Fractions, Decimals, Percents

Question 1

Percentage:

Answer 1

Percentage:

If questions is x is what percent of y, calculation is: x/y

E.g. profit of year 1 is 10, profit of year 2 is 12.
Question: Profit of year 2 is what percent of profit of year 1? 12/10 = 1.2

To convert to percent, multiply by 100: 1.2 x 100=120

What Percent Really Means:
If 75% of the students like chocolate ice cream what that means is that out of every 100 students 75 like chocolate ice cream.

100% of something means 100/100. Therefore 100% = 1

Question 2

Re-expressing Fractions:

Answer 2

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.

Question 3

Subtracting Fractions:

Answer 3

Subtracting Fractions:

If someone spent 5 dollars of a total saving of 12 dollars, the remaining amoung expressed in a fraction is:

7/12

Question 4

Translating decimal numbers into fractions:

Answer 4

Translating decimal numbers into fractions:

0. 5 = ½
1. 8 = 4/5
2. 25 = 5/4

Question 5

Translating percentages into decimal numbers:

Answer 5

Translating percentages into decimal numbers:

• If x is 12.5% higher than y, then: x = y * 1.125 (because it’s 100% plus another 12.5 percent higher)
• if x is 80% higher than y, then: x = y*1.8

To re-express percents as decimals move the decimal point to the left two spaces:

525% = 5.25 52.5% = 0.525

Question 6

Ratio:

Answer 6

Ratio:

E.g. if ration of 3x to 8y is 5 to 7

Translated into: 3x/8y = 5/7

e.g. if ratio of men to women in office is 3:4, then the number of men divided by number of women is 3/4, or 0.75. I.e. there are 0.75 men for every woman.

Question 7

Comparing Positive Fractions in Arithmetic:

Answer 7

If the numerators are the same, the fraction with the smaller denominator will have the larger value since the numerator is divided into a smaller number of parts.

E.g. 4/5 > 4/7

If the denominators are the same, the fraction with the larger numerator will have the larger value.

E.g. 5/8 > 3/8

If neither the numerators nor the denominators are the same, express all fractions in terms of some common denominator. Or multiply the numerator of the left fraction by the denominator of the right fraction and vice versa. Compare the products. If the left product is greater, then the left fraction was greater to start with.

E.g. if 5/7 and 9/11

Compare 55 x 11 and 9 x 7

Because 55 5/7

Another way when finding the common denominator seems time consuming is to convert the fraction into decimals or percents.

E.g. 5/8, 2/3, and 7/11

5/8 = 0.625 , 2/3 = 0.666 , 7/11 = 0.6363

That means: 5/81/2 and 33/6833/68.

Question 8

Common Fraction-to-Decimal Equivalencies:

Answer 8

Common Fraction-to-Decimal Equivalencies:

MEMORIZE THESE!

1/2 =    0.5                            50%
1/3 =    0.333...                     33.3%   
1/4 =    0.25                          25%
1/5 =    0.2                            20%
1/6 =    0.166...                      16.7%
1/7 =    0.143
1/8 =    0.125                         12.5%
1/9 =    0.111...                         11.1%
1/10 =   0.1                              10%
1/11 =    0.0909...
1/20=   0.05                           5%
1/25=   0.04                           4%
1/50=   0.02                           2%
1/100=  0.01                            1%
2/3 =    0.666...                      66.7%
2/5 =    0.4                             40%
2/7 =    0.286
2/9 =    0.222...
2/10 =   0.2
2/11 =    0.1818...
3/2 =    1.5                               150%
3/4 =    0.75                             75%
3/5 =    0.6                               60%
3/7 =    0.429
3/8 =    0.375                           37.5%
3/10 =   0.3                               30%
3/11 =    0.2727...
4/3 =    1.333...                          133%
4/5 =    0.8                                80%
4/7 =    0.571
4/9 =    0.444...
4/10 =   0.4
4/11 =    0.3636...
2/5 =    2.5
5/3 =    1.666...
5/4 =    1.25                                 125%
5/6 =    0.8333...                         83.3%
5/7 =    0.714
5/8 =    0.625                              62.5%
5/9 =    0.555...
5/10 =   0.5
5/11 =    0.4545...
6/5 =    1.2 
6/7 =    0.857
6/10 =   0.6 
6/11 =    0.5454...
7/2 =     3.5
7/3 =     2.333...
7/4 =     1.75                                  175%
7/5 =     1.4
7/6 =     1.1666...
7/8 =     0.875                               87.5%
7/9 =     0.777...
7/10 =    0.7                                    70%
7/11 =     0.6363...
8/3 =     2.666...
8/5 =     1.6 
8/7 =     1.143
8/9 =     0.888...
8/10 =    0.8
8/11 =     0.7272...
9/2 =     4.5 
9/4 =     2.25
9/5 =     1.8
9/7 =     1.286
9/8 =     1.125
9/10 =    0.9                                    90%
9/11 =    0.8181...

