GMAT Math Card 3

A bowl is half full of water. Four cups of water are added, filling the bowl to 70% capacity. How many cups are now in the bowl?

.5x + 4 = 7/10x 2/10x = 4 x = 20 If x is half full +4, then the bowl is currently 14 cups full

Picking smart numbers

0. Recognize you can use smart numbers when the problem doesn't give actual values (fractions are fine) 1. Do not pick 0 or 1 2. For percent problems, pick 100 3. For fraction problems, choose common denominator 3. Do not pick numbers that appear elsewhere in the problem 4. Choose different numbers, even and odd ideally unless too messy

A company sells only apple and cherry pie. What fraction of the total sales was apple? 1. The total sold was 460 last month 2. The company sold 30% more cherry pies than apple

1 is not sufficient. For 2 tho: 1c = 1.3a a/a+c = a/a+1.3a = 1/2.3a or 10/23 of the total was apple

Estimation problems

1. When a problem uses the word approximately, this a clear indicator to estimate 2. When the answer choices are pretty far apart, that's another good time estimate 3. If you round down with one calculation, round up for the next one. 4. If you have to guess on a hard problem, estimate to see if you can eliminate some answers

12.5%

1/8

Circular gears P and Q start rotating at the same time at constant speeds. Gear P makes 10 revolutions per minute, and gear Q makes 40 revolutions per minute. How many seconds after the gears start rotating will gear Q have made exactly 6 more revolutions than gear P ?

10 per minute = 1/6 per second 40 per minute = 2/3 per second after 6 seconds, Gear P does 1 and gear Q does 4 After 12 seconds, Gear P does 2 and gear does 8

Carina has 100 ounces of coffee divided into 5 and 10 ounce bags. If has 2 more 5 ounce bags than 10 ounce bags, how many 10 ounce bags does she have?

100 = 5a + 10b a = b + 2 100 = 5(b+2) + 10b 90 = 15b b = 6

A bank account with $100 earns 8% annual interest, is compounded quarterly. How much money is there after 6 months?

100(1 + 8/4)^0.5*4 100(1.02)^2 First quarter = 102 Second quarter = 102*(1.02) = 2.04 Total = 104.04

If the LCM of a and 12 is 36, what are the possible values of a?

12 = 3*2*2 and 36 = 3*3*2*2, therefore a can must have 2 3's and at most 2 2's. 9,18, 36

5!

120

A circus earned $150K in ticket revenue by selling 1800 VIP and standard tickets. They sold 25% more standard than VIP. If the revenue from standard tickets represents 1/3 of the total revenue, what is the price of a VIP ticket?

1800 = VIP + S 1.25VIP = S VIP = 800 and S = 1000 3S = 15000 S = 50K and VIP = 100K 100K/800 = $125

Alejandro, working alone, can build a doghouse in 4 hours. Betty can build it in 3 hours. If Betty and Carmelo work together and build a dog house twice as fast as Alejandro can alone, how long would it take Carmelo, working alone, to build a doghouse?

2(1/4) = 1/3 + 1/C 1/2 = 1/3 + 1/C 1/6 = 1/C It would take Carmelo 6 hours alone

A bottle is 80% full. The liquid in the bottle consists of 60% guava juice and 40% pineapple juice. The remainder is then filled with 200 ml of rum. How much guava juice is in the bottle?

200 ml represents 20% of the bottle. Therefore the bottle contains 1000 ml 1000*(0.8)*(0.6) = 800*(0.6) = 480

How many different factors does 2000 have?

2000 = 2^4(5^3) 4+1(3+1) = 5*4 = 20 different factors

4!

24

An investment of d dollars at k percent simple annual interest yields $600 interest over a 2-year period. In terms of d, what dollar amount invested at the same rate will yield $2,400 interest over a 3-year period?

