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GMAT - Quant - Content (JB) > 1. FDPs > Flashcards

1. FDPs Flashcards

1
Q

What is the relationship between the numerator and denominator of a fraction?

A

When the denominator is the same: the larger the numerator, the larger the fraction (assuming you have positive numbers everywhere)

When the numerator is the same: the larger the denominator, the smaller the fraction

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2
Q

What is the adding/subtracting rule for fractions with a common denominator?

A

If you are adding or subtracting fractions with the same denominator, then you add or subtract the numerators, leaving the denominators alone

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3
Q

What is a Mixed Number?

A

An integer combined with a proper fraction. A mixed number can also be written as an improper fraction

3 1/2 is a mixed number

This can also be written as an improper fraction 7/2

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4
Q

What is a Proper Fraction?

A

Fractions that fall between 0 and 1

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5
Q

What is an Improper Fraction?

A

Fractions that are greater than 1. An improper fraction can also be written as a mixed

7/2 is an improper fraction

This can also be written as a mixed number 3 ½

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6
Q

What is a complex fraction?

A

A fraction in which there is a sum or difference in the numerator or denominator

Example:
(3+6) / 10
10 / (3+6)

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7
Q

What happens to a positive fraction as you increase the numerator or denominator?

A

As the numerator goes up, the fraction increases in value

As the denominator goes up, the fraction decreases in value

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8
Q

What happens when adding the same number to the numerator and denominator of a positive fraction?

A

Adding the same number to both the numerator and the denominator brings the fraction closer to 1, regardless of the fraction’s value
1/2 < (1+1)/(2+1) = 2/3

Conversely, if the fraction is originally larger than 1, the fraction decreases in value as it approaches 1
3/2 > (3+1)/(2+1) = 4/3

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9
Q

What is the adding/subtracting rule for fractions with an uncommon denominator?

A

If you are adding or subtracting fractions with different denominators, then find a common denominator (i.e. rename the fractions so that they have the same denominator) and then add or subtract the numerators

If you want to give a fraction a different denominator but keep the value the same, then multiply the top and bottom of the fraction by the same number in order to get the LCM; e.g. ¼ = (12) / (42) = 2/8

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10
Q

How do you find the least common multiple? (e.g. 6, 9, 4)

A

Multiply the non-overlapping prime factors

6 = 2 * 3
9 = 3 * 3
4 = 2 * 2

LCM = 2 * 2 * 3 * 3 = 36

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11
Q

What is true of a number that is a multiple of two other numbers?

A

A number that is a multiple of two other numbers must also be a multiple of their LCM

Example: A number (e.g. 36) that is a multiple of 4 and 6 must also be a multiple of their LCM, 12

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12
Q

How and why do you cross multiply fractions?

A

Shortcut to comparing two fractions.

(1) Set the fractions up near each other
(2) Multiply “up” and “across”
(3) Now compare the numbers that you get. The side with the bigger number is bigger

Example: 3/5 vs 4/7

7 * 3 = 21 vs 5 * 4 = 20. Thus, 3/5 is greater than 4/7

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13
Q

How do you compare fractions?

A

Put them in terms of a common denominator or just cross multiply

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14
Q

How do you convert an improper fraction to a mixed number?

A

Actually divide the numerator by the denominator. OR rewrite the numerator as a sum, then split the fraction

Example: 13/4 = 3 ¼

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15
Q

How do you convert a mixed number to an improper fraction?

A

Convert the integer to a fraction over 1, then add it to the fractional part

Example: 7 3/8 = 56/8 + 3/8 = 59/8

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16
Q

How do you simplify a fraction?

A

Cancel out common factors from top and bottom

Example: 14/35 = (27) / (57) = 2/5

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17
Q

How do you multiply fractions? e.g. (20/9) * (6/5)

A

Multiply the tops and bottoms, cancelling common factors first

Example: (20/9) * (6/5) = (4 * 5)/9 * (3 * 2)/5 = (4 * 5 * 3 * 2) / (3 * 3 * 5) = 8/3

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18
Q

What happens when you square a proper fraction?

A

When you square a proper fraction (between 0 and 1), you get a smaller number

Example: (1/3)^2 = 1/9 which is smaller than 1/3

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19
Q

What does Reciprocal mean?

