GMAT Flashcards
What are the 2 top tips for the GMAT?
- Spend extra time on the first 10 questions. They are absolutely crucial
- You are always better off making your best guess ⇒ the best guess can often be determined by which answer option that has the most in common with the rest (the most common answer theorem)
What are the question types in the quantitative test and what share do they account for? (old test)
Data sufficiency questions = 40 %
When are you trying to find a Yes/No answer for data sufficiency questions?
If the question asked starts with “is”, “are” or “does” you look for a yes/no answer rather than a number.
What are integers?
What are whole numbers?
What are irrational numbers?
Numbers that canNOT be expressed as a fraction or ratio.
Examples: π and √2
When do you know that the GMAT is testing you in imaginary numbers?
If they ask you to take any EVEN ROOT OF A NEGATIVE NUMBER.
√-100 or √-8 fx.
What are the prime numbers up until 130
2, 3, 5, 7,
11, 13, 17, 19,
23, 29,
31, 37,
41, 43, 47,
53, 59,
61, 67,
71, 73, 79,
83, 89,
97,
101, 103, 107, 109,
113,
127.
What does “absolute value” mean and how is it expressed?
A number’s absolute value is its distance from zero on the number line, and like any distance, its ALWAYS positive.
Absolute value of 3 is 3. Absolute value of -8 is 8.
What is the order to work in maths?
PEMDAS
Parantheses
Exponents
Multiplication
Division
Addition
Subtraction
What is (x + y)*(x + y)?
(x + y)^2
x^2 + 2xy + y^2
What is (x - y)*(x - y)?
(x - y)^2
x^2 - 2xy + y^2
What is (x + y)*(x - y)?
x^2 - y^2
What is
How do you multiply exponent together? fx: x^3 * x^5
X^3 * X^5
Simply add the exponents together: X^3 * X^5 = X^8
is the same as: (x*x*x) * (x*x*x*x*x) = x^3 * x^5 = X^8
How do you divide exponents together? fx: y^3 / y^5
Simply subtract the denominator from the numerator.
y^3 / y^5 = (y3 - y5) = y^-2
What is the value of one exponent raised to another exponent? such as: (x^6)^3
Just multiply the two exponents
(X^6)^3 = x^18
What is 355^0?
1
ALL real numbers raised to the power of 0 = 1
What is x^0?
1
ALL real numbers raised to the power of 0 = 1
What is 3^-4?
which is = 1 / 3^4
which is 1 / 81
What can be said about “any number greater than 1 raised to a negative exponent”?
It will be a number greater than 0 but less than 1.
What is a number raised to a fractional power? such as X^⅙
ALL real numbers raised to the power of 0 = 1
X^⅙ =
What is the result of 2^2, 2^3, 2^4, 2^5, 2^6, 2^7, 2^2, 2^9, 2^10
What is the result of 3^2, 3^3, 3^4, 3^5?
What is the result of 5^2, 5^3, 5^4?
What is the result of 6^2, 6^3?
What is the result of 7^2, 8^2, 9^2, 10^2, 11^2, 12^2, 13^2, 14^2, 15^2, 16^2?
7^2 = 49
8^2 = 64
9^2 = 81
10^2 = 100
11^2 = 121
12^2 = 144
13^2 = 169
14^2 = 196
15^2 = 225
16^2 = 256
What is the difference between a finding the factors of a number and finding the prime factors? The processes?
Factors = all numbers can be used
Prime factors = only prime factors can be used
What is basis points?
It is 1 of 100.
0.45 % = 45 basis points.
What percent is 600 of 200?
300 %
but percentage increase from 200 to 600 is 200%
What is the percentage increase from 300 to 600?
100 %
but 600 is 300 % of 200.
How can you easily simplify a complex fraction? such as 210/2,772
Divide by 2 if possible, then 3, then 5, then 7
(210 / 2) / (2772 / 2) = 105 / 1386
(105 / 3) / (1386/3) = 35 / 462
(35 /7) / (462/7) = 5 / 66
How do you know if a number is divisible by a prime number?
Take the number, fx. 2772
then write it out as 2+7+7+2 = 14
Any prime number going to 14 will work = either 2 or 7.
What are the six main rules to remember in terms of exponents and roots?
Answer:
Question:
Answer:
(3* 7) + (7* 3) = 21 +21 = 42 = D.
Makes sense, since you have 3 / (1 / 7), as 1/7 naturally goes into 1 seven times. And 3 is in the numerator, so it has to go up to three (which is three times as much as 1, i.e. 3 * 7 = 21).
The same logic can be applied for the other part.
Question:
Answer:
64^k is the same as (4^3)^k
So following the rule of multiplying exponents when one exponent is raised to another, we simply have to figure out what:
3* k > 14
k > 14/3
k > 4,67
K must be B = 5.
Question:
answer:
the prime numbers less than 10 are: 2, 3, 5 , 7
2*3 = 6 so cannot be A
Nothing can be multiplied to B as it has to be DISTINCT numbers = B
2*5 = 10
2*7 = 14
3*5 = 15
What are the rules of the cube root table and how does the table look?
- Take the last digit in a number: for example 6 in 636,056 and find that digit in the last digit column.
- Look at the three first digits in a number and find the digit in the table where they do NOT exceed the cube. For example 636 in 636,056.
So cube root of 636,056 would be 86
What are the solutions to the equation x^2 + -2x = 15?
First, re-order to separate 0.
x^2 - 2x - 15 = 0
Now factor (-15)
- 1 and 15 1 and -15
- 3 and 5 3 and -5
Which pair can you manipulate to get to -2?
-1 + 15 = 14 so no
1 -15 = -14 so no
-3 + 5 = 2 so no
3 - 5 = -2 so YES
(x - 3) * (x + 5)
x + 3 = 0, then x = -3
x - 5 = 0, then x = 5
What is the median of 105, 106, 107, 108, 109, 110, 111, 112, 113?