2400/600 = 4 times the interest in 3 years instead of 2 years 4*2/3 = 8d/3

In a box containing action figures of the three Fates of Greek mythology, there are 3 figures of Clotho for every two figures of Atropos, and five figures of Clotho for every four figures of Lachesis. a. What is the least number of action figures that could be in a box? b. What is the ratio of Lachesis figures to Atropos figures?

3 C : 2 A 5 C : 4 L Find a common denominator - 15 C : 10 A : 12 L a. Since these are integers, the lease amount of figures in the box is 15+10+12 b. The ratio of L to A is 12:10 = 6:5

Harvey runs a 30 mile course at a constant rate of 4 miles an hour. If Clyde runs the same track at a constant rate and completes the course in 90 fewer minutes, how fast did Clyde run?

30 = 4*h h = 7.5 Clyde does it in 6 hours (90 minutes = 1.5 h) 30/6 = 5 miles per hour

GCF and LCM of 35 and 6

35 = 5,7 6 = 2,3 LCM = 2*3*5*7 =210 GCF = 1

3 Floops = 5 fleeps, 4 fleeps = 7 flaaps, and 2 flaaps = 3 fliips. How many floops are equal to 35 fliips?

35 fliips * (2 flaaps/3 fliips) * (4 fleeps/7 flaaps) * (3 floops/5 fleeps) = 8 floops This is just unit conversion

When positive integer A is divided by positive integer B, the result is 4.35. Which of the following could be the remainder when A is divided by B? A. 13 B. 14 C. 15 D. 16 E. 17

4.35 means 4 and 35/100 which simplifies to 7/20. The choice must have 7 as a prime factor, so the answer is B (14)

An office manager must choose a five digit lock code for the office door. The first and last digits must be odd, and no repetition is allowed. How many different lock codes are possible?

5 slots _ x _ x _ x _ x _ First slot has 5 choices (1,3,5,7,9) and the last slot has four choices (since non-repeating). 5*8*7*6*4 = 6,720

The cost of a plane ticket is increased by 25%, then reduced by 20%. What is the overall price change?

5/4 * 4/5 = 1 No price change overall

62.5%

5/8

Premium chocolate at $5/pound and regular for $4/pound. 7 pounds for $31. How many pounds of the premium?

5p + 4r = 31 p + r = 7 5p + 4(7-p) = 31 p +28 = 31 p = 3

5 = (4x+8y)/(x+y)

5x+5y = 4x+8y x = 3y Choose the answers where 3 of y = 1 of x

65% of students at a particular school have taken languages classes. Of those, 40% have studied more than one language. If there are 300 students, how many have studied more than 1 language?

65% is almost 66% of 300, which equals 200 40% of 200. 10% is 20, and 4*20 = 80

Seven people enter a race. There are 4 types of medals given out. The winner gets platinum, the runner-up gold, the next two get silver, the last three get bronze. What is the number of different way the medals can be awarded?

7!/1!*1!*2!*3! = 7x6x5x4x3!/3!x2 = 7x6x5x2 = 420

A dress is marked up by 16 and 2/3 % to a final price of 140. What is the original price of the dress?

7/6x = 140 x = 120

6!

720

A local card shark sends 3 goons to a national convention. If the shark has 8 goons, how many different groups of goods can she send?

8!/3!5! The three represents the attendees and the five represent the stay-at-homes 8*7*6*5!/5!*3! = 8*7 = 56

cuberoot(0.000000008)

9 decimal spots * 1/3 = 3 8^1/3 = 2 0.002 is the final answer

Perfect cube

All the prime factors are raised to a power that is a multiple of 3. Ex: 10^3(4^6)(7^9) is a perfect cube

What is the ratio of x:y:z? 1. x + y = 2z 2. 2x + 3y = z

Although the math is a bit messy, you basically need to get x/y and y/z to then know the total ratio. For either of the statements alone, you cannot get x/y since they are addition instead of multiplication. Together though, you can substitute for z and then get the ratio of x/y. After that you can substitute for x and get the ratio of y/z. Together sufficient.

denominator 9, 99, 999, etc.

