GMAT Focus prep

How do you make a mixed number as a fraction? Example: 7 2/3

Multiply the integer part of the mixed number by the denominator of the fractional part. (7x3) = 21 Add this product to the numerator. 21 + 2 = 23 Then put the sum over the denominator. 23/3

In the equation y=mx+b, what does "b" represent? What is m? how do you find the slope?

b = y-intercept m = slope Subtract one point's y-coordinate from that of the others. Then subtract the former point's x-coordinate from the latter's m = y2-y1/x2-x1

Saying that the number of pink houses in a city is 0.5% of the number of blue houses means....

the city has 0.5 of a pink house for every 100 blue houses. Since 0.5:100 = 1:200, this is the same as saying the city has 1 pink house for every 200 blue houses.

Saying that the number of blue houses in a city is 150% of the number of red houses means....

the city has 150 blue houses for every 100 red houses. Since 150:100 = 3:2, this is the same as saying the city has 3 blue houses for every 2 red houses.

How do you find the range of a set of numbers?

the greatest value minus the least value

Factor to find the solutions of the equation x^3 − 2x^2 + x = −5(x − 1)^2:

x^3 − 2x^2 + x + 5(x − 1)^2 = 0 x(x^2 − 2x + 1) + 5(x − 1)^2 = 0 x(x − 1)^2 + 5(x − 1)^2 = 0 (x + 5)(x − 1)^2 = 0 x + 5 = 0 or x − 1 = 0 x = −5 or x = 1 so x = -5 or x = 1

Translating Words into Math Operations: x to the power of y x to the yth power

x^y

If x ≠ 0, then ​x^y-z = ____

x^y / x^z

If z ≠ 0, then ​(x/z)^y = ___

x^y / z^y

Translating Words into Math Operations: x multiplied by y the product of x and y x times y

xy

xy+z = ____

xyxz

To add 17.6512 and 653.27, insert ___ to the right of the last digit in 653.27 to line up the decimal points when the numbers are in a column.

zeros 17.6512 + 653.2700 = 670.9212

Suppose a house's price in 2018 was 300% of its price in 2003. By what percent did the price increase?

If n is the price in 2003, the percent increase is [(3n-n)/n] = [2n/n] = 2 = 200%

P(E) = ___ /____

# of outcomes in E / total # of possible outcomes

If z ≠ 0, then ​​x^y/z = _____ = ____

(x^y)^1/z = (x^1/z)^y

Convert 0.321 into a sum and then a fraction

0.321 3 is in the tenths place 3/10 2 is in the hundredths place 2/100 1 is in the thousandths place 1/1000 write as a sum: 3/10 + 2/100 + 1/1000 = 321/1000

F + 5 is a linear polynomial in F, since the highest power of F is ___

1

x^0 = ___

1

How do you find the standard deviation?

1. find the mean 2. subtract the mean from each number 3. square each difference 4. find the average of the squared differences 5. take the non-negative square root of this average

A work problem usually says how fast certain individuals work and asks you to find how fast they work together, or vice versa. example: Suppose one machine takes 4 hours to make 1,000 bolts, and a second machine takes 5 hours to make 1,000 bolts. How many hours do both machines working at the same time take to make 1,000 bolts?

1/4 + 1/5 = 1/h h = how long that much work takes both individuals working at the same time the denominators both go into to 20 so.. 5/20 + 4/20 = 9/20 = 1/h 9h = 20 h = 20/9 = 2 2/9 Working together, the two machines can make 1,000 bolts in 2 2/9 hours

If x ≠ 0, then ​​x^-1 = ___

1/x

19x^2 − 6x + 3 is a quadratic polynomial in x, since the highest power of x is ___

2

Of the 3,600 employees of Company X, ​​ are clerical. If the clerical staff were to be reduced by ​​ what percent of the total number of the remaining employees would then be clerical? 25% 22.2% 20% 12.5% 11.1%

25% Clerical staff = ​​3600(1/3)=1200 Clerical staff lost = ​​ 1200(1/3)=400 Remaining clerical staff = ​​ 1200-400 = 800 Remaining company employees = ​​ 3600 - 400 = 3200 Percent of remaining employees who are clerical staff = ​ 800/3200 = 1/4 = 25%

The ratio of 2 to 3 may be written as...

2:3 2/3 2 to 3

When 28 is divided by 8, the quotient is ___ and the remainder is ___ because?

3 4 28 = (8)(3) + 4.

