GMAT Quant Formulas

6 machines can do a certain job in 4 hours. How many hours would it take 8 similar machines to complete the same job?

#Machines1 x time1 = #Machines2 x time2 6 x 4 = 8 x T2 T2 = 24/8 = 3 hours

How many multiples of 4 are there between 12 and 96, inclusive? A. 21 B. 22 C. 23 D. 24 E. 25

(96-12)/4 + 1 = 22 The correct answer is B.

What is the formula for counting the number of multiples in a given range?

(Last multiple of x in range - first multiple of x in range)/multiple + 1

What is the area of an equilateral triangle with side a?

(a^2 √3)/4

For evenly-spaced sets, the number of terms in set a, a+d, a+2d, a+3d,....b-a is...

(b-a)/d + 1, where b = last term a = first term d = increment between each consecutive pair of elements

What is the sum of the internal angles of any polygon?

(n-2) x 180 n = # of sides of the polygon Triangle: (3-1) x 180 = 180 Quadrilateral: (4-2) x 180 = 360 Pentagon: (5-2) x 180 = 540

1/8 = ?

.125

A certain fruit stand sold apples for $0.70 each and bananas for $0.50 each. If a customer purchased both apples and bananas from the stand for a total of $6.30, what total number of apples and bananas did the customer purchase? (A) 10 (B) 11 (C) 12 (D) 13 (E) 14

.7a + .5b = 6.3 7a + 5b = 63 5b = 63 - 7a 5b = 7(9 - a) (5)(b) = 7 (9 - a) From the equation above (LHS = RHS), b must be a multiple of 7, so let's pick 7. This makes LHS = 35, and for RHS to equal 35, a must be equal to 4, which would yield 7(9 - 4) = 7(5) = 35. Therefore, a = 4, b = 7 , and a+b = 11. (Google this question and include the ans choices)

Only two factorials are odd, what are they?

0! = 1 1! = 1 All other factorials are even

7/8 =

0.875 or 87.5%

For any pair of consecutive integers, the GCF = ?

1

The GCF of 1 and any integer n is...?

1

What is the greatest common factor between two primes?

1

If n is an integer, is n divisible by 6? 1) 5n/6 is an integer 2) 21n/6 is an integer

1) 5n/6 can be rewritten as (n/6)*5, and since n/6 must be an integer, then stmt 1 is suff. 2) 21n/6 reduces to 7n/2, and since n/2 tells us nothing about the divisibility of 6 in n, stmt is NS

What are the patterns of mixture problems?

1. Simple addition of substances - find resulting concentration 2. Some water evaporated - what is the resulting concentration of salt as a percentage of what's left? 3. A certain fraction of x% salt solution is replaced with y% salt solution - what fraction of the original solution was replaced?

What are properties of perfect squares for the GMAT?

1. The number of factors must be odd 2. The number of DISTINCT prime factors of a perfect square will always appear in pairs, so there must be an even number of them

Tell me about the product of three consecutive integers

1. The product is always divisible by 6 2. Can be expressed as (n-1)n(n+1) = n^3 - n

What is the difference between "x is 200% greater than y" and "x is 200% of y"?

1. x = y+ [(200/100) * y] = y + 2y = 3y 2. x = (200/100) * y = 2y

How are 1.25 and 0.8 related, and why is that interesting?

1.25 = 5/4, and 0.8 = 4/5. When they're multiplied the result is 1. 1.25 x 0.8 = 1

For the GMAT, what is Sqrt(3)?

1.7

Pump A can fill a tank in 4 hours. Pump B can fill a tank in 5 hours. How long would it take A & B to fill 3/4 of the tank?

1/4 + 1/5 = 1/w 9/20 = 1/w w = 20/9 (to fill an entire tank) To fill just 3/4 of the tank: 3/4 * 20/9 = 5/3 of an hour

12.5% = .125 = ? in fraction form

1/8

What is the remainder when 100x + 10y + z is divided by 7?

100x + 10y + z 98x + 2x + 7y + 3y + z 7(14x + y) + 2x + 3y + z Since 7(14x + y) is a multiple of 7, we need to know what x, y, and z are in order to answer the question.

What is the fraction equivalent of pi?

22/7 = 3.14

What is 216?

2^3 x 3^3 8 x 27

Machine A, working alone at a constant rate, can complete a certain production lot in x hours. Machine B, working alone at a constant rate, can complete 1/5 of the same production lot in y hours. Machines A and B, working together, can complete 1/2 of the same production lot in z hours. What is the value of y in terms of x and z?

2xz / (5x - 10z) The key to these work/rate problems is to get to the rate as quickly as possible...

