Algebra Flashcards
1
Q
Re-expressing Fractions:
A
Re-expressing Fractions:
If something costs 1/3 less than a specific amount then another way of expressing it is it costs 2/3 of that specific amount.
2
Q
Reverse FOIL
A
Reverse FOIL
x^2 + 7x +6
same as: (x+1)(x+6)
Because between 7x and 6 there is a + you know that the two sings in parenthesis have to be the same. The sign before 7x tells you then which sign to use. If the sign between 7x and 6 was negative you would know that the signs in the new equation have to be different in both parenthesis.
Check by multiplying to see if it equals 7x:
1x + 6x = 7x
3
Q
Combining equations:
A
Combining equations:
e.g. 4x + y = 8 and y - 3x = 7
Look for similar expressions in both equations! Then you can combine:
4x + y = 8 \+-3x + y = 7 ------------- x+2y=15
4
Q
Combining Expression About Remainders:
A
If question says:
n/3 = x + 1 and n/7 = x + 1 the two statements are equivalent to: x/21 = x + 1 because 3 and 7 have no factors in common.
5
Q
Rules of Positive and Negative Multiplication:
A
Rules of Positive and Negative Multiplication:
Positive x Positive = Positive
Negative x Negative = Positive
Negative x Positive = Negative
(x – 3)2 = always positive
(x + 3)2 = always positive
(x + 3)3 = could be positive or negative depending on x
(x – 3)3 = could be positive or negative depending on x
6
Q
Rules of Odd and Even in Equations:
A
Rules of Odd and Even in Equations:
(odd x odd) = odd (even x even) = even (odd x even) = even (odd + odd) = even (odd + even) = odd (even + even) = even
You also need to know this for Number Properties questions so know this cold.
There are no odd and even rules for division, mainly because there is no guarantee that the result will be an integer. For example, if 3z = 6, then z is the even integer 2. However, if 3z = 2, then z = 2/3, which is not an integer at all.
7
Q
Square vs. Cube:
A
Square vs. Cube:
if x^3 = x, then x = -1, 0, or 1
if x^2 = x, then x = 0 or 1
8
Q
Rules of Square Roots:
A
Every positive square has two square roots, a positive and a negative one.
√-36 and √36 = -6^2 and 6^2
9
Q
Square Roots and Perfect Squares:
A
If Square root of a variable is an integer it means that the variable is a perfect square, like 16, 25, or 49.
10
Q
Simplifying Equations 1:
A
Simplifying Equations:
How to get rid of a square root:
E.g. 2c = √a/b
Rise to square on both sides to get rid (2c)2 = (√a/b)2
4c2 = a/b^2
11
Q
Simplifying Equations 2:
A
Simplifying Equations:
How to get rid of fractions:
E.g. 4c = a/b
To solve for a =cross-multiply by b 4cb = a
Then divide by 4c to solve for b b = a/4c
12
Q
Multiplying a Sum Equation:
A
Multiplying a sum equation:
If s = u - 10
Then s x 5 = (u-10) x 5
Don’t forget that the multiplication applies to the entire sum equation. You can’t do u – 10 x 5.
13
Q
Units Digits in Multiples:
A
Units Digits in Multiples:
Multiples of 5 have units digits of: 0 or 5
14
Q
Cubes of Numbers:
A
Cubes of Numbers:
Cubes of numbers with units digits of 0 or 5 keep the same units digit. E.g. cube of 10 is 1000, cube of 5 is 125)
15
Q
Word Translation into Math:
A
Word Translation into Math:
equals, is, was, will be, has, costs, adds up to…: =
times, of, multiplied by, product of, twice, by: x
divided by, per out of, each, ratio: /
plus, added to, and, sum combined: +
minus, subtract from, smaller than, less than,
fewer, decreased by, difference between: -
a number, how much, how many, what x, n, etc.
16
Q
Algebraic Vocabulary:
A
Algebraic Vocabulary:
- A TERM is a numerical constant or product of numerical constant and one or more variables.
E.g. 5, 3x, 4x^3y - An ALGEBRAIC EXPRESSION is combination of one or more terms, separated by + or -.
E.g. 4ab + 5cd - A COEFFICIENT is the numerical constant in a term like:
E.g. 3 in 3xy
E.g. 1 in just x - A CONSTANT TERM is a number without any variables:
E.g. just 3 - A MONOMIAL is an expression with just term:
E.g. 3xy - A BINOMIAL is an expression with two terms:
E.g. 3xy + 2ax - A TRINOMIAL is an expression with three terms:
E.g. 3xy + 2ax + x^3 - A POLYNOMIAL is general name for expressions with more than one term.
17
Q
Order of Operations (PEMDAS):
A
Order of Operations (PEMDAS):
P = Parentheses E = Exponents M = Multiplication (in order from left to right) D = Division (in order from left to right) A = Addition (in order from left to right) S = Subtraction (in order from left to right)
18
Q
Laws of Arithmetic Operations:
A
Laws of Arithmetic Operations:
COMMUTATIVE LAW:
2x - 5y = -5y + 2x
5a * 3b = 3b * 5a = 15ab
ASSOCIATIVE LAW:
2x - 3x + 5y + 2y = (2x -3x) + (5y + 2y) = -x + 7y
(-2x) (1/2 x) (3y) (-2y) = (-x^2) (-6y^2) = 6x^2y^2
Process of simplifying expressions by subtracting or adding together terms with same variable component is called COMBINING LIKE TERMS.
