PI(3r)^2
—————— = 9
The area of Circle “D”: PI(r)^2
Last year Kate spent between 1/4 and 1/3 of her gross income on her mortgage payments. If Kate spent $13,470 on her mortgage payments last year, which of the following could have been her gross income last year? Indicate ALL such gross incomes: A) $40,200 B) $43,350 C) $47,256 D) $51,996 E) $53,808
B, C, D y E
El secreto está en utilizar ambas fracciones del “gross income X”, 1/4 y 1/3, igual a $13,470.
1/4x = $13,470 = 53,880
1/3x = $13,470 = 40,410
El rango de posibles cantitades de “gross income”, va a partir de 40,410 hasta 53,880: B, C, D y E.
GEOMETRY
En problemas de trángulos recuerda que aunque no me den las medidas del área, y ésta esté dividida en partes iguales…
Debo considerar la opción de que ambas porciones de un trángulo que está dividido en dos (las bases son iguales), tendrían la misma área.
SKILL: PARA ENCONTRAR LA “MEDIAN” DE UN GRUPO DE DATOS…
Debo ordenar los números “from least to greatest”.
In the expression
3s + 4
---------
3
Can I cancel out the 3 with the "3s"?No, because there is a “plus - addition” sign.
We actually have to divide each term by 3:
s + 4/3
How many 2-digit positive integers are there such that the product of their two digits is 24?
Four.
En preguntas de este tipo tengo aue enlistar todos los factores de 24:
1 * 24 = 24 —–> Eliminado porque es un “2-digit” number.
2 * 12 = 24 —–> Eliminado porque es un “2-digit” number.
3 * 8 = 24 —–> Sí aplica y como dos números diferentes: 38 y 83
4 * 6 = 24 —–> Sí aplica y como dos números diferentes: 46 y 64
At a certain fruit stand, the price of an apple is twice the price of an orange. For which of the following combinations of apples and oranges is the total price equal to the total price of 20 oranges? Indicate ALL such combinations: a) 2 apples and 16 oranges. b) 3 apples and 14 oranges. c) 4 apples and 10 oranges d) 6 apples and 8 oranges.
A, B, D.
El secreto está en probar cada una de las respuestas.
Orange = x
Apple = 2x
La respuesta correcta debe resultar en el precio total de veinte naranjas = 20x
a) 4x + 16x = 20x —–> Aplica como respuesta correcta.
b) 6x + 14x = 20x —–> Aplica como respuesta correcta.
c) 8x + 10x = 18x —–> No aplica como respuesta correcta.
d) 12x + 8x = 20x —–> Aplica como respuesta correcta.
A third-grade teacher has “n” boxes, each containing 12 pencils. After the teacher gives “p” pencils to each student in the class, the teacher has “t” pencils left over. Which of the following represents the number of students in the class?
a) 12n-t/p
b) 12n+t/p
A.
Secreto 1: No quedarme sólo con las variables que tengo en el problema. En este caso, si me piden la expresión que represente el número de estudiantes de la clase, debo incluir una variable adicional para obtenerlo, por ejemplo: “x”. Entonces el Maestro dio a sus estudiantes, “px” pencils.
Entonces, si sabemos que el maestro tiene “12n” cajas de lápices y regaló “px” = 12n - px = t —–> Then solve for “x”= 12n-t/p
If “x”> 0, then
(Square Root of 4x + Square Root of 9x) ^2 =
25x
El secreto inicial es simplificar una raíz cuadrada a la vez:
Square Root of 4x = 2Raíz Cuadrada de “x”
Square Root of 9x = 3Raíz Cuadrada de “x”
Entonces tenemos la siguiente expresión:
(2Raíz Cuadrada de “x” + 3Raíz Cuadrada de “x”)^2
Ataco el paréntesis y luego elevo al cuadrado^2:
(Raíz Cuadrada de 5x)^2 = 25x
Which of the following points are on the graphs of both the equation
y = x + 2, and the equation y = x^2.
Se contesta como una "quadratic equation". x^2 = x + 2 x^2 - x - 2 = 0 (x - 2) (x +1) = 0 x = 2, -1 Finalmente, sustituyo los dos valores de "x" que conseguí, en la expresión original, y = x + 2. y = 4 y = 1 Los puntos finales, son: (2,4) y (-1,1)
COORDINATE GEOMETRY
Recuerda que entre dos líneas, aunque no tenga sus puntos en el plano carteciano, puedo saber cuál que “Slope”, es más grande.
The line that rises faster (or is steeper), es la que tiene una “greater slope”.
Machine R, working alone at a constant Rate, produces “x” units of a product in 30 minutes, and Machine S, working alone at a constant Rate, produces “x” units of the product in 48 minutes, where “x” is a positive integer. What is the number of units of the product that Machine R, working alone at its constant Rate, produces in 3 hours.
6x
El secreto es la multiplicación de la proporción x/30 minutos, "por" los minutos solicitados por el problema: en este caso, 3 horas =
180 minutos.
x
------------- * 3 * 60 = 6x
30 minutesGEOMETRY
Si estoy trabajando con un Cuadrilátero inscrito dentro de un Círculo, del cual solo conozco su Diámetro: por ejemplo 10, y que no forma parte del Cuadrilátero (el diámetro), debo recordar:
Que es probable que no tenga suficiente información para calcular el Área de dicho Cuadrilátero. No puedo asumir la posición de cada uno de los puntos que lo conforman (al Cuadrilátero).
COORDINATE GEOMETRY
Para encontrar valores de Funciones, llamadas en inglés “Functions f”, debo recordar:
Que debo ubicar en el Plano Cartesiano los “puntos” que indica el problema. El dato numérico que acompaña a la Función es una coordenada que se sitúa en el eje de la “x”.