Question 9

Mixed Numbers in Arithmetic:

Answer 9

Fractions where the numerator is greater than the denominator may be converted into a mixed number consisting of an integer and a fraction.

Question 10

Decimals in Arithmetic:

Answer 10

Decimals can be converted to common fractions with a power of 10 in the denominator.

E.g. 0.053 = 53/10^3 = 53/1000

Question 11

Digits in specific places on the GMAT:

Answer 11

Memorize the places of digits:

2,678,891,315.2467

2 = billions
6 = millions 
7 = millions
8 = millions
8 = thousands
9= thousands
1 = thousands
3 = hundreds
1 = tens
5 = units
2 = tenths
4 = hundredths
6 = thousandths
7 = thousandths

When a GMAT question specifies that a variable is a digit, the only possible values are the integers 0 to 9.

In the number 452 you have two ones, five tens, and 4 hundreds. So you can rewrite the number as:

4x100 + 5x10 + 2x1 = 452

In the number 0.8347:

8 is in the tenth place, giving it the value of 8 tenths, or 8/10
3 has the value of 3 hundredths, or 3/100
4 has the value of 4 thousandths or 4/1000
7 has the value of 7 ten-thousandths or 7/10000

0. 8 can be expressed as eight tenths of a dollar, or 8/10 of a dollar or 80/100 or 80 cents.
1. 03 can be expressed as three hundredths of a dollar, or 3/100 or 3 cents.

Question 12

Comparing Decimals in Arithmetic:

Answer 12

To compare decimals find out which denominator you need (10, 100, 1000 etc.) and add zeros after the last digit to the right of the decimal point until all the decimals have the same number of digits. That makes sure that all fractions have the same denominator.

E.g. compare 0.7 to 0.077

means compare 700/1000 to 77/1000

Question 13

Adding and Subtracting Decimals in Arithmetic:

Answer 13

When adding or subtracting decimals make sure that the decimal points are lined up, one under the other.

Question 14

Multiplying decimals:

Answer 14

To multiply two decimals, initially multiply them as you would integers and ignore the decimal places. The number places in the product will be the sum of the number of decimal places in the factors that are multiplied together.

E.g. 0.675 x 0.42

multiply 675 by 42 and then add 5 decimal points counting from right to left.

Question 15

Dividing decimals:

Answer 15

Method 1: First move the decimal by as many point as needed for both values so that the divisor (the number that a the other number is divided by) becomes an integer. This doesn’t change the value of the quotient. Then carry out the division as you would with integers.

E.g. if 0.675/0.25
then: 67.5/25

Method 2: Turn division problem into fraction. It is best when the numbers have common factors. Move the decimal point in numerator and denominator an equal number of places until the fraction is only made out of integers. Then cancel common factors.