DISTRIBUTIVE LAW:
3a(2b - 5c) = (3a x 2b) - (3a x 5c) = 6ab - 15ac
For Product of two binomials apply distributive law twice:
(y+5) (y-2)
= y(y-2) + 5(y-2)
= y^2 - 2y + 5y -10
= y^2 +3y -10
So multiply the First terms first, then the Outer terms, then the Inner terms, and finally the Last terms ⇒ FOIL
19
Q
Factoring Algebraic Expressions:
A
Factoring Algebraic Expressions:
FACTORING A POLYNOMIAL: express polynomial as product of two or more simpler expressions.
For that you can use a COMMON MONOMIAL FACTOR:
if there is a monomial factor common to every term in the polynomial, it can be factored out by using the distributive law:
2a + 6ac = 2a (1 + 3c)
because 2a is great common factor of 2a and 6ac
Remember: Making Problems look more complicated than they are by distributing a common factor is a classic GMAT trick. ⇒ Whenever algebra looks scary, check whether common factors could be factored out.
20
Q
Linear/First Degree Equations:
A
Linear/First Degree Equations:
Equations in which all variables are raised to the first power, i.e. there are no squares or cubes.
21
Q
Equations with Fractions:
A
Equations with Fractions:
GMAT loves to make algebra problems look harder than they need to be by using fractions. ⇒ Whenever you see a fraction in an algebraic question, always get rid of the fraction as first step.
22
Q
Getting Rid of Negative Terms in Fractions:
A
Getting Rid of Negative Terms in Fractions:
if -VR/VT - P
Then multiply both the numerator and denominator of the fraction by -1
That gives you: VR/-VT + P
Which is the same as: VR/P - VT
23
Q
Getting Rid of Fractions in Equations:
A
Getting Rid of Fractions in Equations:
When getting rid of fractions in equations, be careful to transfer all negative signs correctly across parentheses.
E.g. x - [2 - x^2/x] = y/x
multiply both sides by x to get rid of fractions:
= x*x - x [2 - x^2]/x = y
now if you get rid of the x in the second expression on the left you still will have a -1 there:
= x^2 -1 [2 - x^2] = y
now get rid of parentheses by multiplying -1 by every term in it.
= x^2 - 2 + x^2 = y
= 2x^2 - 2 = Y
24
Q
Factoring Quadratic Expressions:
A
Factoring Quadratic Expressions:
When you are asked to solve for a variable that is squared, the most efficient solution is typically to factor the equation into two binomials. Factoring quadratic expressions mean using reverse FOIL to make an quadratic expression simpler. Essentially you’re factoring a polynomial into a product of two binomials. The product of the first term in each binomial must equal the first term in the polynomial. The product of the last term in each binomial must equal the last term of the polynomial. The sum of the remaining products must equal the second term of the polynomial.
How to use reverse FOIL:
- write down (x ) (x )
You now have to fill in the missing term in both parentheses. - Find the missing term in both parentheses. Remember that the product of the two missing terms will be the last term in the polynomial and the sum of the two missing terms will be the coefficient of the second term of the polynomial.E.g. x^2 -3x +2 = (x ) (x )
You know that the product of the terms you fill in has to be two and the sum -3. Through trial and error try to come up with the right answer. four constants would multiply to make 2 either 2 and 1 or -2 and -1. 2 and 1 would add up to +3 so that couldn’t be it. but -2 + -1 would add up to -3 so the answer is:
x^2 -3x +2 = (x - 2) (x - 1)
Rules for signs: If the coefficient of the constant (the last term) is negative, then you binomials will have different signs (one + and one -). If the coefficient is positive, then your binomials will both have the same sign as the coefficient in the middle term (two + or two -).
25
Classic Quadratics:
Classic Quadratics:
There are some patterns called "classic quadratics" which you can factor more quickly by recognizing the pattern than using reverse FOIL. The three patterns are:
1. Difference of Two Perfect Squares
2. Polynomials of the Form a^2 + 2ab + b^2
3. Polynomials of the Form a^2 - 2ab + b^2
The GMAT uses all three forms over and over so practice thoroughly. Notice that all 3 forms begin and end in a perfect square:
a^2 - b^2
a^2 + 2ab + b^2
a^2 - 2ab + b^2
26
Classic Quadratics: Difference of Two Perfect Squares
Classic Quadratics: Difference of Two Perfect Squares
The different of two squares can be factored into the product of two binomials:
a^2 - b^2 = (a + b) (a - b)
d^2 - 16 = (d + 4) (d - 4)
27
Classic Quadratics: Polynomials of the Form a^2 + 2ab + b^2
Classic Quadratics: Polynomials of the Form a^2 + 2ab + b^2
Any polynomial of this form is equivalent to the square of a binomial. The binomial is the sum of two terms:
a^2 + 2ab + b^2 = (a + b)^2
x^2 + 6x + 9 = (x + 3)^2
Example of a more complicated example that might come up on the GMAT:
4x^4 + 52X^2 + 169
if you recognize that 169 is 13^2 and 4x^4 is (2x^2)^2 you know you can factor this to:
(2x^2 + 13) (2x^2 + 13) or (2x^2 + 13)^2
28
Classic Quadratics: Polynomials of the Form a^2 - 2ab + b^2
Classic Quadratics: Polynomials of the Form a^2 - 2ab + b^2
Any polynomial of this form is equivalent to the square of a binomial. The binomial is the difference of two terms:
a^2 - 2ab + b^2 = (a - b)^2
x^2 - 10x + 25 = (x - 5)^2
29
Solving Quadratic Equations:
if expression X^2 - 3x + 2 is set to 0 it's a quadratic equation. You can find the value or values for x that make the equation work by using the factored form of the equation that you get through reverse FOIL:
X^2 - 3x + 2 = 0
(x - 2) (x - 1) = 0
A product is 0 if at least one of the terms is 0. That means in this example x = 2 or x = 1. Plugging both in will show that both make the equation work so both are values of x.