Of the 700 members of a certain organization, 120 are Lawyers. Two members of the organization will be selected at random. Which of the following is closest to the probability that neither of the members selected will be a Lawyer?
Aproximádamente, 0.7.
El secreto está en la doble división de las siguientes cantidades:
1. De mi universo de “No Abogados”, que es de 580 personas, tengo que seleccionar a 2 = 580!/2!*578! Este dato es el numerador de la proporción que necesito.
2. El denominador de la proporción que necesito, viene de las 2 personas que selecciono de entre el universo total de Miembros de la Organización: 700!/2!698!
Entonces, mi resultado final es: 0.68, ó aproximadamente 0.7
COORDINATE GEOMETRY
Cuál es el punto medio del segmento de una línea cuyos puntos finales son: (2,9) y (2,0).
(2, 4.5), o bien, (2, 9/2)
Solo necesito obtener el promedio de los dos valores de “x” y el promedio de los dos valores de “y”.
2+2/2 = 2
9+0/2 = 4.5
COORDINATE PLANE
Line “k” lies in the xy-plane. The x-intercept of line “k” is -4, and line “k” passes through the midpoint of the line segment whose endpoints are (2,9) and (2,0). What is the slope of the line “k”?
3/4
1. Obtengo el punto medio de las coordenadas (2,9) y (2,0).
2+2/2 = 2
9+0/2 = 4.5 ó simplemente 9/2.
2. Si x-intercept es -4, entonces puedo asignar a “y”, el valor de 0.
3. Encontrar la “Slope”, por medio de la formula:
“Y”dos - “Y”uno / “X”dos - “X”uno
9/2 - 0 / 2 - (-4) = 3/4
AREA OF A PARALLELOGRAM
Area = Base * Height
GEOMETRY
Para calcular el área de un Paralelogramo (Parallelogram), debo considerar:
- La altura se identifica al dibujar una línea “punteada”, recta, desde una de las esquinas de la figura.
- Esta línea “punteada” formará un triángulo recto.
- La Hipotenusa de este nuevo triángulo recto, será más grande que sus lados perpendiculares. Con esta información ya puedo estimar una posible área.
- Si conozco un ángulo, puedo calcular los otros tres: si sé que un ángulo mide 125 grados, su opuesto también medirá 125 grados.
- Como los ángulos interiores del paralelogramo suman 360 grados, los otros dos ángulos opuestos medirán 55 grados cada uno.
CÓMO SUMO LOS ENTEROS DE 1 A “N”
Con la fórmula:
n (n + 1)
———-
2
SKILL
RECUERDA QUE CUANDO UNA “INEQUALITY” ES MULTIPLICADA O DIVIDIDA ENTR UN NUMERO NEGATIVO…
LA DIRECCIÓN DE LA “INEQUALITY” CAMBIA.
ESTO ES QUE DEBO CAMBIAR LA DIRECCIÓN DEL SIGNO.
If 1 + x + x^2 + x^3 = 60, then the average (arithmetic mean) of x, x^2, x^3, and x^4 is equal to which of the following?
A) 12x
B) 15x
C) 20x
B) 15x
Usualmente, cuando veo potencias de tres o mayores, no tengo que resolver por "x".
x + x^2 + x^3 + x^4
-------------------------
4
Factorizo, una "x".
x (1 + x + x^2 + x^3)COORDINATE GEOMETRY Los Paralelogramos (Parallelogram) que "viven" en el Plano Cartesiano, y del cual deseo saber las coordenadas de uno de sus puntos, debo pensar en...
Triángulos Rectos, cuya una de sus esquinas sea el punto que deseo encontrar.
A partir del Paralelogramo, encontraré dos triangulos rectos y las medidas de uno, son exáctamente las medidas del segundo triángulo recto.
SKILL
¿El símbolo ‘PI’, se puede elevar al cuadrado?
Sí. Es legar indicar una potencia al símbolo que representa “PI”.
What is 22/17 to the nearest 0.01?
1.29
The complete quotient is: 1.29411765
¿Cómo calcular el ratio aproximado de la siguiente proporción?
34
Opciones de respuesta:
A) 8 to 1
B) 10 to 1
Multiplicando.
Pruebo cada una de las opciones de respuesta que me dan y veo cuál es la más cercana (porque en la pregunta me piden “aproximado”).
A) 8 to 1 —> Multiplico 8 * 34 = 272
B) 10 to 1 —> Multiplico 10 * 34 = 340
272 es más cercano al numerador de la proporción y por ello es la respuesta correcta.
SKILL
Recuerda que el lado opuesto al mayor ángulo de un triángulo…
Es el lado más largo.
Por el contrario, el lado opuesto a un ángulo menor, tendrá una longitud menor.
Si x > y, ¿qué puedo decir de x/y?
Que
x
— > 1
y
“p” is the probability that event “E” will occur; and “s” is the probability that event “E” will not occur.
Which quantity is bigger:
A) p + s
B) ps
A) p + s
“p + s” siempre será igual a 1. Es decir, “p” y “s” son decimales entre 0 y 1.
Entonces, “s” = 1-p
B) “ps” = p (1-p) = p-p^2
Basta con sustituir posibles valores para “p” (que caigan dentro del rango 0-1), para confirmar que p-p^2 es igual a una cantidad menor a 1. Por ello, la opción A) p + s, es la respuesta correcta.
STATISTICS
Recuerda que frente a un problema de Standard Deviation, en una distribución normal, el porcentaje de valores se encuentra a dos desviaciones estándar y éstos representan:
El 95% de la información.
The total amount that Mary paid for a book was equal to the price of the book plus a sales tax that was 4% of the price of the book. Mary paid for the book with a $10 bill and received the correct change, which was less than $3.
Which of the following statements must be true?
A) The price of the book was less than $9.50
B) The price of the book was greater than $6.90
C) the sales tax was less than $0.45
C
Secreto: Si detecto rangos en precios, vale la pena considerar “inequalities”.