The middle value which in this range of 9 values will be the 5th value = 109.
What is the median of 105, 108, 110, 112, 114, 115?
6 different numbers so the median is the average of the 3rd and 4th number.
(110 + 112) / 2 = 111
What is the mode?
The most common element in a set
NOTE!!! A set can have more than one mode.
NOTE NOTE!!! A set can also NOT have a mode if no number occurs more than once
What is the range?
The difference between the largest and the smallest number.
How is the standard deviation calculated? and how is it useful to know?
The standard deviation is the square root of the result of the summation of the squares of the differences between the individual values of the set and the mean, divided by the number of items in the set.
You will not have to calculate this on the GMAT.
But… you might be asked which set of numbers has the largest standard deviation.
What is the formula for calculating the third side of a right triangle?
A^2 + B^2 = C^2 (pythagoras)
What are parallel lines?
Lines that if extended to infinity, would never intersect.
How do you calculate the area of a triangle?
½ * Base * Height
How do isosceles triangles look and what are their properties?
- Two sides of equal length
- The two angles opposite the sides of equal length are also of equal degree
How do equilateral triangles look and what are their properties?
- All sides of equal length
- All angles have 60 degrees
How do you calculate the area of an equilateral triangle?
(√3 / 4) * s^2
(1. 732 / 4) * s^2
0. 433 * s^2
How do right triangles look and what are their properties?
- One angle is 90 degrees
- The third side (hypotenuse) can be calculated by A^2 + B^2 = C^2
How do right isosceles triangles look and what are their properties?
- one angle with 90 degrees
- Two sides of equal length
What is the 3-4-5 triangle type?
A common ratio used on the GMAT.
RIGHT triangles with the ratio of 3,4 and 5.
- A is 3, B is 4 then C = 5
- A is 6, B is 8, then C = 10
- A is 9, B is 12, then C = 15
- A is 12, B is 16, then C = 20
What is the 5-12-13 triangle type?
A common RIGHT triangle ratio:
- A is 5, B is 12, then C = 13
- A is 10, B is 24, Then C = 26
What is the 30 degrees, 60 degrees and 90 degrees triangle?
The 1 - √3 - 2
- If A is 1, B is √3, then c = 2
- If A is 5, B is 5*√3, then c = 10
- If A is 8, B is 8* √3, then c = 16
What is the perimeter?
How far there is around a triangle, square etc.
The sum of the sides.
How do a rhombus look and what are their properties?
A rhombus is a quadrilateral with sides of equal length and two sets of opposite angles with equal measures.
How do a parallelogram look and what are their properties?
Two sets of parallel lines
The two sets have different length
How do you calculate the area of a parallelogram?
If you recognize that it is simply a square and two triangles of equal size, the calculation simply become
AREA = BASE * HEIGHT
How do trapezoids look and what are their properties?
One pair of parallel lines
The two other sides have equal length but are not parallel
How do you calculate the area of a trapezoid?
Recognize that a trapezoid is a square with two equal triangles.
AREA = HEIGHT * ((A + C) / 2)
Thus we just take the height and the average of the length of the two parallel lines. Then it is essentially just as calculating a rectangle.
How do you calculate the circumference?
The circumference is the distance around a circle (what we call a perimeter in a polygon).
It is simply:
Diameter * π
or: 2 * r * π
What is the area of a circle?
A = r^2 * π
How do you calculate the volume and surface area of a right circular cylinder?
How do you calculate the volume and surface area of a square pyramid?
How do you calculate the volume and the surface area of a cube?
How do you calculate the volume and the surface area of a rectangular solid?
How do you calculate the volume of a general cone/pyramid?
For a right circular cone, you would obviously find A by taking r^2*π
How do you calculate the volume and surface area of a sphere?
What is the result of “even*even”?
Even
4*4 = 16
What is the result of “even*odd”?
Even
4*5 = 20
What is the result of “odd*odd”?
odd
13*13 = 169
What is the result of “even + even”?
Even
22 + 36 = 58
What is the result of “odd + odd”?
Even
17 + 19 = 38
What is the result of “odd + even”?
odd
21 + 22 = 43
What is the result of “negative * negative”?
Positive
-4 * -5 = 20
What is the result of “positive * negative”?
Negative
5 * -6 = -30
What is the result of “positive + negative”?
It depends.
on which one is biggest.
When an odd number is raised to an exponent, what will it result in?
It will still be an ODD number.
13*13 = 169.
No matter if it is raised to 2, 3, 4, 5, 6 etc., it will always be an odd number
When an even number is raised to an exponent, what will it result in?
It will still be an EVEN number.
12*12 = 144.
No matter if it is raised to 2, 3, 4, 5, 6 etc., it will always be an even number
What is the result of “non-integer * non-integer”?
It will always be a NON-INTEGER
0.5 * 0.5 = 0.25.
What is the result of “integer * non-integer”?
It depends.
What is the key component of combinatorial math and what is the formula used?
Factorials.
n! = n* (n - 1) * (n- 2) * - (n - 3) * ….
So
- 2! = 2* 1 = 2
- 3! = 3 * 2 * 1 = 6
- 4! = 4 * 3 * 2 * 1 = 24
- 5! = 5 * 4 ‘ 3 * 2 *1 = 120
- 6! = 6* 5 * 4* 3 * 2 ‘ 1 = 720
- 7! = 7 * 6 * 5 * 4 * 3 * 2 * 1 = 5,040
What is the formula for combinations and what does combinations refer to?
Combinations regards groupings where ORDER DOES NOT MATTER
What is the formula for permutations and what does permutations refer to?
Permutation regards groupings where ORDER MATTERS.
How do you multiply fractions? for example 4/9 x 3/2?