A

The product of a number and its reciprocal is always 1. If you want to get the reciprocal of an integer, put that integer on the denominator of a fraction with numerator of 1. If you want the reciprocal of a fraction, flip the fraction

NOTE: the product of a number and its reciprocal is 1

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20
Q

How do you divide by a fraction (e.g. 5/6 divided by 4/7)?

A

If you divide something by a fraction, then multiply by that fraction’s reciprocal

Example: (5/6) / (4/7) = (5/6) * (7/4) = 35/42 = 5/6

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21
Q

How do you simplify a fraction with addition or subtraction in the numerator, e.g. (5x + 10y) / 25y?

A

Pull out a factor from the entire numerator and cancel that factor with one in the denominator.
Example: (5x + 10y) / 25y = 5(x + 2y) / 5(5y) = (x + 2y) / (5y)

OR you might split the fraction into two fractions
Example: (a + b) / c = (a/c) + (b/c)

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22
Q

How do you simplify a fraction with addition or subtraction in the denominator, e.g. 3y / (y^2 + xy)?

A

Pull out a factor from the entire denominator and cancel that factor with one in the numerator…but NEVER split the fraction in two!

Example: 3y / (y^2 + xy) = (y * 3) / y(y + x) = 3 / (y + x)

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23
Q

What are the basic rules of fractions?

A
  • Addition: find a common denominator, then add numerators
  • Subtraction: find a common denominator, then subtract numerators
  • Multiplication: multiply tops and multiply bottoms, then subtract numerators
  • Division: flip, then multiply

*Whenever necessary, treat the numerators and denominators as if they have parentheses around them

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24
Q

Ch 4, Q 14. Add or subtract the following fraction such that it is in its most simplified form. (ab)/(cb) + (a/c) – (a^2 * b^3)/(abc)

A

= a/c + a/c + (ab^2)/c
= (2a – ab^2) / c
= a(2 – b^2) / c

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25
Q

Ch 4, Q 14. Add or subtract the following fraction such that it is in its most simplified form. 24/3Sqrt(2) – 4/Sqrt(2)

A

= [24Sqrt(2) / 6] – [4Sqrt(2) / 2]
= 4Sqrt(2) – 2Sqrt(2)
= 2Sqrt(2)

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26
Q

Ch 4, Q 35. Simplify the following expression. (3ab^2 * Sqrt(50)) / Sqrt(18a^2)

A

= (3 * a * b * b * 5Sqrt(2)) / Sqrt(2 * 9 * a * a

= 5b^2

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27
Q

Ch 4, Q 44. Multiply or divide the following fraction such that it is in its most simplified form. (x * y^3 * z^4)/(x^3 * y^4 * z^2) * (x^3 * y^5 * z^2)/(x^6 * y^2 * z)

A

= (x * x^3)/(x^3 * x^6) * (y^3 * y^5)/(y^4 * y^2) * (z^4 * z^2)/(z^2 * z)
= x^(1+3-3-6) * y^(3+5-4-2) * z^(4+2-2-1)
= x^(-5) * y^2 * z^3
= (y^2 * z^3) / x^5

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28
Q

How do you convert a decimal to a percent and vice versa?

A

Decimal to percent: move the decimal point two places to the right
Percent to decimal: move the decimal point two places to the left

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29
Q

How do you convert a decimal to a fraction?

A

Put the digits to the right of the decimal point over the appropriate power of 10, then simplify.

Example: 0.036 = 36 / 1000 = 9 / 250

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30
Q

How do you convert a percent to a fraction?

A

Write the percentage over 100, then simplify

Example: 4% = 4 / 100 = 1 / 25

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31
Q

How do you convert a fraction to a decimal or percentage?

A

Long divide the numerator by the denominator OR convert the denominator to a power of 10, if the denominator only contains 2’s and 5’s as factors

Example: 7/8 = (7125) / (8125) = 875/1,000 = 0.875

You multiply 8 (=2^3) by 125 (=5^3) to get 1,000 (=10^3)

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32
Q

What is 1/8, 2/8, 3/8, 4/8, 5/8, 6/8, 7/8 and 8/8?