E.g. 0.675/0.25
make it: 675/250 = 27/10 = 2.7

Question 16

Base of 10:

Answer 16

The exponent of a base of 10 tells you how many zeros the number would contain if written out:

E.g. 10^6 = 1,000,000 (6 zeros)

When multiplying a number by a power of 10, move the decimal point to the right the same number of places as in the exponent, i.e. the number of zeros in that power of 10.

E.g. 4 x 10^4 = 40,000

When dividing by a power of 10, move the decimal point the corresponding number of places to the left.

E.g. 4/10^4 = 0.0004

Note: dividing by 10^4 is the same as multiplying by 10^-4.

Question 17

Ratios:

Answer 17

A ratio is a comparison of two quantities by division.

Ratios can be written as:

Fraction x/y
with colon (x:y)
in English “the ratio of x to y”

Ratios should be reduced to lowest terms just like fractions:

16/12 = 4/3

In a ratio of two numbers the numerator is often associated with the word “of” and the denominator with the word “to.”

Ratio: of…/to….

Question 18

Ratios With More Than Two Terms:

Answer 18

GMAT questions with three or more ratios are represented with colons.

E.g. x:y:z

They are governed by the same principles as two-term ratios. They are usually ratios of various parts and the sum of them is usually the whole.

E.g.

x: (x+y+z) = x/(x+y+z)
y: (x+y+z) = y/(x+y+z)
etc.

You can also determine various two-term ratios among the parts given in ratios with more than two parts.

E.g.

x: y = x/y
x: z = x/z
etc.

And you can find ratios of one variable to a combination of two others.

E.g.
x = ratio of children
y = ratio of men
z = ratio of women

i.e. ratio of children to adults: x:(y+z) or x/(y+z)

Question 19

Ratio vs. Actual Numbers:

Answer 19

Don’t forget that a ratio is reduced to it’s simplest form. If ratio of games won to lost is 5:3 it doesn’t necessarily mean that team has won exactly 5 games, it could be 50 games won and 30 games lost. Unless you know the actual number of games played, or games won or lost, you can’t know the actual values of the ratio.

E.g. if in a class of 30 students the ratio of boys int class to students in class is 2:5 then the number of boys in class is:

ratio boys/ratio students multiplied by number of students:

2/5 x 30 = 12

But remember: if you are given one value, you can make deductions about other values, set up equations and solve for the other values.

E.g. A homecoming party at a college is initially attended by students and alumni in ratio of 1:5. Later 36 students come and the ratio changes to 1:2. If number of alumni didn’t change, how many people were there at the beginning of the party?

Let’s make x the total number of students.

Beginning of party: 1x:5x (S:A)
Later: 1x:2X
1(x + 36): 2(x + 36)
x + 36 : 2x + 27

Now you have two equations for A that are the same:
5x and (2x + 72)
You can now write the equation:

 5x = 2x + 72 
   x = 24

That means at the beginning there were 24 students, (24x5) Alumni, and altogether 144 people.

Question 20

Finding Part or Whole Values in Ratios:

Answer 20

If you are not given the actual values of parts of the ratio or of the whole number, you can still find the solutions by setting up and solving equations. Here it’s important to remember:

The actual number will always equal the figures in the ratio multiplied by the same (unknown) factor.

E.g. in a shop the ratio of used to new cars is 2:5. The actual number of the cars is not given but we know that it will equal the figures in the ratio multiplied by an unknown.

I.e. used: 2x new: 5x

That means if there are 20 used cars there will be 100 new cars.

Question 21

Guessing With Ratios:

Answer 21

Many GMAT ratio questions involve things like students, women, children, cars etc. All of these can only exist as integers. If that’s the case the factor by which the ratio is multiplied to become an actual number must be an integer as well.

I.e. if used cars:new cars = 2:5
You know the actual number of used cars is 2x and actual number of new cars is 5x. You also know that x can only be an integer because there is no 1/2 car or so.

Therefore a great tip here: The actual number of something must be a multiple of its value in the ratio.