REMEMBER: Quadratic equations usually have two roots, or solutions. If both factors involving a variable are the same then there is only one distinct root. E.g. (x + 2)^2
30
Squares from 2 to 23:
Squares from 2 to 23:
```
2^2 = 4
3^2 = 9
4^2 = 16
5^2 = 25
6^2 = 36
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
17^2 = 289
18^2 = 324
19^2 = 361
20^2 = 400
21^2 = 441
22^2 = 484
23^2 = 529
```
31
Finding all Variables in Linear Equations:
REMEMBER: If a problems has multiple variables, you need as many distinct equations as you have variables to find unique numerical values for all variables. Distinct means every equation provides new, different information.
There are two approaches to combine the equations to simplify calculations:
1. Isolate one variable in one equation and plug it into the other equation.
2. Add or subtract whole equations from each other to eliminate one of the variables.
Example for 2:
Find values of x and y if 4x + 3y = 27 and 3x - 6y = -21
If you look closely you can multiply the first equation to get rid of the y:
2(4x + 3y) = 2*27
= 8x + 6y = 54
Now add new and second equation:
8x + 6y = 54
+ 3x - 6y = -21
----------------------
11x = 33
x = 3
x can now be plugged into either equation to find y.
⇒ Always look for ways in which you can add or subtract equations and change one equation (on both sides) to change the subtraction and addition in a way to solve for one variable.
Steps to take in these questions:
1. Check how many distinct linear equations and how many variables. If two distinct equations and two distinct variables you can solve for the variables.
2. Choose the technique. If equations look easy to simplify, substituting one equation into the other will be the most efficient approach.
3. Remember that if in the question stem is says that you are looking for x then you have to simplify one equation for to solve for y first and then plug that into the other so that you have eliminated the y and can now solve for x.
E.g. after simplifying and solving for y you might have: y = x - 5 then you can plug that into the other equation to solve for x.
32
Special Cases in Systems of Linear Equations:
In some cases you don't need to solve for equations to find the value of each variable but you are asked to figure out relationships between variables, ratios, difference etc. Most frequently they ask for sums, differences, averages, and ratios and most of these special cases in systems of linear equations appear in the DS section.
REMEMBER: Don't assume that you can't answer a question just because there are more variables than equations because by simplifying the equation you may be able to cancel out one or more variables.
How to solve:
- Use critical thinking and pattern recognition to solve problems involving special cases in systems of linear equations.
- When you are asked to solve for the sum/difference/product/quotient of variables, you may not need to solve for each variable.
- Make sure to cancel out or combine variables before you determine that there are more variables than equations.
33
Simplifying expressions in DS questions:
GMAT questions rarely give you algebraic statements in their most useful form. It's a safe bet that you'll need to simplify or re-express almost all the algebra you see on Test Day. Doing so will help steer clear of common wrong answers. In DS questions you may also see that simplifying the equation in statement results in the same equation as you were given in the other statement so that both statements are really giving you the same equation but in a concealed way.
34
Absolute Value:
Absolute value of a number is the number's distance from zero on the number line. So, because 5 and -5 are the same distance from zero, both numbers have the same absolute value of 5. So, since the absolute value is a distance, it is always nonnegative.
E.g. both 3 and -3 are three units from zero, so the absolute for either one is 3.
The absolute value of a number is denoted by two vertical lines:
E.g. I-3I = 3
I3I = 3
Also: if |a|>|b| then you know that a is father away from 0 than b. That means that a couldn't be 0 because then b couldn't be closer to 0 than a because there is no value closer to 0 than 0.
35
Largests/Smallest Absolute Value Questions with Variables:
If you are asked for the largest or smallest absolute value of answer choices that contain variables, it's efficient to pick numbers and go through every answer choice quickly.
E.g. If - 2 smaller or same as x smaller or same as 2 then which of the following has the largest possible absolute value?
```
3x - 1
x^2 - x
3 - x
x - 3
x^2 + 1
```
Pick numbers that are the highest values on the range, so -2 and 2 and go through every answer choice.
36
Absolute value of Expressions with Variables that are Negatives:
Expressions with variables that are negatives of each other will always produce the same absolute value.
E.g. x - 3 and 3 - x are negatives of each other. No matter what you fill in for x, it will produce the same absolute value (distance from 0).