4/9 * 3/2
4*2 / 9*2 = 12/18 = 2/3
What should you ask yourself for inference questions?
- If the statements in the passage are true, what else has be true?
- When having found potential answers, ask yourself “what if the opposite of this answer were true?”
ALWAYS Pick the moderate statement over the extreme.
What are the three main ways in which an answer can be wrong for sentence correction questions?
It violates a rule of grammar
It is worded in an unclear way
It is worded in a “nonstandard” way; i.e. it sounds funny
Which words should you look for to avoid being trapped by pronouns errors?
- He
- She
- It
- they
- them
Ask yourself whether
- is it clear who the referent is?
- is it the right pronoun for that referent?
What are some of the most frequent quantity-words used in the GMAT?
What are some of the most frequent countable/non-countable words used in the GMAT?
Countable words should be used when you can normally assign a whole number to it. Such as six cars.
Non-countable items are for example a little soup as you cannot quantify “a little soup”.
A cup of soup, of course, would be countable = 1.
What is the most common error in sentence correction questions?
No error at all.
20 % has no errors at all.
What are 3 points that should raise an alarm for sentence correction questions?
A sentence does not have to be wrong when it includes any of the following but it often will be:
- Any answer that includes “being”
- Passive verb forms when active verbs would work just as well
- Words ending in “-ing” when there are simpler choices
If you have to give your best guess for sentence correction questions, which two rules should you follow?
- Pick the shortest answer
- Pick the “most common” answer.
- The one that has the most in common with the other options
What are good examples of phrases in which the writer CONTINUES his/her line of thought?
What are good examples of phrases in which the writer CHANGEs his/her line of thought?
How should your essay outline ideally look?
- Paragraph 1: INTRODUCTION
- State argument in a clear statement
- Acknowledge that the opposite position has merits, but that for the following X reasons (summarize your points), my position is the correct one
- Paragraph 2: REASON 1, with supporting evidence/examples/facts
- Paragraph 3: REASON 2, with supporting evidence/examples/facts
- Paragraph 4: REASON 3, with supporting evidence/examples/facts
- Paragraph 5: CONCLUSION
- For the X reasons previously stated, my argument is the correct one
- The argument would have been more persuasive if it ______.
What does bisect mean?
Divide (a line, angle, or shape) into TWO EXACTLY EQUAL parts.
XB bisects ABY only if it splits ABY into two exactly equal parts, meaning that there must be a 45 degree angle on both sides (given that ABY is a 90 degree angle).
How can you derive the side of a square from its diagonal?
The side of a square is it’s diagonal divided by 2 (derived from pythagoras)
How do you quickly add to fractions with different denominators?
Multiply the numerator of the first fraction with the denominator of the second fraction, which is equal to the numerator of the new fraction.
The same applies for the second part, multiply the numerator of the second fraction with the denominator of the first fraction, which is equal to the numerator of the new second fraction.
Lastly, you multiply the two denominators, which is equal to the new common denominator for the two new fractions.
Now you can just add them together normally.
How do you quickly subtract two fractions with different denominators?
Same methods as how you add them, you just subtract instead when you get the common denominator.
How do you multiply two fractions?
How do you divide a number (or fraction) with a fraction?
What does “place value” refer to?
What are inequalities in math?
Statements that involve “”.
What is a “perfect square”?
What is a “perfect cube”?
What is LCD and what is the LCD of two numbers?
What are the results of (-3)^2 and -3^2 ?
(-3)^2 = -3 * -3 = 9
-3^2 = - 3*3 = - 9
How can you simplify exponent expression so they are easier to work with?
For negative exponents, you can always divide that expression by one and change the negative exponent to a positive.
Also, you can always move exponents from the numerator to denominator or vice versa by changing the sign.
- = 1 / 2^3 = 1 / 8
- ) (3^3 * 1) / 1 = 9 / 1 = 9
When do you make ADDITION and when do you make MULTIPLICATION of exponents?
How do you add or subtract terms when the bases are different? (working with exponents)
How can you rewrite the √5^12 ?
How do you multiply and divide square roots?
What should you do when you get to a result that is not an answer option when working with square root results such as √50 or √12?
How do you ADD or SUBTRACT under the square root symbol?
Do NOT get confused or fall in the trap of using the rules from multiplication and division under the square root symbol.
Instead…
What is 7^6 * 7^9?
7^15
when multiplying roots you can just add the roots together (when the base is the same)
What is (a^3)^2?
A^6
when multiplying roots you can just add the roots together (when the base is the same)
What is (3^2)^-3 ?
3^-6
What is 5^5 / 5^3?
5^2 = 25
When dividing roots, you can subtract them as long as the base is the same
What is 11^4 / 11^x ?
11^(4-x)
what is (5^2)^x ?
25^x OR 5^2*x
what is (5^6 * 5^4x) / 5^4 ?
We can simplify to:
5^2 * 5^4x
And then simplify to:
5^(4x + 2)
What is y^7 * y^8 * y^-6?
y^7+8-6 = y^9
What is y^4 / y^-3?
Y^(4-(-3) = y^7
What is (3^2x * 3^6x) / 3^-3y?
3^8x / 3^-3y
3^8x + 3^3y OR …. 3^(8x + 3y)
What is 27^⅓ + 9^½ + (3/9^0)
3 + 3 + 3/1 = 9
What is -3^4?
-3 * -3 * -3 * -3 = -81
Which of the following 5 options are equal to (⅖)^-3?
- -(⅔)^3
- (⅖)^⅓
- (- ⅖ )^3
- (5/2)^⅓
- (5/2)^3
The answer is option 5 such that (⅖)^-3 = (5/2)^3.
You can shift around the numerator and denominator IF you also change the plus/minus sign in the root.
What is 8^3 * 2^6?