What are the most commonly-occurring right triangles on the GMAT?

3-4-5 5-12-13 8-15-17 7-24-25

.375 in fraction form =

3/8

18^2 =

324

What are some special cyclicities that remain the same, no matter what the number is raised to?

5^n = _____5 for all values of n 6^n = _____6 for all values of n

All prime numbers above 3 are of the form...

6n+1 or 6n-1

83.33% =

83 1/3 = 250/3

250 / 3 =

83.33

If 945s is a perfect square then s can be which of the following numbers? 21 35 54 105 150

945 = 5x189 3x63 3x21 3x7 7x1 To complete the Perfect Square, we need 3, 7, and 5 3x5x7 = 105 - D

If 2 < x < 4, what is the median of the numbers 0, 5, x, 1, 7, and 3? (1) 2x−5=0 (2) 2x2−7x+5=0

Answer is D. The key takeaway from this question is to notice EVERY detail provided in the question stem. My original answer here was A, thinking that the quadratic would be insufficient because of two different answers. However, notice that we are told that x is between 2 and 4. Boom.

If n is not equal to 0, is |n| < 4 ? (1) n^2 > 16 (2) 1/|n| > n

Answer: A

Susie can buy apples from two stores: a supermarket that sells apples only in bundles of 4, and a convenience store that sells single, unbundled apples. If Susie wants to ensure that the total number of apples she buys is a multiple of 5, what is the minimum number of apples she must buy from the convenience store? a. 0 b. 1 c. 2 d. 3 e. 4

Answer: A

If a and b are positive integers such that a/b = 2.86, which of the following must be a divisor of a? a. 10 b. 13 c. 18 d. 26 e. 50

Answer: B a/b=2.86=286/100=143/50 --> b=50a/143=50a/(11∗13) For b to be an integer a must have all the factors of 143 (50 has none of them). Hence a must be divisible by both 11 and 13.

What is the formula for average speed?

Avg Speed = Total distance / Total time

Given that an initial investment of P was made at r% simple annual interest, over t years. What is the balance after t years?

Balance = P(1+ rt/100)

7 friends decide to share the rent an apartment that costs $C, but 2 people backed out. The five remaining friends now have to pay $24 more. What is the original cost of the apartment?

C/5 - C/7 = 24 2C/35 = 24 2C = 35x24 C = (35x24)/2 C = 35x12 C = 420

Express the following in an equation: There are twice as many cows as there are pigs on the farm.

C/P = 2 ... or C = 2P

If z is an integer, is z even? (1) z/2 is even. (2) 3z is even.

D (2) SUFFICIENT: If 3z = even, then z = even/3. Since any even number divided by an odd number produces either an even number or a non-integer, and since z must be an integer, z must be even.

How do you find the point of intersection of two lines?

Express both in the slope-intercept form, y = mx + b, then set both expressions equal to each other, and solve for x. Find 'y' by plugging back into one of the equations. For example, Lines y=−2/3x+2 and y=x+2 intersect at point (0,2)

What is the first thing you do when you see a number line and need to express it with an inequality?

FIRST Find the midpoint, then do |x - {midpoint value}| <= {units from the midpoint} review Magoosh videos

How many integers from 1 to 200, inclusive, are divisible by 3 but not divisible by 7? (A) 38 (B) 57 (C) 58 (D) 60 (E) 66

Find number of integers divisible by 3

Given than n = 20! What is the highest integer k for which 2^k is a factor of n?

Finding the highest powers of a prime number k, in the n! The formula is: n/k + n/k^2 + n/k^3 ... till n>k^x k=20/2 + 20/4 + 20/8 + 20/16 = 10+5+2+1 = 18 (Or you could just list out the multiples of 2 to find out how many twos are in there...)

What is the rule connecting Factors, Multiples, and Multiplication?

If B is a factor of A, and C is any integer, then B is a factor of AxC. 4 is a factor of 12, therefore 4 is also a factor of 24 (12x2), 36 (12x3), 48 (12x4), etc

What is the rule connecting Factors, Multiples, and Addition/Subtraction?

If M is a factor of A, and M is also a factor of B, then M is a factor of A+B, or A-B

What is the relationship between range and standard deviation?

In general, a value that drastically increases the range of a set will also have a large impact on the standard deviation

Express the following mathematically: There are 10 more men than women

M - W = 10 ...or m = w+ 10

Machine A can produce 1200 bolts in 20 hours. Machine B can produce 1200 bolts in 30 hours. Working at their individual rates, how long would it take Machine A and Machine B to produce 1200 bolts?