A 
1/8 = 12.5%
2/8 = 25.0%
3/8 = 37.5%
4/8 = 50.0%
5/8 = 62.5%
6/8 = 75.0%
7/8 = 87.5%
8/8 = 100.0%
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33
Q

What is 1/7?

A

0.14

34
Q

What is 1/9?

A

0.11

35
Q

What happens if you multiply or divide a decimal by a power of 10?

A

Multiply: move the decimal point to the right a number of spaces, corresponding to the exponent of 10 (e.g. 0.007 * 10^2 = 0.7)
Divide: move the decimal point left a number of places, corresponding to the exponent of 10 (e.g. 0.6 / 10^3 = 0.0006)

NOTE: Negative powers of 10 reverse the regular process (e.g. 53.0447 / 10^-2 = 5,304.47)

36
Q

How do you add or subtract decimals?

A

Line up the decimal points vertically

Example: 4.035
+0.120
=4.155

37
Q

How do you multiply two decimals?

A

Ignore the decimal points, multiply the integers, then place the decimal point by counting the original digits on the right

Example:
0.2 * 0.5 = ?
2 * 5 = 10
0.2 * 0.5 = 0.10 (move two decimal places to the left)

If you multiply a decimal and a big number, then you trade decimal places from the big number to the decimal (i.e. move the decimals in the opposite direction). The reason why this trick works is because you are multiplying and then dividing by the same power of 10. In other words, you are trading decimal places in one number for decimal places in another number

Example:
50,000 * 0.007 = 50 * 7 = 350

38
Q

How do you divide two decimals?

A

If you divide two decimals, move the decimal points in the same direction to eliminate decimals as far as you can such that you DIVIDE by whole numbers

Example: 0.002 / 0.0004 = 20 / 4 = 5

39
Q

What is the difference between the process of multiplying and dividing decimals?

A

To simplify multiplication, you can move decimals in opposite directions

But to simplify division, you move decimals in the same direction

40
Q

What do you do if you see “x percent of” a number or “what percentage of”?

A

See 30% of 200: convert 30% into a decimal or fraction, then multiply

Example: 30% of 200 = 0.3(200) = 60

See what percent of 200 is 60

Example: (x/100)200 = 60 -> x = 30%

41
Q

What is the equation for percentage change?

A

Percentage change = Change in Value / Original Value

42
Q

What is the equation for finding the new percentage or new value?

A

New Percent = New Value / Original Value

43
Q

How do you find a percentage change?

A

Divide the change in value by the original value. In other words, (new – old) / old

Example: $230 is what percent more than $200? (230 – 200) / 200 = 30 / 200 = 15%

  • “Percent more than” is just like “percent increase”
  • ”Percent less than” is just like “percent decrease”
  • The original value is always after the “than”. It is the value that you are comparing the other value to.
  • Do not misread “percent of” as a “percent more than” or vice versa
44
Q

What do you do when you have successive percent changes?

The price of a share, originally $50, goes up by 10% on Monday and then 20% on Tuesday. What is the overall change in the price of a share, in dollars?

A

Multiply the original value by the new percents for both percent changes

$50 * 1.1 = $55
$55 * 1.2 = $66
Change in share price = $66 - $50 = $16

45
Q

What do the following words mean in relation to percent word problems: percent, of, is, what?

A 
Percent = divide by 100
Of = multiply
Is = equals
What = unknown value
46
Q

What is simple interest and what is the formula?

A

Interest is computed on the principal only

Simple Interest = Principal * Interest Rate * Time

47
Q

What is compounded interest and what is the formula?

A

Interest is computed on the principal as well as interest

Compound Interest = Principal (1 + r/n)^(nt)
r = annual rate (in decimal form)
n = number of times per year
t = number of years

48
Q

Given $1,000 invested for 2 years, compounded semi-annually, at an annual, or yearly, rate of 4%. Calculate the amount in the account after 2 years.

A

$1,000 is the principal amount, 0.04 is the rate, n is 2 (because it is compounded twice in a year), and t is 2 (because the money is invested for 2 years).