P = Price of the book + 0.04% de tax = 1.04p
Como recibo menos de $3 de cambio = $7
In range of prices problems, remember:
To test “extremes values”. This way I will know the extreme range of possible values, the cheaper and the most expensive.
If x > 1, which quantity is bigger?
x -x
A) ——— B) ———
x +1 1 - x
B)
Primero, hacemos de la expresión “B” una más sencilla, convirtiendo las variables negativas, en positivas, multiplicando numerador y denominador por -1.
1
———
x - 1
Con esta información ya podemos “ver” que A) es más pequeña, porque el numerador es igual en ambos casos y la comparación recae en el demominador.
La cantidad A) es más pequeña porque estamos dividiendo entre un número más grande.
COORDINATE GEOMETRY
Recuerda que para saber la Slope de una línea, a fin de compararla con otra Slope de otra línea (ambas situadas en el mismo plano cartesiano), necesito:
Dos puntos por línea, de tal forma que no se pueda mover su Slope y con ello tener solo una Slope por cada línea.
Cuando trabaje con “inequalities” en fracción, debo recordar que solo puedo hacer una cross-multiplication para aislar una variable, siempre y cuando…
Todos los términos de la inequalidad sean positivos.
Points W, X, Y y Z are on a line, not necessarily in that order. The distance between W and X is 2, the distance between X and Y is 4, and the distance between Y and Z is 9. Which of the following could be the distance between X and Z?
A) 3
B) 5
C) 7
B) 5
En este tipo de ejercicios tengo que buscar acomodar las variables en distintas formas hasta dar con el resultado. El exámen sabe que yo las acomodaré (las variables) en orden (W,X,Y,Z)
y de esa forma no llegaré a la respuesta correcta.
Las líneas las acomodo en bloques, una debajo de la otra.
COMBINATORICS
Un problema de combinación, por ejemplo de dígitos o códigos postales, se resuelve únicamente con multiplicaciones cuando…
… se me permite repetir códigos y dígitos.
Si dos símbolos deben de ser una de las 26 letras de abecedario, solo es cuestión de multiplicar 26*26
LEAST COMMON DENOMINATOR
¿Cuál es el Least Common Denominator de las siguientes fracciones?
1/6 + 1/30
1/6 * 1/30 =
Paso 1. Prime Factorization: 1/23 1/235
Paso 1. Me hace falta un 5 en el denominador izquierdo. Entonces multiplico un 5 por el numerador y denominador de la fracción izquierda = 5/23*5
5/30 + 1/30 = 6/30 = 3/15 = 1/5
Otro ejemplo: p + p/2 + p/4 = 1
¿Cuánto es P?
P = 4/7
SET A: 0.2, 0.4, 0.6, 0.8 SET B: 2, 4, 6, 8 Which quantity is greater: A) Standard Deviation of Set A B) Standard Deviation of Set B
B
For the purpose of the GRE we can think about Standard Deviation as being a measurement of how much the numbers in a set differ from the Mean of that set.
A) Mean of 0.5
B) Mean of 5
Andy drove from Townville to Villageton at an average speed of 40 miles per hour. He then drove from Villageton to Townville at an Average Speed of 60 miles per hour. What is the Average Speed of Andy’s entire trip in miles per hour?
48
1. Por cada viaje, debo calcular t = d/s
d/40 + d/60 = 100d/2400 = d/24
2. Con el Tiempo total ya puedo realizar la división:
Total Distance / Total Time: 2d/1 / d/24 = 48
*Como Distancia es “D” la Total Distance es 2D
x^4 = y^16, then y =
Fourth Root of "x" 1. Simplificar y^16 a sólo "y". "Elevo ambos términos a la potencia 1/16 (x^4)^1/16 = (y^16)^1/16 x^1/4 = y^1 y = Fourth Square Root of "x"
What is the y-intercept of the graph of the equation y = 2|4x-4| - 10?
-2
The”x” coordinate of any “y” intercept is always equal to zero.
So we need to plug in 0 for “x” and solve for “y”:
y = 2|4(0)-4| - 10?
y = 2(4)-10 = -2
What is greater 0.24 or 0.2?
0.24
8.0 is approximately what percent of 5.1?
A) 40%
B) 57%
C) 64%
B) 57% We first find the difference: 8.0 - 5.1 = 2.9 2.9 ----- * 100 = 58
What is the number of integers between 100 and 500 that are multiples of 11?
36
- Primer múltiplo de 11, después de 100: 11*10 = 110
- Último múltiplo de 11, antes de 500: 11*45 = 495
- Resta. Último múltiplo - Primer múltiplo = 385
- 385/11 = 35
- 35+1= 36
In a regular 9-sided polygon, what is the value of each angle?
140
(n-2) * 180 = 7*180 = 1,260
Finally, in order to get each angle measurement I have to divide by the number of sides: 1,260/9 = 140
The function "f" is defined for all numbers "x" by f(x) = x^2 + x. If "t" is a number such that f(2t) = 30, which two of the following could be the number "t"? A) -5 B) -3 C) -1/2 D) 2 E) 5/2
B) -3 E) 5/2
1. Reemplazo “x” por “2t” = (2t)^2 + 2t = 30
2. En este caso, también puedo considerar x^2 + x = 30
3. x^2 + x - 30 = 0
4. (x + 6) (x - 5) = -6, 5
5. Reemplazo “x” por “2t” —> 2t = -6 2t = 5
t = -3 t = 5/2
8.0 is approximately what percent of 5.1?
A) 40%
B) 57%
C) 64%
B) 57% We first find the difference: 8.0 - 5.1 = 2.9 2.9 ----- * 100 = 58
What is the number of integers between 100 and 500 that are multiples of 11?