Machine A 1200 bolts/ 20 hours = 60 bolts/hr Machine B 1200 bolts/ 30 hours = 40 bolts/hr Combined rate = 60+40 = 100 bolts/hr Rate x Time = Work Time = Work / Rate = 1200/100 = 12 hrs

Solution A contains 18 liters, and 12% alcohol. Solution B contains 6 liters, and 24% alcohol. If they are combined, what is the concentration of alcohol in the result?

Mixtures (alligation) Aa + Bb = Cc 18(.12) + 6(.24) = 24(x) Solve for x

What is the smallest possible common multiple of two integers which are both greater than 250? A. 251 B. 252 C. 502 D. 750 E. 884

Notice that we are not told that the two integers must be distinct. So, if they both equal to 251 then the LCM of 251 and 251 is 251 itself. Answer: A.

A store owner purchased an item for $90 and then set the selling price equal to the purchase price plus a markup that was 40% of the selling price. What was the selling price of the item?

P = S - C Let the selling price, S = $90 + 0.4S S - .4S = 90 .6S = 90 S = 90 /.6 S = 90 * 10/6 S = 900/6 S = 150

In United States currency, a nickel is worth 5 cents, a penny is worth 1 cent, and a dime is worth 10 cents. 100 cents equals one dollar. If a hand purse contains 6 nickels, 5 pennies and 4 dimes, what is the probability of picking a coin other than a nickel twice in a row if the first coin picked is not put back?

Probability of a non-nickel on first pick = (5 pennies + 4 dimes) / 15 coins = 3/5 Probability of a non-nickel on second pick = (8 non-nickel coins) / 14 coins = 4/7 Notice that for the second pick both the non-nickel pool and the total coin pool diminished by one coin after a non-nickel was selected on the first pick. Total probability = 3/5 x 4/7 = 12/35. The correct answer is B.

What is the formula for gross profit?

Profit = Selling price - Cost price P = S - C

A box contains 18 red socks, 16 green socks, and 12 blue socks. What is the least number of socks that must be selected to guarantee that the selection contains at least 5 pairs of socks? (A pair of socks is two socks of the same color) A) 11 B) 12 C) 13 D) 15 E) 17

RGB RGB RGB RG R

Jane has a box full of rings, each of which is red, green, blue, or yellow. She takes out one ring at a time without looking. How many rings must Jane remove from the box to be certain that there will be five rings of the same color? (There are at least five rings of each color) A) 6 B) 13 C) 16 D) 17 E) 20

RGBY 4444 The next one HAS to be one of the 4 colors, so the total # is 17 - D

One of the most fundamental concepts in Work problems is that...

Rates are additive, but time is not. In all the work/machine/pump problems, you're always adding the individual rates. You don't add time or any other property.

When x is divided by 5, the remainder is 3. When y is divided by 5, the remainder is 4. What is the remainder when xy is divided by 5?

Shortcut: Multiply both remainders and divide by 5: 4x3 = 12 12 / 5 = 2 rem. 2

Length of sides of a RIGHT isosceles triangle?

Sides x : x : x√2 Angles 45 : 45 : 90 degrees

How do you combine ratios? For example: H:F:R 2:7:-- --3:5

Since F has the common terms, multiply the first row by 3, and the second row by 7: H:F:R 2:7:-- (x3) --3:5 (x7) H:F:R 6:21:-- --:21:35 So the final ratio becomes: H:F:R 6:21:35

In a class of 100 students, 40 students are taking French and 65 students are taking German. What is the greatest number of students that could be taking both French and German? In a class of 100 students, 40 students are taking French and 65 students are taking German. What is the least number of students that could be taking both French and German?

Start with (from a Venn diagram): 100 = 40 - x + x + 65 - x + y

What is the greatest value of m such that 4^m is a factor of 30! ? (A) 13 (B) 12 (C) 11 (D) 7 (E) 6

The easiest way to do this is the following: Divide 30 by 2. You get 15 Divide 15 by 2. You get 7. (Ignore remainder) Divide 7 by 2. You get 3. Divide 3 by 2. You get 1. Add 15+7+3+1 = 26 Since greatest power of 2 in 30! is 26, greatest power of 4 in 30! will be 13. NOTE: If the question says 6^m, then find the number of 3s and that will be your answer (because to make a 6, you need a 3 and a 2. You have definitely more 2s in 10! than 3s. So number of 3s is your limiting condition.)

How do you solve an infinite radical problem? For example: If x>0 x=sqrt(6+sqrt(6+sqrt(6+...), what is x?