$1,000 * (1 + (0.04 / 2))^(2 * 2) = $1,082.43

49
Q

How do you handle ratios? (e.g. ratio of apples to bananas is 7 to 10, and you know the number of apples is 63)

A

Write the proportion of x to y as a fraction. Then write each quantity in terms of an unknown multiplier

Example:
7x/10x
Thus, 7x = 63 and x = 9
Thus, 10x = 90, which is the number of bananas

50
Q

How do you handle two parts that make a whole? (e.g. ratio of lefties to righties is 3 to 4)

A

Use the Part:Part:Whole ratio and use the unknown multiplier as needed

Example:
3x lefties : 4x righties : 7x people

51
Q

What does a constant ratio indicate?

A

If two quantities have a constant ratio, then they are directly proportional to each other

52
Q

What is the unknown multiplier for ratios and when should it be used? (e.g. ratio of men to women in a room is 3:4 and there are 56 people in total)

A

Instead of representing the number of men as M, represent it at 3x, where x is some unknown positive number. Likewise, instead of representing the number of women as W, represent it as 4x, where x is the same unknown number. The variable x is the unknown multiplier, which allows you to reduce the number of variables making the algebra easier

3x + 4x = 56
7x = 56
X = 8

Number of men = 3x = 24
Number of women = 4x = 32

*Use the Unknown Multiplier technique when neither quantity in the ratio is already equal to a number or a variable expression

53
Q

How do you change ratios to have common terms? (e.g. three Cats for every two Antelopes, and five Cats for every four Lamas)

A

You cannot instantly combine these ratios into a single ratio of all three quantities, because the terms for C are different. However, you can fix that problem by multiplying each ratio by the right number, making C’s into the least common multiple of the current values
C:A:L
3:2
5:4

C:A:L

15: 10 (multiply by 5)
15: 12 (multiply by 3)

Combined ratio is 15:10:12

54
Q

What is sufficient if a DS question asks for the relative value of two pieces of a ratio?

A

ANY statement that gives the relative value of ANY two pieces of the ratio will be sufficient

55
Q

What is sufficient if a DS question asks for the concrete value of one element of a ratio?

A

You will need BOTH the concrete value of another element of the ratio AND the relative value of two elements of the ratio

56
Q

What are smart numbers and when do you use them?

A

Sometimes, fraction problems on the GMAT include unspecified numerical amounts; often these unspecified amounts are described by variables. In these cases, pick real numbers to stand in for the variables. To make the computation easier, choose Smart Numbers equal to common multiples of the denominators of the fractions in the problem

  • Do pick smart numbers when no amounts are given in the problem
  • Do not pick smart numbers when any amount or total is given
57
Q

What are benchmark values and when do you use them?

A

Use benchmark values in order to estimate fractions. These are simple fractions with which you are already familiar

Example: what is 10/22 of 5/18 of 2,000?
10/22 ~ 1/2
5/18 ~ 1/4
(10/22)(5/18)(2,000) ~ 250

58
Q

What is the heavy division shortcut? e.g. what is 1,530,794 / (31.49 * 10^4)

A

Move the decimals in the same direction and round to whole numbers

(1) Set up the division problem in fraction form
(2) Rewrite the problem, eliminating powers of 10
1,530,794 / 314,900
(3) Your goal is to get a single digit to the left of the decimal in the denominator
15.30794 / 3.14900
(4) Focus only on the whole number parts of the numerator and denominator and solve
1,530,794 / 314,900 ~ 15 / 3 = 5

59
Q

What are the place values of 1,234.567

A 
1 = thousands
2 = hundreds
3 = tens 
4 = units or ones
5 = tenths 
6 = hundredths 
7 = thousandths
60
Q

What are the most common powers of 10?

A 
10^3 = 1,000
10^2 = 100
10^1 = 10
10^0 = 1
10^-1 = 0.1
10^-2 = 0.01
10^-3 = 0.001
61
Q

What is the rule of rounding to the nearest place value? (e.g. 3.681 to nearest tenth and hundreth)

A

(1) find the digit located in the specified place value. Digit in tenths place is 6
(2) Look at the right-digit-neighbor (digit immediately to the right) of the digit in question. In this case it is 8.