(2^3)^3 * 2^6
2^9 * 2^6
2^9+6
2^15
What is 25^4 * 125^3?
(5^2)^4 * (5^3)^3
5^8 * 5*9
5*17
What is 9^-2 * 27^2?
(3^2)^-2 * (3^3)^2
3^-4 * 3^6
3^-4+6
3^2
9
What is 6^3 + 3^3?
6^3 + 3^3
(2 * 3)^3 + 3^3
2^3 * 3*3 + 3*3
3^3 (2^3 + 1)
3^3 (9)
3^3 (3^2)
3^5
What is (4^8 - 8^4) / (2^4 + 4^2)?
(4^8 - 8^4) / (2^4 + 4^2)
((2^2)^8 - (2^3)^4 ) / ( 2^4 + (2^2)^2 )
(2^16 - 2^12) / (2^4 + 2^4)
(2^12 (2^4 - 1)) / (2^4 (1+1))
(2^12 (15)) / (2^4 (2))
(2^12 (15)) / 2^5
2^7 * (15)
What is √5 * √45?
√5*45
√225
15
What is √3 * √27?
√3*27
√81
9
what is √5000 / √50?
√5000/50
√100
10
What is (√54 * √3) / √2 ?
√54 * 3 / √2
√168 / √2
√81
9
What is √32?
Divide up 32…..
32 = 2 * 16
√16 * √2
4 * √2
what is √180?
Divide up 180
2 * 90
3 * 60
4 * 45
5 * 36
The easiest is 5 * 36
√5 * √36
√5 * 6
What is √135?
Divide up 135
3 * 45
6 * 22.5
9 * 15
√9 * √15
3 * √15
What is √35^2 - 21^2?
Find a common ground…
35 = 5 * 7….. and 21 = 3 * 7
√7^2 (5^2 - 3^2)
√7^2 (25-9)
√7^2 (16)
√7^2 (4^2)
7 * 4
28
What is √6 * (5^6 + 5^7)?
√6* (5^6(1+5))
√6 * (5^6 (6))
√6^2 * 5^6
6 * 5^3
6 * 125
750
what is √(11^4 - 11^2)/ 30?
(11^2 (121 - 1)) / 30
(11^2 (120)) / 30
11^2 * 4
11^2 * 2^2
11 * 2
22
What is √5^7 - 5^5 + 5^4?
√5^4(5^3 - 5^1 + 5^0)
√5^4 (125 - 5 + 1)
√5^4 (121)
√5^4 (11^2)
5^2 * 11
25 * 11
275
What is the reciprocal?
The number (B) for any number (A) that makes A*B = 1.
5 = Reciprocal is ⅕ as 5*⅕ = 1
⅔ = reciprocal is 3/2 as ⅔ * 3/2 = 1
what is the reciprocal of -⅚?
The reciprocal will be the opposite so switching:
6/-5
(-5 / 6) * (6 / -5) = 1
How do you divide something by a fraction?
You simply multiply by that fraction’s reciprocal = you simply flip the numerator and denominator of that fraction.
What must you be aware of if there is addition or subtraction in the fraction’s NUMERATOR?
You MUST divide with the ENTIRE denominator. Thus, you have to find common factors and simplify.
You can split up into two fractions
What must you be aware of if there is addition or subtraction in the fraction’s DENOMINATOR?
You are NOT allowed to split up into two fractions in the same way. You ALWAYS have to divide the numerator by the ENTIRE denominator.
What is a/12 - b/6 - b/4?
First, find a common denominator
a/12 - 2b/12 - 3b/12
next, add it together:
(a - 5b) / 12
What is √3/2 - √2/3?
What is 0.0004 * 0.032?
Calculate it as… 4 * 34 = 128.
Next… count the numbers AFTER the decimal.
0.0004 * 0.032 (7 numbers)
Now move the decimal point 7 to the left from 128.00
0.0000128
What is 80 % of 75 % of 120?
8/10 * ¾ * 120
8/1 * ¾ * 12
4/1 * 3/1 * 12
4 * 3 * 12
72
What is 72.12 * 10^-4?
Move the decimal 4 times to the left.
72.12
=
0.007212
what is 12.6 / 0.3?
Multiply both the numerator and denominator by 10 to get
126 / 3
42
what is 2.10 * 0.08?
Take it as
21 * 8 = 168
Now count the numbers AFTER the decimal points.
2.1 * 0.08 (3 numbers).
Move the decimal 3 to the left.
168.00 becomes 0.168.
What is 0.49 / 0.07?
Multiply both the numerator and denominator by 100 to get
49 / 7
7
What is 1.5 / ( ⅝ - 50 %)?
1.5 / (0.125)
Rewrite to fractions:
3/2 / ⅛
Use the reciprocal rule:
3/2 * 8/1 = 24/2 = 12
What is x when 30 % less than x is 63?
0.7x = 63
7x = 630
630/7 = 90
What percentage decrease is x IF 75 - x % is 54?
75 - x/100 *75 = 54
75-54 = 75x/100
21 = 75x/100
21 = 3x/4
7 = x/4
4*7 = 28 = x
28 %.
10 % of 30 % of what number is 200 % of 6?
200 % of 6 = 12
10 % of 30 % = 3 % of x
3% of x = 12
100/3 * 12 = x
100 * 4 = x
x = 400.
If there are 24 white marbles and 36 marbles in a bag, what is the ratio of blue to white marbles?
White = 24 = 2/5
Blue = 36 = 3/5
Total = 60 = 5/5
BLUE 3 to WHITE 2
3:2
Arjun has a record of winning 3 tennis matches for every 2 he loses. If he played 30 matches, how many did he win?
Win 3 = 3:5
Lose 2 = 2:5
Total 5 = 5:5
If 30 matches, he would win ⅗ * 30 = 18 matches.