37
Distributive Law in Arithmetic:
You may distribute a factor among the terms being added or subtracted.
E.g. a(b + c) = ab + ac
Same in division:
E.g. 4 + 6/2 = 4/2 + 6/2
But not when sum or difference is in the denominator:
E.g. 9/4+ 5 is not the same as 9/4 + 9/5.
38
Factoring in Arithmetic:
Use factoring as the distributive law in its reverse form in order to simplify calculations:
E.g. 11 + 22 + 33 + 44 = 11 (1 + 2 + 3 + 4)
39
Equivalent Fractions in Arithmetic:
multiplying or dividing the numerator and denominator of a fraction by the same number (any number other than 0) will leave the fraction unchanged.
E.g. 1/2 = 1x2/2x2
You can use this technique to get rid of decimals in the denominator:
2/0.5 = 2 x 2/0.5 x 2 = 4/1
40
Canceling and Reducing Fractions in Arithmetic:
Put fractions on the GMAT in their lowest form.
41
Dividing Fractions in Arithmetic:
To divide one fraction by another, just multiply the first fraction by the reciprocal of the divisor (second fraction)
E.g. 4/3 / 4/9 = 4/3 x 9/4
42
Exponents:
In the term 3x^2, 3 is the coefficient, x is the base, and 2 is the exponent.
Remember that in 3x^2 only the x is being squared. If the whole term was to be squared you would write: (3x)^2.
Remember that in 3x^2 you raise to the power before you multiply by 3.
A number multiplied by itself twice is called square.
A number multiplied by itself three times is called cube.
Any number raised to the first power equals itself:
E.g. a^1 = a
Any number except zero that is raised to the zero power is equal 1:
E.g. a^0 = 1
0^0 is undefined.
ALSO REMEMBER:
x^r+s = (x^r) (X^s)
43
Multiplying with Different Exponents:
To multiply two terms with the same base, keep the base and add the exponents:
E.g. 2^2 x 2^3 = 2^5
44
Dividing with Exponents:
To divide two terms with same base, keep base and subtract the exponent of the denominator from the exponent of the numerator:
E.g. 4^4/4^2 = 4^4-2 = 4^2
45
Exponents of Exponents:
To raise a power to another power, multiply exponents:
E.g. (3^2)^3 = 3^6
46
Multiplying with Same Exponents:
To multiply two terms with the same exponent but different bases, multiply the bases together and keep the exponent:
E.g. (2^3) (3^3) = 6^3.
47
Negative Exponents:
Negative exponents indicate a reciprocal. To get an equivalent expression, take the reciprocal of the base and change the sign of the exponent:
E.g. a^-n = 1/a^n or (1/a)^n
48
Raising a Fraction of an Exponent:
2 ways:
1. Raise numerator and denominator to the exponent separately.
2. multiply the whole fraction by itself the number of times indicated by the exponent.
E.g. (2/3)2 = 2^2/3^2 = 4/9
or (2/3)2 = 2/3 x 2/3 = 4/9
Note: Raising a fraction less than 1 to a positive exponent greater than 1 results in a smaller number. The higher the exponent, the smaller the result.
49
Exponents and negative and positive signs of Base:
When raised to an odd power, positive numbers yield positive results and negative numbers yield negative results.
E.g. 2^3 = 8
-2^3 = -8
50
Radicals, Fractions of Exponents:
A fractional exponent indicates a root.
E.g. (a)^1/n = n√a (read "the nth root of a). Symbol √ is called radical.
E.g. 8^1/3 = 3√8 = 2
E.g. a^1/2 = √a
If no specific "n" is present, the radical sign means a square root.
If the numerator in the fractional exponent is a different integer, raise the resulting term to that exponent.
E.g. 8^2/3 = (3√8)^2 = 2^2 = 4
E.g. a^b/c = (c√a)^b
51
Positive and Negative Square Root:
By convention √ means the positive square root only. Even though there are two different numbers whose square is 9 (3 and -3), when you see √9 on the GMAT, it refers to the positive number 3 only. There is a negative sign in front of the radical if it refers to the negative square root.
E.g. √9 = 3
-√9 = -3
52
Adding and Subtracting Radicals:
Treat radicals in much the same way as you would variables. I.e. only like radicals can be added to or subtracted from one another.
E.g. 2√3 + 4√2 - √2 - 3√3 = (4√2 - √2) + (2√3 - 3√3)
= 3√2 - √3
53
Multiplication and Division of Radicals:
To multiply or divide one radical by another, multiply or divide the numbers outside the radical and those inside.
E.g. (6√3) (2√5) = (6x2)√3x5 = 12√15
E.g. 12√15 / 2√5 = (12/2)√15/5) = 6√3
54
Factoring out Perfect Squares in Radicals:
If the number inside the radical is a perfect square, the expression can be simplified by factoring out the perfect square:
E.g. √72 = √36x2 = √36 x √2 = 6√2
55
No Addition Split Up inside Radicals:
You can't split up addition underneath a radical sign, although GMAT will try to trick you into thinking you can.