36
- Primer múltiplo de 11, después de 100: 11*10 = 110
- Último múltiplo de 11, antes de 500: 11*45 = 495
- Resta. Último múltiplo - Primer múltiplo = 385
- 385/11 = 35
- 35+1= 36
8.0 is approximately what percent of 5.1?
A) 40%
B) 57%
C) 64%
B) 57% We first find the difference: 8.0 - 5.1 = 2.9 2.9 ----- * 100 = 58
COORDINATE PLANE
In the xy-plane, the point (1,2) is on line “j”, and the point (2,1) is on line “k”. Each of the lines has a positive slope.
Which quantity is greater:
A) The slope of line “j”
B) The slope of line “k”
The relationship cannot be determined from the information given.
Porque solo tengo un punto de la línea, lo cual permite que “juegue” con la pendiente como yo quiera.
A certain identification code is a list of five symbols: SSDDD. Each of the first 2 symbols must be one of the 26 letters of the English alphabet, and each of the last 3 symbols must be one of the 10 digits. (Repeated letters and digits are allowed).
What is the total number of different identification codes?
676,000
Como la repetición de números y letras está permitida:
26 x 26 x 10 x 10 x 10 = 676,000
PROBABILITY
¿Qué significa que los experimeitos en una probabilidad son “mutually exclusive”?
Que una probabilidad no depende de la otra para suceder. Son independientes.
Por ejemplo: “A certain experiment has three possible outcomes. The lutcomes are “mutually exclusive” and have probabilities “p”, “p/2”, and “p/4”, respectively. What is the value of “p”?
Respuesta: p + p/2 + p/4 = 1
ODD + ODD =
EVEN
ODD - ODD =
EVEN
ODD + EVEN =
ODD
ODD - EVEN =
ODD
ODD * ODD =
ODD
ODD * EVEN =
EVEN
EVEN * EVEN =
EVEN
In the first half of the year, a team won 60 percent of the games it played. In the second half of last year, the team played 20 games, winning 3 of them. If the team won 50 percent of the games it played last year, what was the total number of games the team played last year.
90
El secreto está en establecer a “x” como el número de juegos jugados durante el primer semestre, de tal forma que el total anual es:
x + 20 = ?
Entonces, la relación de planteamientos matemáticos se establece por medio de los juegos ganados.
0.60x + 3 = 0.50 (x + 20)
x = 70
70 + 20 = 90
1st half of the year: 60% games won.
2nd half of the year: 20 games. 3 won, 17 lost.
GEOMETRY
The Perimeter of Square “S” is 40. Square “T” is inscribed in square “S”. What is the least possible area of square “T”?
50
1. Reconoce que el Cuadrado “T” se encuentra dentro del Cuadrado “S”, en forma de Rombo, que toca 4 puntos del Cuadrado “S”.
2. Reconoce que el Cuadrado “T” contiene dos triángulos especiales 45-45-90 —> x : x : xSquare Root of 2
3. La Hipotenusa es “xSquare Root of 2” = 10
x = 10 / Square Root of 2
Y como el área de un Cuadrado es Lado^2:
(10 / Square Root of 2)^2 = 100 / 2 = 50
A survey asked 1,150 people to choose their favorite laundry detergent from brands A, B and C. Of the people surveyed, "x" percent chose A as their favorite brand. If "x" is rounded to the nearest integer, the result is 3. Which of the following could be the number of people who chose A as their favorite brand? A) 25 B) 30 C) 35 D) 40 E) 45
B) 30 C) 35 D) 40
1. Identifica el rango de posibles valores de “x” (If “x” is rounded to the nearest integer, the result is 3): 2.5
RECUERDA
En problemas de Secuencias (A, Sub 1; A, Sub 2; etc), dadas por una función establecida, como A, Sub k = 1/k - 1/k+1…
me pueden preguntar: “what is the sum of the 100 terms of this sequence”.
Son problemas candidatos a encontrar patrones, por lo que debo encontrar por lo menos los primeros A, Sub 1; A, Sub 2; A, Sub 3; y también dos números antes de llegar a al último número.
2^11 * 5^11 =
10^11
5^6 * 2^6 =
10^6
10^11 * 10^6 =
10^17
GEOMETRY
En un problema de Triángulos, recuerda:
Siempre hacer el inventario de lo que tengo, incluyendo encontrar los ángulos internos de los triángulos, ya que ésto me puede ayudar a identificar un triángulo especial 30:60:90 y con ello, sus proporciones.
A desert outpost (Avanzada) has a water supply that is sufficient to last 21 days for 15 people. At the same average rate of water consumption per person, how many days would the water supply last for 9 people?
35
1. Asigno una variable a la cantidad de agua que se toma en 21 días: “w”. Entonces w/21
2. La cantidad de agua, w/21, se sabe que es tomada y distribuida entre 15 personas.
Entonces, w/21 / 15/1 —> w/2115 —> w/315, lo cual significa que una persona tendría suficiente agua para 315 días.
3. Nueve personas tendrán suficiente agua por 1/9315 = 35
COORDINATE GEOMETRY
The x-intercept of a line given by an equation is the value of “x” when “y” equals 0.
Therefore, the y-intercept of the line is the value of “y” when “x” equals 0.
1 - x 1
—— = —
x - 1 x
Quantity A = x
Quantity B = -1/2
Quantity B = -1/2
El secreto está en desaparecer la fracción por medio de cross-multiplication.
x(1-x) = 1(x-1)
x - x^2 = x - 1
Se van las “x”…
-x^2 = -1 —> Multiplico por -1 —> x^2 = 1
Entonces, “x” puede ser 1, o bien, -1 (nota que no se integra ninguna raíz cuadrada como parte del despeje de “x”.
Al sustituir cada uno de los posibles valores de “x” = 1, -1, descubro que “x” es -1, y por lo tanto, más pequeño que -1/2.