What is x and y IF -2x + y = 1 AND 2(x - 3y) = 2
2(x - 3y) = 2
becomes
2x - 6y = 2
We can now eliminate x by adding the two equations together
2x - 6y = 2
- 2x +y = 1
- 5y = 3
y = -⅗
and x can be found substituting the number for y in to any of the equations.
- 2x + -⅗ = 1
- 2x = 1 ⅗
- 2x = 8/5
- x = ⅘
x = - ⅘
If 4^y = 8^y+1, then 2^y is what?
4^y = 8^y+1
Find a common base…
(2^2)^y = (2^3)^y+1
when having a common base we can now simply set the exponents equal to each other and solve…
2y = 3(y+1)
2y = 3y + 3
y = -3
so… 2^y = 2^-3 = ⅛
If (2x + 6(9-2x)) / (x-4) = -3, what is x?
First, multiply both sides with the denominator such that…
2x + 6(9 - 2x) = -3(x-4)
2x + 54 - 12x = -3x + 12
- 7x = -42
- x = -6
x = 6
If a/c + b/3c = 1, what is “c” in terms of a and b?
a/c + b/3c = 1
First get a common denominator
3a / 3c + b/3c = 1
Now multiply the denominator to both sides
3a + b = 1*3c
3a + b = 3c
c = a + b/3
What is the value of P when c = 100 and P = 300*c^2 - c?
P = 300 * 100^2 - 100
P = 300* 10,000 - 100
P = 3,000,000 - 100
P = 2,999,900
What is ((√3x + 1) / 2) - 1 = 3. what is x?
multiply by 2 to remove the denominator.
√3x + 1 -2 = 6
√3x + 1 = 8
square both sides
3x + 1 = 64
3x = 63
x = 21
What is x if 13 / (x + 13) = 1
13/(x + 13) = 1
13 = 1(x + 13)
13 = x + 13
x = 0
2h - 4k = 0 and
k = h -3
2h - 4(h - 3) = 0
2h - 4h + 12 = 0
12 = 2h
h = 6
k = h - 3
k = 6 - 3 = 6
What is b and g if…12b = 2g and 4g - 3b = 63?
12b = 2g
6b = g
4g - 3b = 63
substitute g with 6b…
4*6b -3b = 63
24b - 3b = 63
21b = 63
b = 3
then g will be…
6b = g
g = 6*3 = 18.
If y - 2x - 1 = 0
and… x -3y - 1 = 0
what is the values of x and y?
First, rewrite the two equations for them to be as similar as possible…
y - 2x = 1
-3y + x = 1
now multiply the second equation with 2 to get an equal value of x for the two equations.
2(-3y + x) = 1*2
-6y + 2x = 2
Now add the two equations to eliminate x.
y - 2x = 1
- 6y + 2x = 2
- 5y = 3
y = - ⅗
2x = 1 + y
2x = 1 + 3/5
2x = 8/5
x = 4/5
If 2p = (m - 5) / (n +2), what is m in terms of n and p?
We need to isolate m…
2p (n+2) = m - 5
2pn + 4p + 5 = m
What is the value of z if… 2^x+y = √(z - 2) and… x = 2 - y
2^x+y can be re-written as we know that x = 2 - y
2^(2-y)+y
so…
2^(2-y)+y = √(z - 2)
2^2 = √(z - 2)
Now we square both sides…
(2^2)^2 = z - 2
16 = z - 2
z = 18.
Rewrite x^2 + 14x + 33
To rewrite, we want it in the form…
(x ….)(x….).
to figure out what should be in the parentheses besides x, we have to look at the number “33” and figure out its factors AND find a set of factors that can yield 14 as well.
The factors of 33 are: 1 and 33. 3 and 11.
1 and 33 can never become 14.
3 and 11 can be added to become 14 so that is the set of factors we want.
(x + 3) (x + 11) = x^2 + 14x + 33
what is x IF x^2 - 15x = - 26?
first, recognize that there most likely will be two different solutions as we have x^2.
x^2 - 15x = -26
x^2 -15x + 26 = 0
(x - 2) (x - 13) = 0
x - 2 = 0, then x = 2
x - 13 = 0, then x = 13.
X is either 2 or 13.
What is x IF x^3 - 2x^2 = 3x?
First recognize that we might have three different solutions as x^3.
Now factor out x.
x^3 - 2x^2 = 3x
x^3 - 2x^2 - 3x
x (x^2 - 2x - 3)
x ( (x - 3) (x + 1)) = 0
x = 0, then x = 0
x - 3 = 0, then x = 3
x + 1 = 0, then x = -1
What is ( x^2 + 7x + 12) / (x + 3) = ?
Rewrite the numerator
x^2 + 7x + 12
((x + 3) (x + 4)) / (x + 3)
We can remove (x + 3) from both numerator and denominator
We get…
(x + 4)
what is y IF …. y^2 - 11y + 30 = 0?
y^2 - 11y + 30 = 0
(y - 5) (y -6) = 0
y - 5 = 0, then y = 5
y - 6 = 0, then y = 6
what is x IF…. x^3 - 5x^2 + 4x = 0?
x^3 - 5x^2 + 4x = 0
x (x^2 - 5x + 4) = 0
x ( (x - 1)(x - 4) ) = 0
x = 0, then x = 0
x - 1 = 0, then x = 1
x- 4 = 0, then x = 4
what can you reduce the following expression to? (a^2 - b^2) / (a - b)
(a^2 - b^2) / (a - b)
Write out the numerator…
((a + b) ( a - b)) / ( a- b)
Reduce…
(a + b).