Let's solve these two equations for x and y: (1) 3x + 2y = 11 (2) x − y = 2

3(2 +y) + 2y = 11 6 + 3y + 2y = 11 6 + 5y = 11 5y = 5 y = 1 since y = 1, we find x - 1 = 2, so x = 2 + 1 = 3

One way to solve for an unknown in a proportion is to cross multiply, then solve the resulting equation. To solve for n in the proportion ​​, cross multiply to get....

3n = 24, then divide both sides by 3 to find n = 8

Multiply each term of one expression by each term of the other: (3x − 4)(9y + x) =

3x(9y + x) − 4(9y + x) 3x(9y) + 3x(x) − 4(9y) − 4(x) 27xy + 3x^2 − 36y − 4x

To solve the inequality 3x − 2 > 5 for x, isolate x:

3x-2 > 5 3x > 7 x > 7/3

Divide out any common factors: (3xy - 9y) / (x-3) =

3y(x-3) / (x-3) = 3y(1) = 3y

The ratio of the number of months with exactly 30 days to the number of months with exactly 31 days is...

4:7, NOT 7:4

To solve the inequality ​​5x-1/-2 < 3 isolate x

5x-1 > -6 5x > -5 x > -1

To divide 698.12 by 12.4, first move the decimal points in both the divisor 12.4 and the dividend 698.12 one place to the right to make the divisor an integer. That is, replace 698.12 / 12.4 with ____

6981.2 / 124

The mixed number 7 2/3 = ___

7 + 2/3

How do you add or subtract two fractions with different denominators?

Express them as fractions with the same denominator

How do you find the percent increase/decrease from one quantity to another? Example: A price increases from $24 to $30

Find the amount of increase/decrease. 30 - 24 = $6 Divide the increase/decrease amount by the original quantity. 6/24 = 1/4 = 0.25 Write this quotient as a percent. 25%

If 5 shirts cost a total of $44, then what is the total cost of 8 shirts at the same cost per shirt?

If c is the cost of the 8 shirts, then 5/44 = 8/c​ Cross multiplying gives ​​5c = 8 x 44 = 352, so ​​c = 352/5 = 70.4. Thus, the 8 shirts cost a total of $70.40.

A price is discounted 20%. Then this reduced price is discounted another 30%. These two discounts together make an overall discount of what percent?

If p is the original price, then 0.8p is the price after the first discount. The price after the second discount is 0.7p (0.7)(0.8) p = 0.56p. The overall discount is 100% − 56% = 44%.

How do you add or subtract 2 fractions with the same denominator?

Just add/subtract the numerators, leaving the denominators the same

How do you multiply fractions?

Multiply numerators and multiply denominators. Simplify if you can.

The probability of A occurring if B occurs is written as _____

P(A\B)

The probability of an event E is written as ____ and is a number between 0 and 1, inclusive.

P(E)

Events E and F are mutually exclusive if no outcomes are in E ∩ F. Then the event "E and F" is impossible: P(E and F) = 0. The special addition rule for the probability of two mutually exclusive events is P(E or F) = _____.

P(E) + P(F)

Two events A and B are independent if neither changes the other's probability. The multiplication rule for independent events E and F is P(Eand F) = ____

P(E) P(F).

Dividing a positive integer y by a positive integer x, and then rounding down to the nearest nonnegative integer, gives the ____ of the division.

quotient

If $8,000 is invested at 6% simple annual interest, how much interest is earned after 3 months?

Since the annual interest rate is 6%, the interest for 1 year is (0.06)($8,000) = $480. A year has 12 months, so the interest earned in 3 months is (3/12) x ($480) = $120

Consider the two equations 3x + 4y = 17 and 6x + 8y = 35. 3x + 4y = 17 implies 6x + 8y = 34, what does this mean compared to the second equation? can the values of x and y satisfy both equations at once?

This contradicts the second equation So, no values of x and y can satisfy both equations at once.

when ​​, each possible choice of k objects out of n objects is a ____ of n objects taken k at a time. The number of these combinations is written as (n k) ​​. This is also the number of k-element subsets of a set with n elements, since the combinations simply are these subsets. It can be calculated as (n k) = n!/k!(n-k)!

combination

The reciprocal of a fraction n/d is ___

d/n

A ____ is an equation between ratios

proportion

What is the mode of numbers?

the number/s that occur/s most often

When a smaller integer is divided by a larger integer, the quotient is 0 and the remainder is the smaller integer. When 5 is divided by 7, what is the quotient and remainder?

the quotient is 0 the remainder is 5 since 5 = (7)(0) + 5.