```
E.g. √100 = 10
√100 = √36+64
√100 ≠ √36 + √64
√100 ≠ 6 + 8
√100 ≠ 14
```
That means:
√a + √b ≠ √a+b
√a - √b ≠ √a-b
56
Exponents Under Radical Signs:
If exponent is under square root sign divide the exponent by 2:
E.g. √13^4 = 13^2
E.g. √a^2 = IaI
If exponent is under cube root sign divide the exponent by 3:
E.g. 3√7^6 = 7^2
57
Decimals Under Radicals:
If a decimal is under a square root, take the square root of the number and divide the number of decimal places by 2:
E.g. √0.0009 = 0.03
If a decimal is under a cube root, take the cube root of the number and divide the number of decimal places by 3:
E.g. √0.000125 = 0.05
58
Raising Square Roots to Power of 2 and Cube Roots to Power of 3:
If you raise a square root to the power of 2 you are essentially getting rid of the square root.
E.g. (√3)^2 = 3
If you raise a cube root to the power of 3 you are essentially getting rid of the cube root.
E.g. (√3)^3 = 3
59
Multiplying identical Square Roots:
If you multiply two identical square roots you are essentially getting rid of the square root:
(√3) (√3) = 3
60
Absolute Value and Variables:
If IzI = 3 then z could be -3 or 3 because all the equation told you is that z is three points from zero.
Another way to write it:
+z = 3 that means z = 3
-z = 3 that means z = -3
61
Expressions with Absolute Values:
In expressions with absolute values, treat the absolute value bars as parentheses and figure out the value of what's inside before you perform the operation. GMAT will punish you with wrong answer if you perform these operations in wrong order.
Incorrect: I-3I + I5I = I-3 + 5I = I2I = 2
Correct: I-3I + I5I = 3 + 5 = 8
62
Absolute Value Expressions with Variables:
In expressions like this:
Ix-3I = 3
you don't treat it any different from this:
IxI = 3
Think of it like this:
I--chunk--I = 3
That means:
--chunk-- = 3 or --chunk-- = -3
So:
x - 3 = 3 that is x = 6
OR x - 3 = -3 that is x = 0
63
Absolute Value Expressions with Variables and Inequalities:
In absolute value expressions with variables and inequalities it's key to remember that when considering the possible negative value, the inequality sign changes direction:
E.g. if Ix-3I > 3
then x - 3 > 3 OR x - 3 6 OR x
64
Solving Inequalities:
To solve inequalities, use the same methods as used in solving equations with one exception:
If the inequality is multiplied or divided by a negative number, the direction of the inequality is reversed.
E.g. given 4-3x -2 (this is the range, so the answer)
Note that the solution set to an inequality is not a single value but a range of possible values. It helps to sketch a number line to visualize the range of the inequality.
REMEMBER: Don't multiply or divide inequality by variable unless you know the sign because depending on the sign, the inequality sign may need to be flipped.
65
Caution: Multiplying or Dividing by Variables in Inequalities:
Equation: 3b 2b^2
because of the square you don't need to add the negative sign on the right side for when b is negative because negative multiplied by negative will always be positive so it won't change the sign on that side of the equation
Then solve for b for both possibilities to find the range of values for b.
Manhattan GMAT says: Do not multiply or divide an inequality by a variable unless you know the sign of the number that the variable stands for.
66
Three-Part or Compound Inequalities:
In three-part or compound inequalities you perform calculations like on regular inequalities but what you do to one part of the inequality you have to do to ALL parts of the inequality.
67
Changing Direction of Inequalities:
You often have to simplify inequalities in order to evaluate them. Remember that in equalities the signs change if you multiply both sides by a negative number but not if you multiply with a positive number.
E.g. in inequality a/b > c/b:
If b = positive the direction of the inequality remains the same and you can multiply both sides by b to get a > c.
But if b = negative then the direction of the inequality changes when we multiply by b and we get:
a
68
Creating Compound Inequalities:
Sometimes GMTA will give you multiple inequalities and to solve the questions you have to convert the several inequalities to a compound inequality, a series of inequalities strung together.
69
Combining Inequalities:
In Some GMAT questions you have to combine inequalities to get to the solution.
Is a + 2b b
Let's say you already know the statements individually are not sufficient and are looking at them together now. You have to add them together here to be able to answer. First line the inequalities in the statements up so they all face the same direction. Then add. You have to add the second inequality twice to get to 2b and 2d
a
70
Square-Rooting Inequalities:
Just like equations involving even exponents, inequalities involving even exponents require you to consider 2 scenarios.
If x^2 -2.
So, always always consider both scenarios when you have variables with square roots in inequalities.
BUT REMEMBER: You can only take square root of an inequality for which both sides are definitely not negative because you can't take square root of a negative number. So, you know that in x^2
71
Minimizing and Maximizing Values in Equalities:
In Optimization problems o GMAT (ask for maximum possible value of variables) you have to focus on largest and smallest values for each of the variable to get to solution.
E.g. If -7≤a≤6 and -7≤b≤8, what is the maximum possible value for ab?
for a: for b:
max: 6 max: 8
min: -7 min: -7
maximum possible value for ab: (-7)(-7) = 49
72
Inequalities and Absolute Values:
Absolute value can be interpreted as simply distance on the number line. So, for a simple absolute value expression like |x|, you are evaluating distance from 0. You have to look at two possible solutions because it's an absolute value.