“WHAT PERCENT GREATER” problem skill…
Primero se realiza la Resta o diferencia de cantidades: Mayor - Menor = Resultado
El Resultado se “divide” entre la cantidad Menor y se multiplica por 100.
GEOMETRY
The height of an equilateral triangle with sides of length “r” is:
*r = Radius
Square Root of 3
———————-r
2
GEOMETRY
Area of an Equilateral Triangle:
4
*a^2 = Side
If a^-1 = 1/a, 1/a^-1 =
a
So, 1/4^-1 = 4
EXPONENTS
3^-1
——- =
4^-1
4/3
NUMBERS
Which quantity is greater?
A) The sum of the odd integers from 1 to 199.
B) The sum of the even integers from 2 to 198.
A
Sum of terms: (Mean) (Number of Terms)
Mean (or Average) = Sum of Terms / Number of Terms
Mean = Median -> 1 + 199 = 200 —> 200/2 = 100
Number of Terms = 199 - 1 = 198 —> 198/2 = 99 + 1 = 100
Sum of Terms of Quantity A: (100) (100) = 10,000
*I have to do the same for Quantity B (9,900) and compare both.
GRAPH
Recuerda que en una gráfica circular / pastel, puedo tener cada “rebanada” dividida en porcentajes. ¿Qué debo responder cuando me preguntan: “In the circle graphs, the degree measure of the central angle of the sector representing the number of workers unemployed for 11 to 14 weeks is how much greater in the Manufacturing industry graph than in the service industry graph?”
Si la rebanada de pastel en cuestión representa, digamos, el 10%, debo de obtener el 10% de 360.
Respuesta Oficial: “Recall that in a Circle Graph the degree measure of the central angle of a Sector representing “n” percent of the Data is equal to “n” percent of 360.
*the degree measure: “la medida en grados”.
RATIO La proporción 2:1.28 es cercana a cuál de las siguientes proporciones: A) 5 to 4 B) 6 to 5 C) 3 to 2 D) 5 to 2
C) 3 to 2
- Divido 2/1.28 = 1.5625
- Divido cada una de las opciones que me dan como opción de respuesta para saber qué resultado es el más cercano a mi primera división.
- 3/2 = 1.5 y por eso es la respuesta correcta.
2/3 is what fraction of its reciprocal?
2 --- 3 4 -------- = -------- 3 9 --- 2
SCIENTIFIC NOTATION
38300 =
3.83 x 10^4
When the positive integer “n” is divided by 3, the reminder is 2 and when “n” is divided by 5, the reminder is 1. What is the least possible value of “n”?
11
El secreto está en recordar que la respuesta correcta debe de ser el primer número común entre la división con 3 y 5.
n/3 = el primer número de la lista es 2 y posteriormente solo sumo 3 a cada término subsecuente, 5, 8, 11, 14…
n/5 = el primer término es 1, y voy sumando 5: 6, 11, 16…
El único número compartido es 11.
What is “Nth” term formula?
Nos ayuda a encontrar el número de términos en un Set:
a-Sub n = a + (n-1) * d
Resuelvo por “n”.
d = difference entre los números del Set.
Otra versión de la fórmula: n = aSub”n” - a / d + 1
If n^2 = (2^4) (3^4), what would be “n”?
(2^2) (3^2)
Half the exponents.
What is the Least Common Multiple of 24 and 108 in Prime Factors?
(2^3) (3^3)
24: (2^3) (3)
108: (2^2) (3^3)
Tomo los exponentes más grandes.
Which of the following numbers is NOT the sum of three consecutive odd integers? A) 15 B) 75 C) 123 D) 297 E) 313
E
El secreto está en reconocer que lo que busco es un NO múltiplo de 3, lo cual podemos saber gracias a un buen planteamiento:
x-2 + x + x+2 = 3x
¿Qué significa?
“A tax rate of 5 1/2 dollars per hundred dollars”
“A tax rate of 55 dollars per thousand dollars”
Which tax is greater?
5.5 55
—– and ——
100 1000
Both taxes are equal = 0.055
¿Cuánto es, aproximadamente, el 70% de 9 horas 40 minutos?
6 3/4
9 horas 40 minutos = 9 horas y 40/60 = 2/3
9 2/3 = 29 / 3
70/100 * 29/3 =
¿Qué cantidad es más grande?
7.9 ó 7.9999
7.9999
Which quantity is greater?
A) -2x^2
B) x^2/3
*x is not equal to 0
B) x^2/3
Debo darme cuenta que no importa cuánto valga x, x^2 siempre será positivo, por lo que la expresión completa x^2/3 siempre será positiva.
Por su parte A, -2x^2, es lo mismo que -(2x^2), entonces el término de la expresión dentro del paréntesis siempre será psitivo pero el signo “menos” de afuera la haría negativa.
The scale drawing of a certain room is a rectangle 5 inches by 6 inches. If the drawing has the scale 1/4 inch to 1 foot, which of the following will give the actual number of square feet of the room?
A) 90
B) 120
C) 480
C) 480
El secreto es primero realizar la multiplicación que elimine la fracción “1/4 Inch”. Es decir, multiplicar TODO por 4.
4 (1/4) = 4 (1)
1 Inch = 4 Foot
Posteriormente, ya me voy a la figura para multiplicar cada dato INDIVIDUAL “por” 4. 6x4 = 24 5x4 = 20
Finalmente: 24 x 20 = 480
Which quantity is greater?
A) 1/0.83
B) 1
A) 1/0.83
El secreto es darme cuenta que la división entre 1 y un número MENOR a 1, resulta en una cantidad mayor a 1, en este caso 1.204.
RECUERDA
Cuando un entero se divide entre un decimal, el resultado es un número mayor que el Numerador.