If the right-digit-neighbor is 5 or greater, round the digit in question UP. Otherwise, leave the digit the alone and ignore the other digits to the right

  1. 681 to nearest tenth = 3.7
  2. 681 to nearest hundredth = 3.68
62
Q

How do you handle a decimal to a larger power? e.g. 0.5^4

A

Rewrite the decimal as the product of an integer and a power of 10, and then apply the exponent

Example: (0.5)^4 = (5 * 10^-1)^4 = 5^4 * 10^-4 = 625 * 10^-4 = 0.0625

63
Q

What is the shortcut for counting decimal places for powers and roots of decimals?

A

The number of decimal places in the result of a cubed decimal is 3 times the number of decimal places in the original decimal
(0.04)^2 = 0.000064

Likewise, the number of decimal places in a cube root is 1/3 the number of decimal places in the original decimal
Curt(0.000000008) = 0.002

64
Q

What is the general rule of when to use fractions, decimals or percents?

A

Prefer fractions for doing multiplication or division, but prefer decimals and percents for doing addition or subtraction, for estimating numbers, or for comparing numbers

65
Q

How do you determine if a fraction has terminating decimals?

A

The decimals of a fraction terminate if the fraction’s denominator in fully reduced form must have prime factorization that consists of only 2’s and/or 5’s

66
Q

How do you determine if a fraction has repeating decimals?

A

If the denominator of the fraction is 9, 99, 999 or another number equal to a power of 10 minus 1, then the numerator gives you the repeating digits

67
Q

What is the last digit shortcut? e.g. what is the units digit of (7^2)(9^2)(3^3)??

A

To find the units digit of a product or a sum of integers, only pay attention to the units digits of the numbers you are working with and drop any other digits.

Step 1: 77 = 49
Step 2: 99 = 81
Step 3: 333 = 27
Step 4: 917 = 63

Multiply the last digits of each of the products. The units digit of the final product is 3.

68
Q

How do you determine a ratio with four different terms?

A

Because the ratio involves four different terms, using variables will generally just make this kind of problem harder. Instead we can put all of the information into a table and then use that table to merge the ratios by finding common terms.

The next step is to start merging rows by multiplying the ratios in order to get the numbers in each column to match up. This works because the ratio can be thought of as fractions, and it does not change the value of the fraction to multiply both the numerator and denominator by the same number.

69
Q

Can you find the percent change in net income given an increase in revenue and expenses?

A

No because you do not know the starting or ending net income values. Thus, you cannot calculate the change in net income or the original net income value

70
Q

When is a ratio of two variables sufficient to answer the question?

A

If the question asks for the percentage of something and not the actual values, then a ratio will be sufficient to answer the question

71
Q 
FDP Strategy Guide, Ch 7. A and are both two-digit numbers, with A > B. If A and B contain the same digits, but in reverse order, what integer must be a factor of (A – B)?
(A) 4
(B) 5
(C) 6
(D) 8
(E) 9
A

To solve this problem, assign two variables to be the digits in A and B: x and y. Let A = xy (not the product of x and y).
A = 10x + y
B = 10y + x
A – B = 10x + y – (10y + x) = 9x – 9y = 9(x – y)

Clearly 9 must be a factor of (A – B). The correct answer is 9

NOTE: in general, for unknown digits problems, be ready to create variables (such as x, y, z) to represent the unknown digits. Recognize that each unknown is restricted to at most 10 possible values (0 through 9)

72
Q

FDP Strategy Guide, Ch 7. In the multiplication below, each letter stands for a different non-zero digit, with A*B

A

Apply the given constraint that A*B

73
Q

FDP Strategy Guide, Ch 7, Q 8. A professional gambler has won 40% of his 25 poker games for the week so far. If, all of a sudden, his luck changes and he begins winning 80% of the time, how many more games must he play to end up winning 60% of his games for that week?

A

This is a weighted averages problem. You can set up a table to calculate the number of games he must play to obtain a weighted average win rate of 60%.
Columns: Original + Change = New
Rows: Win + Lose = Total

Win – Original 0.4(25) = 10, Change 0.8x, New 10 + 0.8x
Lose – Original 15, Change 0.2x, New 15 + 0.2x
Total – Original 25, Change x, New 25 + x

(10 + 0.8x)/(25 + x) = 0.6
(10 + 0.8x) = 0.6(25 + x)
0.2x = 5
x = 25

74
Q

FDP Strategy Guide, CH 7, Q 9. A feed store sells two varieties of birdseed: Brand A, which is 40% millet and 60% sunflower, and Brand B, which is 65% millet and 35% sunflower. If a customer purchases a mix of the two types of birdseed that is 50% millet, what percent of the mix is Brand A?