E.g. in |x| -5 so write: -5-7
So range of possible values: -7
73
Reciprocals of Inequalities:
General rule: If you don't know the signs of variables you cannot take reciprocals.
If you know the signs, flip the inequality unless if the variables have different signs.
General rule for inequalities for if you know sign: if x 1/y when x and y are both positive. Sign is flipped.
E.g. if 31/5
1/x > 1/y when x and y are both negative. Sign is flipped.
E.g. if -51/-3
1/x
74
Squaring Inequalities:
You cannot square both sides of an inequality unless you know the signs of both sides of the inequality.
Rules:
1. If both sides are known to be negative, flip the inequality sign when you square.
E.g. if x 9
But if you had x>-3 you couldn't square because x could be positive or negative and so x^23, then left side must be positive.
x^2 > 9
But if you had xy^2 x^2
75
Distinct Integers in Number Properties:
- Distinct integers simply means different, no integer appears more than once. But don't automatically assume that integers are distinct unless it's said so. That's often the mistake when we have variables, x and y don't necessarily have to represent different values.
- All integers are either odd or even. 0 is an even integer. When a number is classified as odd or even it must also be an integer. Non-integers are not classified as odd or even.
- All integers ending with the digit 0, 2, 4, 6, 8 are considered "even numbers." All integers ending with the digit 1, 3, 5, 7, 9 are considered "odd numbers."
- Integers can either be positive or negative. Only 0 is neither positive nor negative.
76
Zero in Number Properties:
- 0 is the only number that is neither positive nor negative.
- You have to pay attention as GMAT will test your attention to the right detail regarding the number 0.
- A question that asks about negative numbers does not include 0 as a possibility, but a question that asks for non-positive numbers does include 0.
- There will be a wrong answer in NP questions on GMAT for considering 0 when you shouldn't or not considering 0 when you should have.
77
Six Areas of Number Properties that show up on GMAT:
- Integers and Non-Integers
- Odds and Evens
- Positives and Negatives
- Factors and Multiples
- Remainders and Primes
- Sequences
78
Integers and Non-Integers in Number Properties:
- Never assume that a number is an integers unless GMAT specifically says so. GMAT puts in traps like that. It could be a fraction in those instances.
- Integers are all numbers that are no fractions or decimals. Negative numbers and 0 are also integers.
- When a number is added to, subtracted from, or multiplied by an integer, the result is an integer. But if an integer is divided by an integer, the result could be an integer or a fraction/decimal.
- Picking Numbers can make Number Property questions about integers easier to answer. SO PICK NUMBERS WHEN POSSIBLE OR WHEN EASIEST.
79
Rules for Integers:
- All even numbers can be evenly divided by 2. I.e. if 2z is always an even integer then z has to be an integer.
- If z/3 is an integer then z has to be a multiple of 3 and therefore an integer.
- If z/2 is NOT an integer, then z could either be an integer, e.g. 1 in which case you would have 1/2 which is not an integer, or z could be a non-integer, e.g. 1/2 in which case you would have an non-integer as a result as well.
- If you are asked if e.g. √d is an integer then you can rephrase the question to: Is d a perfect square? Whenever they give you a √ value and say that the result is an integer, it means that whatever is under the √ is a perfect square.
80
Odds and Evens in Number Properties:
- Only integers are labeled odd and even, fractions are not.
- Even numbers are integers that are divisible by 2. And a number needs just a single factor of 2 to be even so the product of an even number and any integer (even or not) is always even. I.e. even x odd = even.
- Odd numbers are integers that are not divisible by 2.
- Both odd and even numbers may be negative.
- 0 is an even number.
81
Exponent Rules for Odds and Evens:
Odd^any positive integer = Odd
| Even^any positive integer = Even
82
Division by Zero:
Division by 0 is undefined. When given an algebraic expression make sure that the denominator is not zero. Fraction 0/0 is also undefined. All fractions with denominator 0 are undefined.
E.g. 4/0 = undefined
-1/0 undefined
83
Reciprocal of Numbers:
The reciprocal of a number is 1 divided by the number. Zero has no reciprocal as 1/0 is undefined.
84
Reciprocal of Numbers between 0 and 1:
The reciprocal of a number between 0 and 1 is greater than the number itself because:
1/2/3 = 1 x 3/2 = 3/2 3/2 > 2/3
85
Reciprocal of a Fraction:
To get the reciprocal of a fraction just switch the numerator and denominator.
E.g. reciprocal of 2/3 is 3/2.
The product of a number and its reciprocal must always equal 1.
E.g. 2/3 x 3/2 = 1
86
Reciprocal of Number between -1 and 0:
The reciprocal of a number between -1 and 0 is less than the number itself.
E.g. reciprocal of -2/3 is 1/-2/3 = -3/2 -3/2
87
Square of Number between 0 and 1:
The square of a number between 0 and 1 is less than the number itself.
E.g. (1/2)^2 = 1/2 x 1/2 = 1/4 1/4
88
Multiplying Numbers by Fractions:
Multiplying any positive number by a fraction between 0 and 1 gives a product smaller than the original numbers.