If 0 q
7
- — = 14
0. 5
RECUERDA
Ocasionalmente un problema me pedirá saber cuáles son los números que dividen a un número de “x” cantidad de dígitos.
Un posible planteamiento para un número de 5 dígitos es:
Número = abcde
= 10,000a + 1,000b + 100c + 10d + e
Número = 38,754
= 30,000
= 3 x 10,000 + ...LEAST COMMON MULTIPLE OF:
6 and 8 ?
24
6: 2 x 3
8: 2 x 2 x 2
LCM: 2 x 2 x 2 x 3 = 24
The jewels in a crown consist of Diamonds, Rubies, and Emeralds. If the Ratio of Diamonds to Rubies is 5:6 and the Ratio of Rubies to Emeralds is 8:3, what is the least number of jewels that could be in the tiara (ornamental band / high diadem)?
53
El secreto está en obtener el LCM de las cantidades correspondientes a los Rubies 6 y 8:
6: 2 x 3
8: 2 x 2 x 2
LCM: 2 x 2 x 2 x 3 = 24
Diamonds Rubies Emeralds
5 6
8 3
LCM of 6 and 8 = 24
Ahora que sé el LCM, ¿qué sigue?
1. Encontrar el número que al ser multiplicado por 6 (Rubies), resulte en 24, o sea, 4. Dicho número 4, será multiplicado por 5 (Diamonds):
5 x 4 = 20
- Encontrar el número que al ser multiplicado por 8 (Rubies), resulte en 24, o sea, 3. Dicho número 3, será multiplicado por 3 (Emeralds):
3 x 3 = 9
SUMA TOTAL: 20 + 24 + 9 = 53
FORMULA FOR SUM OF NATURAL NUMBERS
1,2,3,4,5,6,7,8 …
2
Find the sum of even numbers from 1 to 50.
650
Step 1: Total Numbers from 1 to 50 / 2
50/2 = 25
Step 2: (25) (25 + 1) = 650
*Another example:
2 + 4 + 6 + 8 + 10 = n (n + 1) = 5 (6) = 30
¿De qué otra forma puedo escribir 72.42?
72 + 42/100
PROBABILITY
If one number is chosen at random from the first 1,000 positive integers, what is the probability that the number chosen is a multiple of both 2 and 8?
1/8
Muy importante reconocer que si un número es múltiplo de 8, automáticamente también será múltiplo de 2. Entonces sólo calculo cuántos múltiplos de 8 existen en los primeros 1000 enteros positivos:
Divido: 1000 / 8 = 125
125 1
——– = ——-
1000 8
Which of the following is equal to 1 ---(20^8) 4 A) 5^8 B) 4(20^7) C) 5(20^7)
C) 5(20^7)
El secreto está en desglosar 20^8 y PLANTEAR la división entre 4.
20 x 20 x 20 x 20 x 20 x 20 x 20 x 20
———————————————————–
4
Sólamente uno de los ocho 20 del Numerador se puede “ir” con el 4 del denominador, resultando:
5 x 20 x 20 x 20 x 20 x 20 x 20 x 20, ó bien,
5 (20^7)
ALGEBRA Simplify the following expression: x^2 - 16 ------------ = x - 4
x + 4 El secreto está en reconocer que el numerador es Difference of Squares! x^2 - 4^2 (x - 4) (x + 4) ------------ = --------------- = x + 4 x - 4 (x - 4)
Two cars started from the same point and traveled on a straight course in opposite directions for exactly 2 hours, at which time they were 208 miles apart. If one car traveled, on average, 8 miles per hour faster than the other car, what was the Average Speed of each car for the 2-hour trip?
48 y 56
D = R * T
Car A R 2
Car B R+8 2
2R + 2R + 16 = 208
R = 48
Car A = 48
Car B = 48 + 8 = 56
TRANSLATE TO MATH
“A group can charter a particular aircraft at a fixed total cost. If 36 people charter the aircraft rather than 40 people, then the cost per person is greater by $12”.
C = Fixed Total Cost
C C
—– = —– + 12
36 40
C = 4,320
FIND THE SLOPE AND y - intercept OF THE LINE WITH EQUATION
2y + x = 6
m (which is the Slope) = -1/2 y-intercept = 3
Debo reescribir la expresión original 2y + x = 6 en la forma
y = mx + b
*m = Slope
*b = y-intercept
A) 2y + x = 6
B) Despejo “y” = 2y = -x + 6 —> y = -x/2 + 3
-x/2 es en realidad -1/2, entonces, “m” es -1/2
C) y-intercept es 3
COORDINATE GEOMETRY
Find the equation of the line passing through the point (3,2) with
y-intercept 1
y = 1/3x + 1
Como sé que el y-intercept es 1, puedo inferir que la coordenada total es (0,1).
Ya con los dos puntos puedo hacer la resta
“Diferencia de Y”/”Diferencia de X” = 2-1/3-0 = 1/3
COORDINATE GEOMETRY
FIND THE y-intercept OF A LINE WITH SLOPE 3 THAT PASSES THROUGH THE POINT (-2,1)
7 y = 3x + b I know from the information given in the problem that when "x" is -2, "y" has to be equal to 1. 1 = 3(-2) + b 1 + 6 = b b = 7
COORDINATE PLANE
For the parabola y = x^2 - 4x - 12 in the xy-plane, find the following:
A) The x-intercepts
B) The y-intercept
C) Coordinates of the Vertex
A) Recuerda que “y” sería 0. Posteriormente, resuelvo la Cuadrática:
(x - 6) (x + 2) = 6 y -2. Por lo tanto, ya tengo dos coordenas de la parábola: (6, 0) y (-2, 0).
B) El y-intercept es el punto en el que la parábola toca el Eje de las “y”.