What is a quotient?

the result of division

An object's average travel speed = ____ / ____

total travel distance / total travel time

If y = 3 what happens to xy? what words?

triple x

The ____ of two sets A and B is the set of all elements that are each in A or in B or both. The union is written as A ∪ B.

union Example: ​​{3,4} ∪ {4,5,6} = {3,4,5,6}

If a quadratic expression isn't easily factored, we can still find its roots with the quadratic formula: If ax2 + bx + c = 0 and a ≠ 0, the roots are

x = -b +/- squareroot b^2-4ac / 2a

An ______ is any number in the set {. . . −3, −2, −1, 0, 1, 2, 3, . . .}. For any ____ n, the numbers in the set {n, n + 1, n + 2, n + 3,...} are consecutive.

integer

The _____ of two sets A and B is the set of all elements that are each in both A and B. The intersection is written as A ∩ B.

intersection Example: {3,4} ∩ {4,5,6} = {4}

How do you divide fractions?

invert the second fraction (reciprocal) and multiply

If x and y are both positive, then xy is ____. If x and y are both negative, then xy is ____. If x is positive and y is negative, then xy is ____. If xy = 0, then x = ___ or y = ___, or ___.

positive positive negative then x = 0 or y = 0, or both.

The distance an object travels = ____ x ____

rate x time

How do you find the average or mean of a set of numbers?

sum of the numbers divided by how many numbers are in that set

If y = 2 or y = 3 what happens to x^y? what words?

x squared x cube

Translating Words into Math Operations: x added to y x increased by y x more than y x plus y the sum of x and y the total of x and y

x+y

A sequence is an algebraic function whose domain includes only ____ integers.

positive

If x and y are both positive, then x + y is _____. If x and y are both negative, then x + y is _____.

positive negative

Translating Words into Math Operations: x decreased by y difference of x and y y fewer than x y less than x x minus y x reduced by y y subtracted from x

x-y

Translating Words into Math Operations: x divided by y x over y the quotient of x and y the ratio of x to y

x/y

If (2^x)(2^y) = 8 and (9^x)(3^y) = 81, then (x, y) = (1, 2) (2, 1) (1, 1) (2, 2) (1, 3)

(2, 1) (2^x)(2^y) = 8 2^x+y = 2^3 x+y = 3 y= 3-x (9^x)(3^y)=81 (3^2x)(3^y) = 81 (3^2x)(3^y) = 3^4 3^2x+y = 3^4 2x+y = 4 Substitute 3 - x for y in this equation, and solve for x 2x + 3 - x = 4 x + 3 = 4 x = 1 Substitute 1 for x in the first equation, and solve for y y = 3 - 1 y = 2

If ​​x/y = 2/3​​, then ​​ = x-y/x = ____ -1/2 -1/3 1/3 1/2 5/2

-1/2 x-y/x = x/x - y/x = 1 - y/x if x/y = 2/3, then y/x = 3/2 thus, x-y/x = 1 - 3/2 = -1/2

Check whether 8/36 and 14/63 are equivalent. What's the greatest common divisor?

8/36 gcd is 4 which reduces the fraction to 2/9 14/63 gcd is 7 which reduces the fraction to 2/9 Both reduced are identical = they are equivalent

what is an integer?

A WHOLE number (not a fraction or decimal!) that can be positive or negative

What is the quadratic equation?

A equation in the form Ax^2+Bx+C=0

what is a polynomial?

A monomial or the sum of monomials

What is a prime number?

A number divisible only by 1 and itself. e.g. 7 is a prime number as it's factors are 1 and 7. 9 is not a prime number as it's factors are 1, 3 and 9.

If a set of n objects is being ordered from 1st to nth, there are n choices for the 1st object, n − 1 choices left for the 2nd object, n − 2 choices left for the 3rd object, and so on, until only 1 choice is left for the nth object. So, by the multiplication principle, a set of n objects has n(n − 1)(n − 2) . . . (3)(2)(1) = n! permutations. Example: The set of letters A, B, and C has 3! = 6 permutations:

ABC, ACB, BAC, BCA, CAB, and CBA

Add 3/5 and 4/7

Both denominators go into 35 you get 21/35 + 20/35 = 41/35

Another way to remove one unknown and solve for x and y is to make the coefficients of one unknown the same in both equations (ignoring the sign). (1) 6x + 5y = 29 (2) 4x − 3y = −6

Multiply equation (1) by 3 and equation (2) by 5 to get 18x + 15 y = 87 20x − 15 y = −30 Add the two equations to remove y. This gives us 38x = 57, or x = 3/2 Substituting 3/2​​ for x in either original equation gives y = 4.