E.g. 6 x 1/4 = 6/4 = 3/2 3/2 -6
89
Multiplication and Division of Negative Numbers:
The product or quotient of two negative numbers is positive:
E.g. (-2) x (-3) = 6 -6/-3 = 2
90
Multiplication and Division of Numbers with Opposite Signs:
The product or quotient of two numbers with opposite signs is negative.
E.g. (-2) x 3 = 6 -6/3 = -2
Important: It doesn't matter whether the negative sign is in the numerator or denominator.
2/-3 = -2/3 = -(2/3)
91
Re-Expressing Numbers with Exponents:
Remember that if one base is a factor of another base you can easily rewrite the expression:
9^10 is the same as (3x3)^10 = (3^2)^10 = 3^20
92
Proportions Questions on GMAT:
Proportions show up on GMAT in form of fractions, ratios, decimals and percents. It's important that you know how to quickly convert among all these forms of the same value as GMAT often mixes these different formats in one question so you want to bring them all to the same format for easier calculations.
E.g. 45% is:
Decimal: 0.45
Fraction: 45/100
Ratio: 45:100
REMEMBER: Percent simply means "out of 100" (that should help with the conversion)
93
Proportions:
A proportion is a comparison of two ratios. Usually, a proportion consists of an equation in which two ratios (expressed in fractions) are set equal to each other.
E.g. 6/11 = m/33
6 x 33/11 = m
6 x 3 = m
It's usually easier to express proportions as fractions rather than ratios in GMAT questions.
Remember to write proportions the right way. You can write them so that units of the same type are either one above the other, or directly across from one another:
E.g. ratio of T-Shirts and sweaters is 4/5 and you have 12 T-Shirts. How many Sweaters do you have?
2 Ways to express this:
1. 4/5 = 12/T i.e T-Shirts/Sweaters = T-Shirts/Sweaters
OR
2. 4/12 = 5/Sweaters
i.e. T-Shirts/T-Shirts = Sweaters/Sweaters
T = 15
94
Part-to-Part and Part-to-Whole Ratios:
A ratio can compare either a part to another part or a part to a whole. One type of ratio can be converted to the other only if all parts together equal the whole and there is no overlap among the parts, i.e. if the whole is equal to the sum of its parts.
E.g. ratio of domestic to foreign sales of certain product is 3:5. What fraction of total sales are domestic sales?
domestic + foreign sales is 8. So ratio of domestic to total sales 3:8 or fraction 3/8. I.e. for every 8 sales of product 3 are domestic.
Note that here we knew that total sales is 8 because the product is obviously sold domestically and on foreign markets. Don't assume if you are not sure but here it was obvious.
95
Solving Questions With Missing Part of Ratio:
Set up a proportion by setting two equivalent fractions equal to each other. Then use cross-multiplication to solve for the missing piece.
96
Multiplying by -1 to Change Equations:
Important concept for GMAT:
a - b = -(b - a)
a^2 - b^2 = -(b^2 - a^2)
97
Using Combination to Solve Equations:
Use combination to solve two equations with two variables if you see that that makes the calculation easier. For that the variables must have the same coefficient in both equations so that it will cancel out. You can multiply both sides of one equation by a number to get that situation.
E.g. 8a + 8b = 96
8a + 5b = 84
You can write the two equations underneath each other like this and then subtract the first equation from the second to cancel out 8a. Then solve for b.
98
Radicals Extra - Negative Exponents Under Radicals
REMEMBER:
√10^-2 (where -2 is under the radical) is the same as:
10^-1
And √10^-4 (where -4 is under radical) is 10^-2
So you just divide the exponent by 2 when you get rid of the radical like you always would and leave the exponent in the negative form.
99
Radicals Extra: Get Rid of Radicals in Denominator
Whenever you have radicals in denominator on GMAT try to get rid of them. Often the rid answer on GMAT is the same fraction but simplified to get rid of the radicals in denominator. You simply have to multiply nominator and denominator by same expression as the original radicals with opposite signs. That way you get radicals in nominator but get rid of radicals in denominator.
E.g. 1/√8 + √6 = 1x(√8 - √6)/(√8 + √6)(√8 - √6) = (√8 - √6)/(√8x√8) - (√8x√6) + (√6x√8) - (√6x√6) = (√8 - √6)/8-6 = (√8 - √6)/2 = 1/2 x (√8 - √6)
100
Arithmetic Extra:
Remember that if there is one variable in question stem and the same variable in the answer choices you should pick a number for that variable and the calculations accordingly. If there are two variables in the question stem but only one of them appears in the answer choices, then it's best to set up equations with the variables and do the calculations that way to find the expression that matches one of the answer choices. E.g. a question stem mentions variables x and y but in the answer choices only y appears.
101
Factors of x! Type of Numbers:
Numbers of the type x! are divisible by all all positive integers up to and including x.
E.g. 8! is divisible by all integers up to and including 8 because 8! = 8x7x6x5x4x3x2x1 (which are all factors of 8!)
102
Cubes of Positive Integers to Memorize:
```
1^3 = 1
2^3 = 8
3^3 = 27
4^3 = 64
5^3 = 125
6^3 = 216
7^3 = 343
```
103
Square Root of Negatives:
Square Roots of Negatives don't result in real numbers.