The first thing to do is to express the number, 6 in this case, in terms of two consecutive integers, and take the higher number as the solution. In this case, it would be 3, since 2x3 = 6. Otherwise, Let x = √(6+x), then x^2 = 6+x X^2 - x - 6 = 0 (x+2)(x-3) = 0 x = -2, or 3. Since we only consider the positive root, we take 3 as the answer.

Which of the following fractions has a decimal equivalent that is a terminating decimal? A. 10/189 B. 15/196 C. 16/225 D. 25/144 E. 39/128

The fraction will terminate if and only if the denominator has prime divisors only 2 and 5 or both. A. 10/189 = 10/(3x3x3x7) Non-term. B. 15/196 = 15/(2x2x7x7) Non-term. C. 16/225 = 16/(5x5x3x3) Non-term. D. 25/144 = 25/(2x2x2x7) Non-term. E. 39/128 = 39/(2^7) Terminating because only 2 in denominator.

How many zeros are in 100! ?

The number of trailing zeros in the decimal representation of n!, the factorial of a non-negative integer n, can be determined with this formula: n/5 + n/5^2 + n/5^3+...+n/5^k, where k must be chosen such that 5k≤n For example: How many zeros are in 32!? 32/5 + 32/5^2 = 32/5 + 32/25 = 6 + 1 = 7 How many zeros does 100! end with? 100/5 + 100/5^2 = 20 + 4 = 24

List X contains 8 integers. Is the standard deviation of the integers in list X equal to zero? (1) The range of the integers in list X is 3. (2) The average (arithmetic mean) of the integers in list X is 5.

The only way the Std Dev can equal zero is if the numbers in the set are all the same. Stmt 1 says the range is 3, which means the numbers in the set are NOT the same, hence SUFF. (Google the explanation of why stmt 2 is NS).

What is the remainder when a number is divided by 10?

The same as the units digit. Example: Remainder when 3276 is divided by 10 is 6

What does it mean when the difference between any two elements of a set is equal? Give an example

The standard deviation is zero (0)

How do you quickly find the sum of elements of a set?

The sum of a set of numbers is equal to the average times the number of terms. E.g.: The sum of numbers from 1-10, inclusive, is 10*((10+1)/2) = 55

What is the relationship between the sides of a triangle?

The third side of a triangle must be less than the sum of the other two sides and greater than their difference: |y - z| < x < y + z

Average mileage (mpg) = ?

Total Distance Traveled (miles) / Total gasoline consumed (gallons) Total gasoline consumed = Distance (miles)/mileage (miles/gal)

What does it mean when the mean and median are the same in a given set?

When you have a set of consecutive numbers (integers, evens, odds, multiples), the mean is equal to the median. To locate the median, find the average of the endpoints The numbers in the set are equally spaced, (e.g. consecutive numbers). E.g. The median of numbers from 1-10, inclusive, is 5, and so is the mean.

What is the first thing you do when you see a Units digit problem?

Write down the cyclicity series to see what the repeating pattern is for the unit's digit. For example, 33^43 = ? 33^1 = ___3 33^2 = ___9 33^3 = ___7 33^4 = ___1 33^5 = ___3 <---- starts repeating at this point, so the cyclicity is {3,9,7,1} Next, divide the exponent by 4 (# of times the pattern repeats in the cycle), and count the remainder to get the units digit

In rate problems, can you add rates?

Yes. (rate of machine A) + (rate of machine B) = (rate together)

An apartment that cost $C to rent was originally supposed to be rented by n number of people. x people dropped out. What is the change in the per person cost?

[C/(n-x)] - C/n = ?? C[1/(n-x) - 1/n]

For any two numbers a and b, what is the relationship between their LCM and GCF?

a x b = LCM x GCF

If n is a positive integer and n^2 is divisible by 72, then the largest positive integer that must divide n is? A. 6 B. 12 C. 24 D. 36 E. 48

n^2 = 72 * k = 2^3 * 3^2 * k Notice that for n^2 to be a perfect square, we need one more 2, so that n^2 = 2^4 * 3^2 * k^2 Therefore, n has to be of the form n= 2^2 * 3 = 4*3 = 12

The sum of squares of n consecutive integers

n^3/3 + n^2/2+n6 (Where did I get this from? Might be irrelevant at this point...)

In Number Properties DS questions, what do you do with differences and ratios between variables? For example: Is x/y > 1?

x -y > 0 x > y x/y > 1

What is the length of the sides of a 30-60-90 triangle?

x : x√3 : 2x 30 : 60 : 90

In the equation of a line, what do the different components mean?

y = mx + b m = slope b = y intercept, the point where the line crosses the y-axis (where x = 0) (x,y) is one of the solutions of the line