Any number with a denominator of just 9 or 99 or 999 will result in a repeating decimal of whatever is at the numerator: 2/9 = 0.222222 23/99 = 0.232323 176/999 = 0.176176 Turn ordinary numbers into 99 version: 3/11 = 27/99 = 0.272727

A yearbook committee has to pick a color scheme for the yearbook. There are 7 colors to choose from. How many different color schemes are possible if the committee can select at most 2 colors?

At most two means either 1 OR 2. This means addition. 7!/1!6! + 7!/2!5! = 7 + 21 = 28

A and B are both two digit numbers and A > B. If A and B contain the same digits, but in reverse order, what integer must be a factor of A-B?

Choose values for A and B. 21 and 12. 21-12 = 9, therefore 9 must be a factor. The algebraic way to solve this is 10x+y = 10y+x, 9x = 9y A - B = 9x - 9y = 9(x-y), so 9 must be a factor

non-multiple of N +/- non-multiple of N

Could be either a multiple or non-multiple of N

Finding a repetition of repeating decimals

Do long division to find this. 3/7 for instance: 30/7, 4 remainder 2 - put a 4 20/7 = 2 remainder 6 - put a 2 60/7 = 8 remainder 4 - put a 8 So on and so forth

What is the length of the sequence of different digits in the fraction 3/7?

Do long division: 30/7, 4 remainder 2 - put a 4 20/7 = 2 remainder 6 - put a 2 60/7 = 8 remainder 4 - put a 8 So on and so forth Count up the total spots until there is repetition. 6 digits sequence

Is N divisible by 7? 1. N = x - y, where x and y are integers 2. x is divisible by 7, and y is not divisible by 7

Each statement taken separately is obviously not sufficient. Together, though, you know that a multiple of 7 + a non-multiple of 7 will never be divisible by 7. C is the answer.

What is 10/22 of 5/18 of 2000?

Estimate this problem. 10/22 is almost 1/2, which would be 1000. 5/18 is close to 1/3 but since you rounded up, round down to 1/4. Therefore the answer is roughly 250 (instead of 333)

If x is a positive integer, what is the units digit of x? 1. The units digit of x/10 = 4 2. The tens digit of 10x is 5

For statement 1, 44 would make this true but 45 would also, so this is not sufficient For statement 2, Any number that is the units digit and is multiplied by 10 is therefore the units digit for all. Statement 2 is sufficient.

Is the product of all the elements in Set S negative? 1. All the elements in Set S are negative. 2. There are 5 negative numbers in Set S

For statement 1, we don't know how many there are in the set. If the number is even, then it will be positive and if negative then it will be negative. For statement 2, there could be other numbers, such as 0, which would make it 0. Together, you know how many there are and what the signs are.

Is the integer z divisible by 6? 1. the great common factor of z and 12 is 3 2. the greatest common factor of z and 15 is 15

For statement 1, we need to know what z has in common with 12. 2*3*3 = 12 and if they have a 3 in common, that means z only contains 3 but no 2. Statement 1 is sufficient. For statement two, we know z contains 3 and 5, but we don't know about 2. Insufficient.

50% of 10% of a number is what percent of 100?

Great example to choose a number. Choose 100 50% of 100 = 50 10% of 50 = 5 5/100 = 5% For percent problems, always pick 100

terminating decimal

If the denominator is made up of prime factors 2 and 5, then the fraction will terminate

exponents and roots of decimals

If you have to take a decimal to a higher power or a root, rewrite the decimal as a power of 10. For example: .005 = 5 x 10^-3 Since roots are just fractional exponents, this can be much easier to calculate You can also just multiply the number of decimals: (0.04)^3 = 2 decimal points * 3 = 6 decimals 0.000064

Company Z only sells chairs and tables. What percent of revenue did the company drive from chairs? 1. The price of tables sold was 10% higher than the price of chiars 2. The quantity of tables was 20% less than chairs