What can you reduce the following expression to? (2t - 1 + (2t-1)^2 ) / (2t - 1)
(2t - 1 + (2t-1)^2 ) / (2t - 1)
write out the numerator
(2t - 1 + ((2t-1)(2t-1))) / (2t - 1)
now factor out (2t - 1)…
(2t-1) (1 + (2t - 1)) / (2t-1)
Reduce…
(1 + (2t - 1))
1 + 2t - 1
2t…
What can you reduce the following expression to? …. (3x^2 - 6x - 45) / (3x + 9)
(3x^2 - 6x - 45) / (3x + 9)
factor the numerator / rewrite…
((3x + 9)(x - 5)) / (3x+9)
Reduce…
(x - 5)
What can you reduce the following expression to? (5ab + abc) / (abc^2 + 10abc + 25ab) ?
(5ab + abc) / (abc^2 + 10abc + 25ab) ?
First factor out “ab”.
(ab (5 + c)) / (ab (c^2 + 10c + 25))
reduce…
(5 + c) / (c^2 + 10c + 25)
rewrite the denominator…
(5 + c) / ((c + 5)(c + 5))
Reduce…
1 / (5 + c)…
What is the solution to |−x - 4| > 8 ?
|−x - 4| > 8
- x -4 > 8 -(-x) - (-4) > 8
- x > 12 x + 4 > 8
x < -12 x > 4
what is the solution to 2(x-1)^3 + 3 < 19 ??
2(x-1)^3 + 3 < 19
2(x-1)^3 < 16
(x-1)^3 < 8
x-1 < 2
what is the solution to…. 2 |x + 0.32| = 7
2 |x + 0.32| = 7
x + 0.32 = 3.5
x + 0.32 = 3.5 -x - 0.32 = 3.5
x = 3.18 -x = 3.82
x = -3.82
What is the solution to…. |x/4 + 3| = 0.5
x/4 + 3 = 0.5 -x/4 - 3 = 0.5
x/4 = -2.5 -x/4 = 3.5
x = -10 -x = 14
x = -14
What is the solution to…. |x^3| < 64
x^3 < 64 -x^3 < 64
x < 4 -x < 4
x > -4
A hose is placed into an empty pool and turned on at 2:00 pm. the pool, which holds 680 gallons of water, reaches its capacity at 5:24 pm. How many gallons of water per hour did the hose add to the pool?
680 gallons
Time: 5:24pm - 2:00pm = 3 hours and 24 minutes.
680g / 3:24h
680g / 204m = 3 ⅓
3 ⅓g/m * 60 = 200 gallons of water per hour
40 students in a class of 200 got A’s on their test. 64 got B’s, 18 got D’s and 6 got F’s. If students can only get A, B, C, D or F as grades, what percent of the students got C’s?
a = 40
b = 64
c = ?
d = 18
e = 6
t = 200
40 + 64 + 18 + 6 + c = 200
c = 72
72 / 200 = 36 %.
The temperature in Limerick is ¾ that in Cairo, where the temperature is 8/5 that in Halifax. If the temperature in Limerick is 66 degrees, what is the temperature in Halifax?
L = 66
L = ¾ C
C = 8/5 H
H = ?
L = ¾ (8/5H)
L = 24/20H
L = 6/5H
66 = 6/5H
66* ⅚ = H
H = 55
At a convention of monsters, ⅖ have no horns, 1/7 have one horn, ⅓ have two horns, and the remaining 26 have three or more horns. How many monsters are attending the convention?
⅖ no horns
1/7 one horn
⅓ two horns
26 three or more horns
⅖ + 1/7 + ⅓
35/105 + 15/105 + 35/105 = 92/105
105/105 - 92/105 = 13/105
13/105*m = 26
m = 26 * 105/13
13m = 26*105
13m = 2730
m = 210
Of all the homes on Gotham Street, ⅓ are termite-ridden, and ⅗ of these are collapsing. What fraction of the homes are termite-ridden, but NOT collapsing?
Ft = ⅓
Ft+c = ⅓ * 3/5 = 3/15
Ft-c = Ft - Ft+c
Ft-c = ⅓ - 3/15
Ft-c = 5/15 - 3/15 = 2/15
A plane leaves Chicago in the morning and makes three flights before returning.The first flight has traveled twice as far as the second flight, and the second flight traveled three times as far as the third flight. If the third flight was 45 miles, how many miles was the first flight?
F1 = 2*F2
F2 = 3*F3
F3 = 45
F1 = ?
F1 = 2(3*F3)
F1 = 6*F3
F1 = 6*45
F1 = 270 miles.
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?
Q2 * P + Q2(P+2) = 2400
100*11 + Q2(11+2) = 2400
1100 + Q2(13) = 2400
Q2*13 = 1300
Q2 = 1300
Q = Q1 + Q2
Q = 100 + 100 = 200.
Amar is 30 years younger than Lauri. In 5 years, Lauri will be three times as old as Amar. How old will Lauri be in 10 years?
A = L - 30, L = A + 30
L in + 10y?
L + 5 = 3A
L+5 = 3(L - 30 +5)
L+5 = 3L - 75
80 = 2L
40 = L
L + 10 = ?
40 + 10 = 50.
What is the temperature in Fahrenheit when it is 30 degrees Celsius? C = 5/9(F - 32)
C = 30
C = 5/9(F - 32)
30 = 5/9(F-32)
270 = 5F - 160
430 = 5F
86 = F
Joe’s car can travel 36 miles per gallon of fuel. Approximately how many kilometers can the car travel on 10 liters of fuel? (5 miles = approximately 8 kilometer; 1 gallon = approximately 4 liters).
36m / g * 8km /5m = 288km / 5gallons
288km/5gallons * 1gallon/4liters
288km / 20liters
- 4 kilometers/l
- 4 km/l * 10 liters = 144 kilometers.
Svetlana is running a 10-kilometer race. She runs the first 5 kilometers of the race at a speed of 12 kilometers per hour. At what speed will she have to fun the last 5 kilometers of the race if she wants to complete the 10 kilometers in 55 minutes?