How do you find the median for a set of numbers?

Order the numbers from least to greatest. go to the middle number or if its b/w two numbers you add them and divide by 2

Each of 25 students is taking history, mathematics, or both. If 20 of them are taking history and 18 of them are taking mathematics, how many of them are taking both history and mathematics?

Separate the 25 students into three disjoint sets: the students taking history only, those taking mathematics only, and those taking both history and mathematics. This gives us the Venn diagram below, where n is the number of students taking both courses, 20 − n is the number taking history only, and 18 − n is the number taking mathematics only.

When choosing 1 object apiece out of 3 sets with x, y, and z elements, respectively, xyz different choices are possible. The general multiplication principle also means that when choosing 1 object apiece out of n different sets of exactly m objects apiece, m^n different choices are possible. Example: Each time a coin is flipped, the 2 possible results are heads and tails. In a set of 8 consecutive coin flips, think of each flip as a set of those 2 possible results.

The 8 flips give us 8 of these 2-element sets. So, the set of 8 flips has a total of 2^8 possible results.

In mixture problems, substances with different properties are mixed, and you must find the mixture's properties. If 6 kilograms of nuts that cost $1.20 per kilogram are mixed with 2 kilograms of nuts that cost $1.60 per kilogram, how much does the mixture cost per kilogram?

The 8 kilograms of nuts cost a total of 6($1.20) + 2($1.60) = $10.40. So, the cost per kilogram is ​​$10.40/8 = $1.30

A customer paid $24 for a dress. If the customer got a 25% discount off the original price of the dress, what was the original price before the discount?

The discounted price is (100 − 25 = 75)% of the original price. So, if p is the original price, 0.75p = $24 is the discounted price. Thus, p = ($24 / 0.75) = $32 is the original price before the discount

A certain appliance costs a merchant $30. At what price should the merchant sell the appliance to make a gross profit of 50% of the appliance's cost?

The merchant should sell the appliance for a price s such that s − 30 = (0.5)(30). So, s = $30 + $15 = $45.

Two sets sharing no elements are ___ or mutually exclusive.

disjoint

How do you check whether fractions are equivalent?

divide each fraction's numerator and denominator by the greatest common divisor

If x and y are integers and x ≠ 0, then x is a ____ or ____ of y if ​​y = xn for some integer Then y is divisible by x and is called a ____ of x.

divisor factor multiple

Since 32 divided by 8 has a remainder of 0, 32 is divisible by 8. So 8 is a ____ of 32, and 32 is a ____ of 8.

divisor or factor multiple

If y = 2 what happens to xy? what words?

double x twice x

How many times a value occurs in a data set is its ______ in the set.

frequency

If y = 2 what happens to x/y? what words?

half of x x halved

​​3x^2 / (2x-5) is not a polynomial, because...

it's not a sum of powers of x multiplied by coefficients.

How do you find a certain percent of a number? Example: 20% of 90

multiply the number by the percent expressed as a fraction or decimal 90 (20/100) = 90 (1/5) = 90/5 = 18

To find the remainder of the division, what do you do?

multiply x by the quotient, then subtract the result from y.

For any integer n > 1, n factorial is written as ___ and is the product of all the integers from 1 through n.

n! Example: 2! = 2 x 1 = 2 3! = 3 x 2 x 1 = 6 4! = 4 x 3 x 2 x 1 = 24

what is a coefficient?

number in front of a variable

Given two events E and F in an experiment, the probability that E doesn't occur is P(not E) = 1 − P(E). The probability that "E or F" occurs is P(E or F) = P(E) + P(F) − P(E and F). This is based on the general addition rule for two sets. Example: A 6-sided die rolled once, let E be the event {1, 3, 5} that the outcome is an odd number. Let F be the event {2, 3, 5} that the outcome is a prime number. Then P(E and F) = P(E ∩ F) = P({3,5}) = ___

|{3,5}| / 6 = 2 / 6 = 1 / 3 How? there are 2 diff outcomes in E and there are 6 total possible outcomes P(E) = # of outcomes in E / total # of possible outcomes