Individually they are not sufficient. Together however... Pt = 1.1Pc Qt = 0.8Qc Pt*Qt = 1.1Pc*0.8Qc Rt = 0.88Rc The percent of chair revenue from the total is: Rc/(Rc+Rt) = Rc/(Rc+0.88Rc) = Rc/(1.88Rc) = 1/1.88 = 53%

Al and Barb shared the driving on a certain trip. What fraction of the total distance did Al drive? 1. Al drove 3/4 as much time as Barb did 2. Al's average driving speed was 4/5 of Barb's average

Individually, you cannot figure it out. Together though... Al = 4/5r * 3/4t = 3/5rt Barb = rt Al drove 3/5rt/8/5rt= 3/8 of the total This is the ratio of Al's total to the whole trip distance

How many ways to arrange a group?

N!

N! multiple

N! is a multiple of all integers 1 to N

A pod of 6 dolphins always swim in single file, with 3 females at the front and 3 males at the back. In how many different arrangements can the dolphins swim?

Notice the word AND, meaning multiply. Therefore, the arrangement of 3 females is 3! and 3 males is 3!, therefore 3! x 3! = 36

If Lucy walks to work at a rate of 4 miles per hour, and she walks home the same route at 6 miles per hour, what is her average walking time round trip?

Pick a nice number for the distance one way. 12 3 hours there, 2 hours back 12+12/3+2 = 24/5 = 4.8 miles/hour

When positive integer x is divided by 5, the remainder is 2. When positive integer y is divided by 4, the remainder is 1. Which of the following values cannot be the sum of x and y? A. 12 B. 13 C. 14 D. 16 E. 21

Set up the building the dividend equation x = 5(C) + 2 y = 4(C) + 1 Now come up with the possible values of both. X = 2, 7, 12, 17 Y = 1, 5, 9, 13, 17 The only value that cannot be added up from the two is 14

In isosceles ∆RST what is the measure of angle R? 1. The measure of angle T is 100° 2. The measure of angle S is 40°.

Since it is isosceles, two angle will be the same. Therefore statement 1 is sufficient as angle R = 40 Statement 2 is not sufficient because angle S could equal 40 but either be the same as angle T or the odd angle out

If a and b are both prime numbers greater than 10, which of the following cannot be true? i. ab is an even number ii. the difference between a and b equals 117 iii. the sum of a and b is even

Since prime numbers besides 2 are always odd, then odd*odd = odd. I cannot be true. The difference between odd - odd is always even, so statement ii is not true either. The sum of odd + odd = even, so statement iii is true

Working backwards

Start with answers B and D. If you need a larger or smaller number, then you can pretty safely find out if it is A or C or E. You can test that choice just to be sure.

If wx = y, what is the value of xy ? 1. wx^2 = 16 2. y = 4

Statement 1 is sufficient. We can isolate w as y/x, Then we plug this in for statement 1: (y/x)(x^2) = 16 yx = 16

Is the integer x odd? 1. 2(y+x) is an odd integer 2. 2y is an odd integer

Statement 2 is obviously not enough. Statement 1 means that y is not an integer (3/2 or 4/2 etc). This means that y could be even or odd, so we don't know what happens to x. If we combine the statements, 2y is odd and 2y + x is odd, but that means x could be even or odd (and multiplied by 2 to become even)

What is the units digit of (7^2)(9^2)(3^3)?

Step 1: 7 x 7 = 49 - Drop all except units digit - 9 Step 2: 9 x 9 = 81 - Drop all except units digit - 1 Step 3: 3 x 3 x 3 = 27 - Drop all except units digit - 7 Step 4: 9 x 1 x 7 = 63 The units digit of the final product is 3

Divisible by 6

The number can be divisible by 2 and by 3 48 for example

Factorial problems (grouping) with duplicates

The numerator is the total of the group N! The denominator is each category factorial based on how many duplicates there are: Ex: N!/1!*2!*3!

perfect square prime factorization

The prime factors of a perfect square will always be raised to an even power (2^4)(5^6)(10^4) is a perfect square, even if we cannot calculate it

Rates for having a manuscript typed at a certain typing service are $5 per page for the first time a page is typed and $3 per page each time a page is revised. If a certain manuscript has 100 pages, of which 40 were revised only once, 10 were revised twice, and the rest required no revisions, what was the total cost of having the manuscript typed?