Distance = Rate * Time
5 = 12 * t
5 = 12t
t = 5/12
55 minutes * 1hour/60minutes = 55/60 = 11/12
11/12 - 5/12 = 6/12 = 1/2
The last 5 kilometers will be….
Distance = Rate * Time
5 = r * 0.5
r = 10 km/t.
What is arc length?
A section of circle’s circumference.
what is central angle?
The angle created by any two radii..
Fx. the angle determining what big a share a sector is of a whole circle
What is sector area?
A “wedge” of the circle, composed of two radii and the arc connecting these two radii.
What is the hypotenuse?
The longest side of a RIGHT triangle. The hypotenuse is opposite the right angle.
What are the 5 pythagorean triplets?
8-15-17
5-12-13 and double up: 10-24-26
3-4-5 and double up: 6-8-10
True or false? The point (4,14) is on the curve y = x^2 - 2.
The equation y = 3x + 4 is already in y = mx + b form.
For the point 4, 14. to be on the curve, you can plug in x = 4 into the equation.
y = 4^2 - 2
y = 16 -2
y = 14.
Which is the y-coordinate we were given so the answer is YES.
A sector has a central angle of 270 degrees. If the sector has a radius of 4, what is the area of the sector?
270/360 = ¾ of a circle.
The circles Radius is 4.
So TOTALarea = 4^2*π = 16π
16π * ¾ = 12π.
A sector has a radius of 8 and an area of 8π. What is the arc length of the sector?
SectorRadius = 8
SectorArea = 8π
TOTALarea = 8^2*π = 64π
8π / 64π = ⅛.
TOTALcircumference = D*π = (2*8)*π = 16π
SECTORarclength = 16π * ⅛ = 2π
A sector has an arc length of π/2 and a central angle of 45 degrees. What is the radius of the sector?
SECTORarclength = π/2
SECTORcentralangle = 45
SECTORradius?
45/360 = ⅛. = the sector is ⅛ of the circle
So… π/2 is ⅛ of the circumference of the circle.
8π / 2 = 4π = TOTALcircumference
C = D * π
C = (2*r)*π
So the squareroot(4)*π = 2π.
SECTORradius = 2.
Two sides of a triangle have lengths 4 and 8. Which of the following are possible side lengths of the third? a = 2, b = 4, c = 6, d = 8
8+4 = 12
8-4 = 4
4 < x < 12
only c = 6 and d = 8 works.
Does the point (3, -2) lie on the line y = 2x - 8?
Plug in x = 3 and see if it results in y = -2.
y = 2*3 - 8 = -2.
YES…
For the line y = 4x +2, what is the y-coordinate when x = 3?
Plug in x = 3 and see…
y = 4*3 + 2 = 14.
In the coordinate plane, a circle has center (2, -3) and passes through the point (5, 0). What is the area of the circle?
A = r^2 * π
A = (2 - 5)^2 + (-3 + 0)^2 * π = (9 + 9)π = 18π
What is (0.0036*2.8) / ( (0.04)*(0.1)*(0.003))?
(0.0036*2.8) / ( (0.04)*(0.1)*(0.003))
(36*10^-4) * (28*10^-1) / ( (4*10^-2)*(1*10^-1)*(3*10^-3))
(36*28 / 4*1*3) * 10^(-4-1)-(-2-1-3)
36/3 * 28/4 * 10^-5-(-6)
12*7*10^1 = 840.
What is a quotient and a remainder?
When x and y are positive integers, there exist unique integers q and r, called the quotient and remainder:
y = x*quotient + remainder, and 0 < r < x
What is the quotient and remainder of y =29 when x =3?
y = x*quotient + remainder
29 = 9*3 + 2
Is 0 positive or negative?
It is NEITHER positive or negative. It is just 0.
Are 8/36 and 14/63 equivalent?
Yes…
they are EQUIVALENT as both represent 2/9
What are mixed numbers?
Numbers that consist of both a whole number and a fraction.
fx… 7⅔
7⅔ = 23/3
What is I x + y I equal to?
I x + y I < I x I + I y I
as… if x = 10, y = 2
10+2 = 10 +2 = 12 < 12
and…
if x = 10 and y = -2
then…
10 - 2 = 10 + 2 = 8 < 12
What is the formula for calculating the PERCENTAGE increase/decrease?
Take…
(New number - Original number) / Original number
(30 - 24) / 24 = 6 / 24 = 0.25 = 25 %
What is 0!, 1!, 2! and 3!?
0! = 1 (by definition 0! = 1)
1! = 1
2! = 1*2 = 2
3! = 1*2*3 = 6
What are the permutations of A, B and C?
Essentially how many ways can you order A, B and C?
The answer is “3!”
3(n-1)(n-2) = 3*2*1 = 6
If you have the letters, A, B, C, D and E…, how many combinations can you make when only choosing 2 items at a time?
n = 5
k = 2
5! / (2! (5-2)!)
5! / (2! * 3!)
120 / 2*6
120 / 12
10 combinations
What is P(a) = 0.3 + P(b) = 0.42 IF 1) P(a) and P(b) are mutually exclusive? and 2) if P(a) and P(b) are independent?
P(a or b) = P(a) + P(b) (where A and B are mutually exclusive)
0.3 + 0.42 = 0.72 = 72 %
P(a or b) = P(a) + P(b) - P(a)*P(b) (where A and B are independent events)
0.3 + 0.4 - 0.3*0.42 = 0.7 - 0.126 = 0.574 = 57.4 %
What are the solutions to x(x-3)(x^2+5) / x-4?
A fraction equals 0 if and only if its numerator equals 0.
Thus… x(x-3)(x^2+5) = 0
x=0, x-3=0 or x^2+5=0
x=0 or x=3 but there is no solution to x^2+5=0 as x^2 + 5 > 0 for all real numbers.