E.g. √-3 is not a real number because what squared results in -3? NOTHING!!!
104
Units Digits:
When raising a number to a power, the units digit is influenced only by the units digit of that number. For example 16^2 ends in a 6 because 62 ends in a 6.
105
Units Digits of Consecutive Powers of 7:
Units Digits of Consecutive Powers of 7 follow this pattern:
```
7^1 = 7
7^2 = 9
7^3 = 3
7^4 = 1
```
And then the pattern repeats itself for the consecutive powers of 7.
106
Units Digits in Powers of 2:
Powers of 2:
```
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
2^5 = 32
2^6 = 64
2^7 = 128
2^8 = 256
...
```
So, pattern for units digit is 2, 4, 8, 6 and then it repeats itself.
107
Formula for Sum of Geometric Progression:
A geometric progression or geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
Formula for finding the sum of a geometric series:
Sum = a (1-r^n/1-r)
Where a = first term
r = common ration between terms
n = number of terms
E.g. Sum of 2 + 2^2 + 2^3 ....2^8?
Sum = 2 (1-2^8/1-2) = 2 (1-256/-1) = 2(255) = 510
108
Factors of Perfect Squares:
REMEMBER: For any perfect square the number of factors will always be odd.
E.g. perfect square 36
factors of 36: (36, 1), (9, 4), (6, 6), (12, 3), (2, 18)
You can see there are 9 factors. Reason: One of the pair of factors in perfect square will consist of the same number (in 36 that's 6x6) which is then only counted once and therefore the total number of factors will always be odd.
That also means that any number that's not a perfect square will always have an even number of factors.
109
Prime Factors of Perfect Squares:
A perfect square will always be able to be expressed as the product of an even number of prime factors. That's because a perfect square means taking a number and multiplying it with itself, e.g. 6x6 = 36. That means whatever prime factors are in one part of the perfect square are present in the total perfect square twice, i.e. the number of prime factors gets multiplied by 2 which means that the number has to be even because anything multiplied by 2 will result in an even number.
110
Greatest Common Factor:
The greatest common factor (GCF, or HCF highest common factor) of two integers is the largest integer that divides both of them evenly (i.e. leaving no remainder).
E.g. GCF of 16 and 12 is 4.
111
Least Common Multiple:
The least common multiple (LCM) of two numbers is the smallest number (not zero) that is a multiple of both.
E.g. of the number 10 and 6, 30 is the LCM
10 x 3 = 30
6 x 5 = 30
112
Variables in Square Brackets:
If you see something like this on GMAT:
[x] it means that it is largest integer less than or equal to x.
So [x] = 0
is equivalent to:
0≤x
113
Exponents and Doubling:
Something doubles: x 2
Something quadrouples: x 2^2
Something doubles six times x 2^6 (e.g. doubles every 6 hrs)
114
REMEMBER for Multiplying Number Raised to Power by Decimals:
0.5 x 10^10 = 5 x 10^9
115
Equations With Three Variables:
If you have equations with three variables see if you can first isolate one variable and then put into each of the other two and so that now in those two you only have two variables. Then isolate one variable in one of those two and substitute for the same variable in the other equation. That way you wills solve your first variable. After that just plug that value in one of the two equations with just two variables and solve for the second variable and finally solve for the third variable.
In equations with three variables, like in equations with two variables, use substitution or addition/subtraction of whole equations to solve.
116
Complex Absolute Value Equations:
If you see questions on GMAT with equations that contain two or more variables in more than one absolute value expression, know that the question is most likely testing the concept of positives and negatives.
If there is more than one absolute value expression but only one variable this is how you solve:
E.g. |x-2| = |2x-3|
Test all four cases:
1. both positive:
x-2 = 2x-3 x = 1
2. both negative:
- (x-2) = -(2x-3) x = 1
3. pos/neg:
x-2 = -(2x-3) x = 5/3
4. neg/pos:
- (x-2) = 2x-3 x = 5/3
You see that you only got two different solutions because equations for (1) and (2) and (3) and (4) are the same in this case.
In complex absolute value equations you have to go ahead and test the values you got for x in the end to make sure they are valid:
Test:
For x = 1:
|1-2| = |2x1-3| =
|-1| = |-1|
= 1
For x = 5/3
|5/3-2| = |2(5/3)-3|
|-1/3| = |-1/3|
= 1/3
So both values are valid.
117
GMAT Questions With Integers Constraints:
In some questions all you need to do is find a possible value for a variable, e.g. y, in equations with two variables, e.g. x and y, but the constraint given in the question stem is that both x and y are integers. So, just go through answer choices and plug in different values for y and see for which one x becomes an integer. That value of y is your answer.
118
Units Digits in Consecutive Powers of 3:
```
n^1 = 3
n^2 = 9
n^3 = 7
n^4 = 1
n^5 = 3
n^6 = 9
n^7 = 7
n^8 = 1
n^9 = 3
```
etc.
119
Quadratic Equation Extra:
When you have a quadratic equation that doesn't seem to FOIL right consider factoring out first before using FOIL.
E.g. 2x^2 - 2x - 4 = 0 factor out 2 to be able to FOIL
2 (x^2 - x - 2) = 0
2 ((x-2)(x+1)) = 0
Values for x = 2, -1