The trap here is to just do 40(3) + 10(6) + 5(5) = $430 You have to pay the base of $5 no matter what: 40(8) + 10(11) + 5(5) = $680

If N is a divisor of x and y

Then N is divisor of x + y

A frat has to choose a delegation of 3 senior members and 2 junior members to the assault trial. If the frat has 6 senior members and 5 junior members, how many different delegations are possible?

This is a combination of groups and decision-making. 6!/3!3! x 5!/2!3! = 20 x 10 = 200

Boys and girls in a class write letters. There are twice as many girls as boys in the class, and each girl writers 3 more letters than each boy. If the boys write 24 letters of the 90 total, how many letters does each boy write? A. 3, B. 4, C. 5, D. 6, E. 8

This is a good backsolving problem. Start with B: 24/4 = 6 boys. Therefore 12 girls * (4+3) = 84 84 + 24 <> 90 Now with D: 4 boys. Therefore 8 girls * (6+3) = 72 72 + 24 <> 90, but it's closer Therefore the correct answer must be E

A used-car dealer sold one car at a profit of 25 percent of the dealer's purchase price for that car and sold another car at a loss of 20 percent of the dealer's purchase price for that car. If the dealer sold each car for $20,000, what was the dealer's total profit or loss, in dollars, for the two transactions combined?

This is a trap question. You first need to find the purchase price of both cars. 20000 = 1.25P, P = 16000 20000 = .8P, P = 25000 If they both sold at 20000, then P1 = 4000 and P2 = -5000 The combined is a $1000 loss (-$1000)

What is 37.5% of 240?

This is the same as 3/8 x 240 = 3*30 = 90

Is 10! + 7 a multiple of 7?

This must be a multiple of 7, since 10! is a multiple of 7

AB * CA = DEBC Each letter corresponds to a different non-zero digit, with AxB < 10. What is the 2 digit number AB? A) 23 B) 24 C) 25 D) 32 E) 42

This type of "unknown digit" problem has a method to the madness: 1. Use the answer choices 2. Use the constraints given 3. Focus on the units digit in the product or sum 4. Test the remaining answer choices Choice C is wrong because of the constraint. B*A = C. With this, test answers and eventually you will find the right one.

Are root(3)/2 and (2root3)/2 reciprocals?

To find out if two things are reciprocals, if you multiply them together, you get 1 Also if you flip it over, you get: 2/root(3), and then you multiply top and bottom by root3 to get 2root3/3

Compound interest formula

Total amount = P(1 + r/n)^n*t P is the principal r is the annual rate n is the number of times compounded t is the total years Remember that for short periods, you can just do successive percent changes

Tip for data sufficiency

Try different test cases, especially with numbers that aren't in the problem. Ideally, you want to know why something is always right as opposed to just testing it out, but sometimes this isn't possible in the time frame

(0.5)^4

Turn it into a power of 10: (5 x 10^-1)^4 5^4 = 625 10^-1^4 = 10^-4 0.0625

cuberoot(0.000027)

Turn the decimal in a power of 10: (27 x 10^-6)^1/3 27^1/3 = 3 10^-6^1/3 = 10^-2 0.03

Anagram Grid

Use for factorial problems (groups with specific numbers chosen from them. The total group can be written out 1,2,3,4,5 underneath the numbers, do outcome Y,N,N,Y,N 5!/3!2! = 20

A list of measurements in increasing order is 4, 5, 6, 8, 10, and x. If the median of these measurements is 6/7 times their arithmetic mean, what is the value of x ? 12, 13, 14, 15, 16

We know from the answer choices that the median will be 7 Therefore (4+5+6+8+10+x)/6 = 7(7/6) x+33 = 49 x = 16

Percent problems in words

What = unknown, x Percent = /100 Of = multiply Is = equals

Last digit shortcut

When asked to find the units digit of a complex number, just look at the units digit of the pieces . 45*9*62 becomes 5*9*2 = 90, so the units digit is 0

A bag contains equal numbers of red, green and yellow marbles. If John pulls out three marbles, replacing them after she pulls each one out, what is the probability that at least one will be red?