I.e. the actual value of used cars is a multiple of 2, actual value of new cars is multiple of 5 and actual value of all cars in the shop is a multiple of 7 (5+2). Knowing that can help you get to the right answer by just looking at the answer choices without doing any math.

Question 22

Ratios and Quantities:

Answer 22

Remember not to confuse ratios and quantities. In ratio questions, if you change the actual quantities through multiplication or division, the resulting ratio is the same, regardless of the starting values of the quantities.

E.g. 2:3 is the same as 12:18 (multiplied by 6)

But: The same ratio may not be preserved if you add or subtract from the quantities.

E.g. 2:3 adding 6 on both sides makes it: 8:9 which is not the same ratio as 2:3.

Question 23

Solving Ratio Questions on GMAT:

Answer 23

Remember: Without the actual value of the whole or one of the quantities’ value in a ration, you can’t determine the new values.

Question 24

Percents on GMAT:

Answer 24

Percent is just another word for “per one hundred.”

E.g. 19% is the same as:
19/100
0.19
19 out of every 100 things.

Question 25

Percent Formula for Converting Among Percents, Decimals, Fractions and Ratios:

Answer 25

Percent = Part/Whole x 100%

Remember: Multiplying by 100% is the same as multiplying by 1 as 100% is 1. That means multiplying by 100% will not change the number.

Question 26

Converting Decimal or Fraction into Percent:

Answer 26

Multiply by 100%. (Since multiplying by 100% is same as multiplying by 1 it won’t change the number). Another way to say this: Concert decimals into percents by moving the decimal point to the right two spaces.

E.g. 0.17 x 100% = 17%
1/4 x 100% = 25%

0. 6 = 60%
1. 28 = 28%

Question 27

Converting Percent into Fraction:

Answer 27

Divide by 100%. (Since multiplying by 100% is same as multiplying by 1 it won’t change the number)

E.g. 32% = 32%/100% = 32/100 = 8/25

    1/2% = 1/2%/100% = 1/2 x 1/100 = 1/200

Question 28

Percent Increase and Decrease:

Answer 28

Formulas:

Percent Increase = Amount of increase/Original Whole x 100%

Percent Decrease = Amount of decrease/Original Whole x 100%

New Whole = Original whole +- Amount of Change

Remember: Always put the amount of increase or decrease over the original whole, not over the new whole.

Also, read the question correctly to make sure you are really looking for the percent increase. If the question says “increase” then the percent increase is probably what they are looking for. Otherwise they might be looking for what percent of the original the new amount is which is something different. (that would be calculated by “Original Amount x Y%/100 = New Amount)

(Whenever you’re calculating a percent “discount” you’re actually calculating a percent “decrease”)

E.g. the price of a cup of coffee is increased from 80 to 84 dollars. What is the percent increase?

4/80 = 1/20 = 5/100 = 5%

You may also be asked for the final price. Then the calculation is a bit different.

E.g. If the price of a $30 shirt is decreased by 20%, what is the final price of the shirt?

There are different ways of calculation this. One way:

New Percent = New Value/Original Value.

If something is decreased by 20% the new price is 100-20 = 80% of the original price. So:

80/100 = x/30
80x30/100 = x
24 = x

Question 29

Convert Percent to Decimal:

Answer 29

Divide by 100%. (Since dividing by 100% is same as multiplying by 1 it won’t change the number)

Or even easier: Drop percent sign and move decimal point two places (same as dividing by 100%).

0.8% = 0.008

Question 30

Common Percent-to-Fraction Equivalents to Memorize:

Answer 30

10% =   1/10
20% =  1/5
30% =  3/10
40% =  2/5
50% =  1/2
60% =  3/5
70% =  7/10
80% =  4/5
90% =  9/10
100% = 10/10 = 1

12 1/2% = 1/8
37 1/2% = 3/8
62 1/2% = 5/8
75% =       3/4
87 1/2% =  7/8

16 2/3% = 1/6
33 1/3% = 1/3
66 2/3% = 2/3
83 1/3% = 5/6

Question 31

Solving Percent Problems on GMAT:

Answer 31

Usually you will be given two of the terms from the formula below and asked for the third one. It is usually easiest to change the percent to a common fraction before doing the calculation.