En este caso debo recordar que “x” sería 0, el cual debe ser indicado en la Cuadrática original. y = x^2 - 4x -12 —> y = 0^2 - 4(0) - 12
y = -12. Lo cual me da la coordenada (0, -12)
C) Para encontrar las coordenadas del Vértice, debo encontrar el punto medio entre las dos coordenadas originales (6, 0) y (-2, 0).
- 2 + 6
- ——— = 2
INEQUALITY
6x - 5
2 - 5x
El secreto está en reconocer que el signo de “menos” que se encuentra en el término de la derecha, afecta a todo el término, por lo que otra forma de escribirlo sería:
- (6x - 5/3)
3 (2 - 5x)
Not Brown Hair: 45
Red Hair: 20
“What fraction of those who do not have Brown Hair, have Red Hair?”
20 4
—— = ——-
45 9
*El 45 es el denominador porque en el problema original es el Total. Es decir, los 20 del numerador formaban parte de los 45.
A dealer gives a 25 percent discount on the list price of a radio and still makes a 40 percent profit on the cost of the radio. If the cost of the radio is $30, then the list price of the radio is:
56
"x" = List Price
Profit = Selling Price - Cost
.75x - 30 = 0.40 ( 30)
x = 560
Quantity A: x + x^3 + x^5 + x^7
Factorizando la cantidad “B”, obtengo el mismo bloque de “x” que en “A”.
Quantity A: x + x^3 + x^5 + x^7
Quantity B: x (x + x^3 + x^5 + x^7)
La cantidad A es más grande, dado que estoy trabajando con decimales y esa última multiplicación en realidad haría más pequeña a la cantidad
x + x^3 + x^5 + x^7.
Planteamiento de “T is no more than 20 percent greater than the integer “x”
T
Planteamiento de “T is no smaller than a number 20 percent less than the integer “x”.
T _> 0.80x
GEOMETRY
“Triangles PQR and XYZ are similar. If PQ = 6, PR = 4, and XY = 9, what is the length of side XZ?”
6
La idea es saber que los triángulos similares tienen los mismos ángulos. Por eso a la hora de dibujarlos, las variables de cada vértice deben estar en el mismo orden.
Este problema se resuelve con una proporción:
6 4
—- = —- = Side XZ = 6
9 XZ
COORDINATE GEOMETRY
Para identificar líneas paralelas, a partir de ecuacionos como:
y = 2x + 3, tengo que fijarme en:
El Slope de dicha ecuación.
En este caso (y = 2x + 3), tengo una Slope de 2, por lo que una línea paralela también tendría que tener una Slope de 2, por ejemplo:
y = 2x - 5.
GEOMETRY
¿Cuál es la proporción de un triángulo especial: 30 : 60 : 90, y cómo está asignada en dicha figura?
x : x Square Root of 3 : 2x
Lado 1 Lado 2 Hipotenusa
100
——- =
2 “PI”
“PI”
SIMPLIFY
60^2 - PI (30)^2
——————— =
60^2
4
El secreto está en factorizar un 30^2.
COORDINATE GEOMETRY
Recuerda que cuando en un problema está involuvrado un Círculo y recibo información como la siguiente: “In the rectangular coordinate system, if (3,1) is a point on circle R and the center of R is (2, -3), what is the Radius of R?
Squar Root of 17.
El secreto está en reconocer que la información “if (3,1) is a point on circle R”, significa que (3,1), en un punto que está en la línea / cuerda del círculo, no en un punto aleatorio del círculo. De esta forma sé que la conexión entre el Centro del círculo y el punto (3,1), es el Radio de la figura.
A partir de ahí trabajo con un triángulo recto en mente para encontrar medidas entre un punto y otro (Hipotenusa).
COORDINATE GEOMETRY
Cuál es el área de un tríangulo que se conforma de los puntos: (2,0), (-2,0) y (0, -9/4).
4.5 Base: 4 Height: 9/4 (Para este propósito el signo negativo de la coordenada original -9/4, es irrelevante). 4*9/4 --------- = 4.5 2
SIMPLIFICA
(2-x)^2 = 4x^2
Opción 1: 3x^2 + 4x - 4 = 0 Opción 2: Elevar ambos lados de la expresión a una raíz cuadrada. 2 - x = 2x or - 2x x = 2\3 x = -2
COORDINATE GEOMETRY
For the line with equation y = ax + b (ab is not zero), the x-intercept is twice the y-intercept.
Column A: The Slope of the line
Column B: 1/2
Column B: 1/2
- Pienso en la fórmula: y = ax + b
- y-intercept = b
- x-intercept = 2b
- Llamemos a x-intercept “C”. y-intercept sería 0
- -> Coordenada (C,0)
- 0 = aC + b
- C = -b/a
- -b/a = 2b
- -b = 2ab
- Divido ambos lados por “b”, para desaparecer convertir el lado izquierdo en -1.
- -1 = 2a
- a = -1/2
- -1/2
DATA ANALYSIS
¿Cómo calculo el ángulo de una sección determinada en una gráfica de pastel, cuya rebanada representa 20% del total de datos de dicha gráfica?
20% of 360 grados = 72 grados.
DATA ANALYSIS
¿Qué significa tener un set con una Standard Deviation de cero?
Que todos los números del set son idénticos.
PERMUTATION
In a Permutation problem:
I just do the multiplication, NO DIVISION.
8 x 7 x 6 …
COMBINATION
In a Combination problem…
I do multiplication and division.
8 x 7 x 6
————-
1 x 2 x 3
STATISTICS
The Standard Deviation of the nine numbers Set: 19, 21, 22, 22, 28, 31, 33, 44, 50, is approximately equal to 10.2
Find the Interquartile Range of the 9 numbers.