When you see "at least" then it's easier to do 1 - P(A). 1/3 are red, so 2/3 are not. 2/3 x 2/3 x 2/3 = 8/27 1 - 8/27 = 19/27

Work rate formula

Work completed = rate * time rate = work/time Work completed = (work/time)*time

Is 10! + 15 a multiple of 15?

Yes because 15 is made up of 5*3, which is included in 10!

Given that 6 is a divisor of r and r is a factor of s, is 6 a factor of s?

Yes. When they say divisor here, they mean that it can go into r without a remainder left. This the factor foundation rule

Two dice are rolled. What is the probability that the sum of the rolls is 8?

You need to calculate the numerator and the denominator separately: Denominator: 6*6 = 36 combinations Numerator: 2+6,3+5,4+4,5+3,6+2, 5 combinations 5/36

Multiple of N +/- a non-multiple of N =

a non-multiple of N 6 + 5 = 11 where N = 3

AND vs OR (combinatorics)

and means multiply Or means add Ex: There are 3 main dishes and 2 sides, how many combos are there? steak OR chicken OR fish AND soup OR salad 3 * 2 = 6

even powered prime factorization

if a prime factorization has only numbers raised to an even power, then it is a perfect square 2^45^3 is not a perfect square because the 5 is raised to an odd power

Divisible by 8

last 3 digits are divisible by 8 OR The number can be divided by 2 three times

integer with an odd number of total factors

must be a perfect square

If k^3 is divisible by 240, what is the least possible value of integer k?

prime factorize 240 = 2^4,5^1,3^1. Since it is a cube, you know that each exponent must be multiplied by 3 2^3,2^3,5^3,3^3. If you cube root this, you get 2,2,3,5 which equals 60.

Divisible by 4

the last two digits are divisible by 4 OR The number can be divided by 2 TWICE

The more even numbers you multiply

the powers of 2 the product can be divisible by: 2 x 5 x 6 = this is divisible by 4 guaranteed since 2 and 6 both add factors of 2 to the product 2 x 10 x 6 = this is divisible by 8 as well

perfect square and factor count

the total factors of a perfect square will always be odd

x is 40% of y. 50% of y is 40. 16 is what percent of x?

x = 4/10(y) y = 2*40 = 80 x = 4/10(80) = 32 16 is 50% of x

If the speed of x meters per second is equivalent to the speed of y kilometers per hour, what is y in terms of x ?

x(meters/second) = y(km/hr) x *60s/m * 60m/h *km/1000m = 3600x/1000 = 18x/5

A stereo is marked down by 30% and sold for 84$. What was the presale price of the stereo?

x.7 = 84 7/10x = 84 x = 84(10/7) = 120

A company has two types of machines, type R and type S. Operating at a constant rate, a machine of type R does a certain job in 36 hours and a machine of type S does the same job in 18 hours. If the company used the same number of each type of machine to do the job in 2 hours, how many machines of type R were used?

x/36 + x/18 = 1/2 3x/36 = 1/2 x = 36/(2*3) = 6 6 machines (since the same amount was used for R and S)

For all positive integers m, ~m = 3m, when m is odd ~m = 1/2m when m is even. Which of the following is equivalent to ~9 x~6? ~81 ~54 ~36 ~27 ~18

~9 = 3(9) = 27 ~6 = 1/2(6) = 3 3*27 = 81 ~27 = 3(27) = 81