Percent Formula: Percent x Whole = Part

“Whole” will generally be associated with word “of.”
“Part” will generally be associated with word “is.”
Percent can be represented as ratio of the part of the whole, or the “is to the of.”

Use this formula to plug in variables to find out a certain percent of a number:

      x% of y = x/100 x y. 

Example on GMAT:

13 is 33 1/3% of what number?

You know 33 1/3% is 1/3 (memorize these from other flashcard). So formula is:

     13 = 1/3 x n
      = 39

Question 32

Examples of Percent Questions on GMAT:

Answer 32

Example 1:

13 is 33 1/3% of what number?

You know 33 1/3% is 1/3 (memorize these from other flashcard). So formula is:

     13 = 1/3 x n
      = 39

Example 2:

18 is what percent of 3?

    18 = m% of 3
     m = 6  to turn into % multiply by 100%
    m% = 6 x 100% = 600%

Question 33

“Percent Greater Than” and “Percent Less Than” on GMAT:

Answer 33

Remember: In these questions it’s the number that follows the word “than” to which the percent is applied.

Trick: Convert “Percent greater than” or “Percent less than” to “Percent of.”

I.E. “x% greater than” = “(100 + x)% of”
“x% less than” = “(100 - x)% of”

And “of” always means multiply.

Example 1:
E.g. What number is 15% less than 60?

Dissect the question and turn it into and an equation:

n = 60 - (15% of 60)
n = 60 - (15% x 60)
n = 60 - (15/100 x 60)
n = 60 - 9
n = 51

Example 2:
What is the price of a television that costs 28% more than a $50 radio?

t = $50 + (28% of $50)
t = 50 + (28/100 x 50)
t = 50 + 14
t = 64

Question 34

Combining Percents:

Answer 34

Some GMAT questions ask for more than one percent. They as for a percent of a percent.

Caution: You can’t just add percents, unless the percents are of the same whole.

E.g. The price of a product of $200 is reduced by 20% and that discount price is reduced by another 10%. What’s the final discount price?

2 separate calculations:

1. p = 200 - (20% of 200)
   p = 200 - (1/5 x 200)
   p = 200 - 40
   p = 160
2. fp = 160 - (10% of 160)
   fp = 160 - (1/10 x 160)
   fp = 160 - 16
   fp = 144

You can’t add 20% here together with 10% to make 30% because the two discounts are not of the same whole but the second discount is subtracted from the first reduced price.

Note that two reductions end up actually being less than just the sum of the separate discounts. A 20% reduction followed by a 10% reduction isn’t a total of a 30% reduction but actually 28% because the second reduction is of a smaller starting point. Likewise two percent increases are more than just the sum of the separate percent increases. That’s because the second increase is of a higher starting point because the first increase already happened. You can use this rule to guess on some GMAT questions with combining percents.

Question 35

“Percent Greater Than” and “Percent Less Than” on GMAT:

Answer 35

Remember: In these questions it’s the number that follows the word “than” to which the percent is applied.

Trick: Convert “Percent greater than” or “Percent less than” to “Percent of.”

I.E. “x% greater than” = “(100 + x)% of”
“x% less than” = “(100 - x)% of”

And “of” always means multiply.

Example 1:
E.g. What number is 15% less than 60?