A

This is a weighted averages problem. You can set up a table to calculate the answer, and assume that you purchased 100 lbs of Brand B
Columns: Brand A + Brand B = Total
Rows: Millet + Sunflower = Total

Millet – Brand A 0.4x, Brand B 65, Total 0.5(100 + x)
Sunflower – Brand A 0.6x, Brand B 35, Total 0.5(200 + x)
Total – Brand A x, Brand B 100, Total 100 + x

0.4x + 65 = 0.5(100 + x)
x = 150

% Brand A = x / (100 + x) = 3/5 = 60%

75
Q

FDP Strategy Guide Ch 7, Q 10. A grocery store sells two varieties of jellybean jars, and each type of jellybean jar contains only red and yellow jellybeans. If Jar B contains 20% more red jellybeans than Jar A, but 10% fewer yellow jellybeans, and Jar A contains twice as many red jellybeans as yellow jellybeans, by what percent is the number of jellybeans in Jar B larger than the number of jelly beans in Jar A?

A

This is a weighted average percent change problem. You can set up a table to calculate the answer, and assume that Jar A contains 100 red jellybeans and 50 yellow jellybeans.
Columns: Jar A + Jar B = Total
Rows: Red + Yellow = Total

Red – Jar A 100, Jar B 120, Total 220
Yellow – Jar A 50, Jar B 45, Total 95
Total – Jar A 150, Jar B 165, Total 135

Jar B / Jar A = 165/150 – 1 = 10%

76
Q

FDP Strategy Guide Ch 7, Q 11. Last year, all registered voters in Kumannia voted for the Revolutionary Party or for the Status Quo Party. This year, the number of Revolutionary voters increased 10%, while the number of Status Quo voters increased 5%. No other votes were cast. If the number of total voters increased 8%, what fraction of voters voted Revolutionary this year?

A

This is a weighted average percent change problem. You can set up a table to calculate the answer, and assume that last year there were a total of 100 votes.

Columns: Last year, change, this year
Rows: Rev, SQ, Total

Rev – Last Year x100, Change 0.1(x100), This Year x100 + 10x
SQ – Last Year (1-x)100, Change 0.05(1-x)100, Total (1-x)100 + 5(1-x)
Total – Last Year 100, Change 0.08(100), 108

% Revenue This Year = 0.6(110)/108 = (330/5)/108 = 66/108 = 11/18 = 61%

77
Q

Ch 5, Q 57. 75 reduced by x% is 54. What is x?

A

75 – 75x = 54
21 = 75x
X = 21/75 = 7/25 = 28/100 = 28%

78
Q

Ch 5, Q 63. What percent of y percent of 50 is 40 percent of y?

A 
xy50 = 40y
x50 = 40
x = 4/5 = 80%
79
Q

What does 2(0.1) + 3(0.1)^2 + 2(0.1)^3 equal?

A

Calculate the squared and the cubed term first:
2(0.1) + 3(0.01) + 2(0.001)
Multiply each term by its coefficient, then sum the terms:
0.2 + 0.03 + 0.002

= 0.232

80
Q

FDP Strategy, Ch 6, Q 3. Lisa spends 3/8 of her monthly paycheck on rent and 5/12 on food. Her roommate, Carrie, who earns twice as much as Lisa, spends 1/4 of her monthly paycheck on rent and 1/2 on food. If the two women decide to donate the remainder of their money to charity each month, what fraction of their combined monthly income will they donate?

A

Use smart numbers to solve this problem. Since the denominators in the problem are 8, 12, 4, and 2, assign Lisa a monthly paycheck of $24. Assign her roommate, who earns twice as much, a monthly paycheck of $48.

Lisa:
Rent = (3/8)(24) = 9
Food = (5/12)(24) = 10
Remainder = 24 – 9 – 10 = 5

Carrie:
Rent = (1/4(48) = 12
Food = (1/2)(48) = 24
Remainder = 48 – 12 – 24 = 12

The women will donate a total of $17 out of their combined monthly income of $72

GMAT - Quant - Content (JB) (13 decks)

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