Thus, the two only solutions are x=0 and x=3.
what is the formula used to find the roots of quadratic expressions and when is it relevant?
When you cannot factor a quadratic expression, you can always find its roots with the following formula..
What is the rules regarding lengths of 45-45-90 degrees triangles and 30-60-90 triangles?
If n = 33^43 + 43^33, what is the units digit of n?
Both numbers are raised to 3.. 33 and 43 so they will follow the sequence of 3,9,7,1,3,9,7,1… as 3^1 = 3, 3^2 = 9, 3^3 = 27 and 3^4 = 81.
thus…. the 43rd number will be 7 and the 33rd number will be 3.
7+3 = 10 = 0
If x is a positive integer greater than 1, what is the value of x ?
2x is a common factor of 18 and 24.
x is a factor of 6.
The answer is A.
X > 1, then 2*x = > 4 AND has to be a common factor of 18 and 24.
The factors of 18 are… 1,2,3,6,9,18 and the factors of 24 are 1,2,3,4,6,8,12,24
The only common factor that is >4 is 6.
2x = 6 and then x = 3
2) X is a factor 3 AND greater than 1. It can be either 2, 3 or 6.
If a triangle is circumscribed by a circle such that one of its sides is a diameter of the circle, what can be said about that triangle?
That it will be a right triangle.
How do you calculate the slope of a line using two coordinates?
On a 400-mile trip, Car X traveled half the distance at 40 miles per hour and the other half at 50 mph. What was the average speed of car x?
- First determine the total traveling time
200 miles / 40 miles per hours = 5 hours
+
200 miles / 50 miles per hour = 4 hours
=
9 hours total
400 miles / 9 hours = 44 4/9 miles per hour.
If machine X can produce 1,000 bolts in 4 hours and Machine Y can produce 1,000 bolts in 5 hours, in how many hours can Machines X and Y, working together at these constant rates, produce 1,000 bolts?
1/r + 1/s = 1/h
1/machine x + 1/machine y = 1/combined time
¼ + ⅕ = 1/h
5/20 + 4/20 = 1/h
9/20 = 1/h
9h = 20
h = 20/9
h = 2 2/9
Working together, Machines X and Y can produce 1,000 bolts in 2 hours and 2/9 hours.
How many liter of a solution that is 15 % salt must be added to 5 liters of a solution that is 8 % salt so that the resulting solution is 10 % salt?
0.15n + 0.08(5) = 0.10(n+5) (multiply both sides by 10)
15n + 40 = 10n + 50
5n = 10
n = 2
So add 2 liters and we get a total of 5+2= 7 liters, which now has a salt percentage of 10.
The price of an item is discounted by 20 % and then this reduced price is further discounted by an additional 30 %. These two discounts are equal to an overall discount of what percentage?
The discounted price is…
0.8p
After second discount…
0.7*0.8p = 0.56p.
1-0.56 = 0.44 = 44 % total discount.
Each of 25 people is enrolled in history, mathematics or both. If 20 are enrolled in history and 18 are enrolled in mathematics, how many are enrolled in both history and mathematics?
History = 20-n
Both = n
Mathematics = 18-n
(20 - n) + n + (18 - n) = 25
38 - n = 25
-n = -13
n = 13
In a certain production lot, 40 % of the toys are red and the remaining toys are green. Half of the toys are small and half are large. If 10 % of the toys are red and small, and 40 toys are green and large, how many of the toys are red and large?
If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3k is a factor of p ?
During a trip, Francine traveled x percent of the total distance at an average speed of 40 miles per hour and the rest of the distance at an average speed of 60 miles per hour. In terms of x, what was Francine’s average speed for the entire trip?
A worker carries jugs of liquid soap from a production line to a packing area, carrying 4 jugs per trip. If the jugs are packed into cartons that hold 7 jugs each, how many jugs are needed to fill the last partially filled carton after the worker has made 17 trips?
Carrying 4 jugs per trip, the worker carries a total of 4(17) = 68 jugs in 17 trips. At 7 jugs per carton, these jugs will completely fill 9 cartons with 5 jugs left over since (9)(7) + 5 = 68. To fill the 10th carton, 7 – 5 = 2 jugs are needed.
How many prime numbers between 1 and 100 are factors of 7,150 ?
To find the number of prime numbers between 1 and 100 that are factors of 7,150, find the prime factorization of 7,150 using a method similar to the following:
7,150=10×715=(2×5)×(5×143)=2×5×5×(11×13)7,150=10×715=(2×5)×(5×143)=2×5×5×(11×13)
Thus, 7,150 has four prime factors: 2, 5, 11, and 13.
Team A and Team B are competing against each other in a game of tug-of-war. Team A, consisting of 3 males and 3 females, decides to line up male, female, male, female, male, female. The lineup that Team A chooses will be one of how many different possible lineups?
Any of the 3 males can be first in the line, and any of the 3 females can be second. Either of the 2 remaining males can be next, followed by either of the 2 remaining females. The last 2 places in the line are filled with the only male left followed by the only female left. By the multiplication principle, there are 3 × 3 × 2 × 2 × 1 × 1 = 36 different lineups possible.
In an electric circuit, two resistors with resistances x and y are connected in parallel. In this case, if r is the combined resistance of these two resistors, then the reciprocal of r is equal to the sum of the reciprocals of x and y. What is r in terms of x and y ?
In Country C, the unemployment rate among construction workers dropped from 16 percent on September 1, 1992, to 9 percent on September 1, 1996. If the number of construction workers was 20 percent greater on September 1, 1996, than on September 1, 1992, what was the approximate percent change in the number of unemployed construction workers over this period?
If n = 38 − 28, which of the following is NOT a factor of n ?
- 97
- 65
- 35
- 13
- 5