- Identify Second Quartile (Median) = 28
- First Quartile is the Median of the first half of the Set: 19, 21, 22, 22 = 21.5
- Third Quartile is the Median of the last half of the Set: 31, 33, 44, 50 = 38.5
The Interquartile Range = Third Quartile - First Quartile =
38.5 - 21.5 = 17
STATISTICS
Recuerda que cuando se resta cierto número a cada elemento de un set, ¿qué le sucede a la Desviación Estándar original?
Nada, se queda igual, porque es resta.
Sólo cambia la Desviación Estándar cuando se MULTIPLICA cada elemento del Set por otro número.
STATISTICS
A group of 20 values has a Mean of 85 and a Median of 80. A different group of 30 values has a mean of 75 and a Median of 72.
A) What is the Mean of the 50 values?
B) What is the Median of the 50 values?
A) Debo saber la suma de los números en cada Set y dividirla entre 50.
20 x 85 + 30 x 75 3,950
————————– = ————– = 79
20 + 30 50
B) Sólo porque tenemos el dato de las dos Medians de cada set, no significa que en este caso tenga suficiente información para obtenerla. No tengo el ranking de cada número.
No tengo suficiente información para obtener la Median.
STATISTICS Find the Mean and Median of the values of the random variable X, whose relative frequency distribution is given in the table below. X Relative Frequency ----------------------------------- 0 0.18 1 0.33 2 0.10 3 0.06 4 0.33
Mean:
1. Multiplico cada número de la lista Random Variable X “por” el número que le corresponde a la derecha, en la lista Relative Frequency.
Sumo los resultados y el número final es la Mean.
(0 x 0.18) + (1 x 0.33) + (2 x 0.10) + (3 x 0.06) + (4 x 0.33) = 2.03
Median:
En problemas de Frecuencia debo buscar el valor donde cae el 0.5. Para conseguirlo debo sumar (a partir de cualquier dirección) hasta llegar a dicho valor.
Si empiezo a sumar del 0.18 + 0.33 = 0.51. Por ello, la Median es 1 (en número a la izquierda de 0.33.
Martha invited 4 friends to go with her to the movies. There are 120 different ways in which they can sit together in a row of 5 seats, one person per seat. In how many of those ways is Martha sitting in the middle seat?
24
El 120 viene de la cantidad de personas en las que podemos sentar a cinco personas en una fila de 5 asientos: 5! = 120
La posición de Martha siempre será la misma, por lo que solo me quedan 4 lugares para sentar al resto de sus 4 amigos.
M
—– —– —– —– —–
4 x 3 x 2 x 1 = 4! = 24
How many 3-digit positive integers are odd and do not contain the digit 5?
288
Enlisto todos los números que puedo utilizar para armar el número de tres dígitos.
0, 1, 2, 3, 4, X, 6, 7, 8, 9
Ahora debo colocar en cada espacio, la cantidad de números que podría utilizar para armar el 3-digit number:
*el zero no puede estar en la posición de las centenas, por ello en dicha posición sólo tengo 8 opciones.
En la segunda posición ya juega el 0 y en la tercera, como el 3-digit number es “odd”, solo tengo cuatro opciones (1, 3, 7, 9).
8 x 9 x 4
—– —– —– = 288
From a box of 10 lightbulbs, you are to remove 4. How many different sets of 4 lightbulbs could you remove?
210 Es un problema de "Combination" porque el orden no importa. El planteamiento es: 10C4 (10 chose 4). 10! -------- = 210 6! 4!
A talent contest has 8 contestants. Judges must award prizes for first, second, and third places, with no ties.
A) In how many different ways can the judges award the 3 prizes?
B) How many different groups of 3 people can get prizes?
A) 8 x 7 x 6 = 336
B) Combination Question: 8C3 =
8!
—– = 56
5! 3!
PROBABILIDAD
En el planteamiento P (A or B) = P (A) + P (B) - P (A and B)
P (A) + P (B) - P (A and B)
*Si el problema me dice que los eventos en cuestión son “Mutually Exclusive”, significa que no pueden ocurrir al mismo tiempo. Por ello, en tal caso, la probabilidad, P (A and B), es 0.
PROBABILIDAD
Recuerda que en un par eventos P (C and D) “Independent”, es:
La probabilidad de los eventos individuales multiplicados.
P (C and D) = P (C) * P (D)
Of the 700 members of a certain organization, 120 are lawyers. Two members of the organization will be selected at random. Which of the following is closest to the probability that NEITHER of the members selected will be a lawyer?
A) 0.5
B) 0.6
C) 0.7
C) 0.7
Primero tengo que saber que dos miembros del grupo de 580 personas que NO son abogados, serán seleccionados.
580C2 =
580 x 579
--------------- = 167,910
2!
Posteriormente, divido entre el número de formas que hay para seleccionar a dos personas de entre un grupo de 700 personas.
700C2
700 x 699
-------------- = 244,650
2!
167,910
------------ = 0.68
244,650
0.68 --> C) 0.7PERCENT INCREASE
What is the percent increase from 4,500 sales in cars to 7,000 sales in cars?
55%
7,000 - 4,500 = 2,500
2,500
——————- * 100 = 55%
4,500 (original starting point)
SCIENTIFIC NOTATION
20,000,000,000 =
100,000 =
20,000,000,000 = 20 x 10^9
100,000 = 10^5
*Si los quiero dividir:
20 x 10^9
————— = 20 x 10^4 = 200,000
10^5
STATISTICS If "d" is the Standard Deviation of the numbers in the list above, for which of the following values of "x" would the value of "d" be least? A) 0 B) 5 C) 20 D) 30 E) 80
C) 20
10, 15, 25, 30, x
Poner atención en los números que sé y sacar su promedio:
10 + 15 + 25 + 30
———————- = 20
4
El numero que va a “deviate the least”, es el número más cercano o igual al Average (20) de los números que conozco.
De esta forma si “x” es 20, la desviación estándar va a contribuir menos a la desviación estándar.