Dissect the question and turn it into and an equation:

n = 60 - (15% of 60)
n = 60 - (15% x 60)
n = 60 - (15/100 x 60)
n = 60 - 9
n = 51

You could also just calculate 85% of 60 since 100% - 15% is 85%:
85% x 60 = 85/100 x 60 = 85/10 x 6 = 51

Example 2:
What is the price of a television that costs 28% more than a $50 radio?

t = $50 + (28% of $50)
t = 50 + (28/100 x 50)
t = 50 + 14
t = 64

Question 36

Shortcut to Solving Percent Problems:

Answer 36

If you are being asked 30 is what percent of 50 a quick way of solving is to calculate:

30/50 = 60/100 = 60%

Because when you calculate the long way:

30 = 50 (x/100)
30/50 = x/100
(30/50) 100 = x

you are essentially just calculating 30/50 and converting it into percent by multiplying by 100.

Don’t forget that whenever you have a fraction with denominator 100, the numerator is the percent value.

E.g. 115.5 is what percent of 100?

shortcut calculation: 115.5/100 = 115.5%

So you can ignore the denominator when it’s 100 because that is already the percent form and so the numerator is the value for the percent. That’s because percent means “per 100.”

Question 37

Easy Percent Calculations:

Answer 37

If asked what percent of 30 is 150 you can set equation:

30 x y/100 = 150
3y = 1500
y = 500 and that is your percentage, i.e. 500%

Or you can set up simple equation and convert into % later:

30y = 150
y = 5

to convert to percent: 5 x 100 = 500%

Question 38

Important Squares in Tens:

Answer 38

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

Question 39

Converted Roots into Exponents:

Answer 39

If you want to convert a root into an exponent for easier calculations then take the reciprocal of the root and use as exponent:

E.g. Square Root of 4: √4 and √4 = 4^1/2

Cube root of 8: 3√8 and 3√8 = 8^1/3

Question 40

Interest Rate Formulas:

Answer 40

1. Simple Interest is Interest applied only to the principal, not to the interest that has already accrued.

Simple Interest Formula:
Total of Principal and Interest = Principal x (1 + rt)

where r is the interest rate per time period expressed as a decimal and t equals the number of time periods.

2. Compound Interest is interest applied to the principal and any previously accrued interest.

Compound Interest Formula:
Total of Principal and Interest = Principal x (1 + r/n)^t

3. There could be compound interest with annual interest but payments that are not on an annual basis but for instance on a quarterly basis. In this case r equals the annual interest rate divided by the number of times per year it is applied and t equals the number of years multiplied by the number of times per year that interest is applied.

E.g. for interest that’s compounded quarterly:

Total = $100 x (1 + 0.12/4)^3x4

If someone receives 4% annual interest rate compounded quarterly it means that the person receives 1/4 of his annual interest, or 1%, every quarter, so every 3 months.

With compound interest sometimes it’s easier to use a step by step approach than the formula. Test both out and see which one is better for you.

Step by step approach:

E.g. A bank account with $100 earns 8% interest compounded quarterly. How much is in the account after 6 months?

If the interest is compounded quarterly that means that every quarter the capital increases by 8/4 = 2%. Since we are looking at 6 months, the interest will compound twice in that time period.

1. compound: 100 x 1.02 = 102
2. compound: 102 x 1.02 = 102 + 102 x 2/100 = 102 + 204/100 = 102 + 2.04 = 104.04

Question 41

Rewriting Very Low Decimals:

Answer 41

0.000064 can be rewritten as:

64 x 10^-6 because it’s the same as 64/10^6

Question 42

Decimals with Exponents:

Answer 42

REMEMBER:

0. 1^2 = 0.01
1. 1^3 = 0.001

Every exponent adds a zero when you raise 0.1 to the power.

But for example:

0.7^2 = 0.49 (here you would just do the calculation without decimals and then add the decimal points to the result. I.E 7x7 = 49 and then add 2 decimal points to get 0.49.

Question 43

Percent Increase and Decrease from Constant:

Answer 43

If you have to calculate numbers like 30% decrease of a variable, or 300% increase of a variable, you can make calculations easier by choosing 1 as the variable if that’s permissible. When you then calculate the percent increases and decreases work with decimals rather than fractions for more accurate calculation.

E.g. 30% decrease from 1 = 0.7 (if you had chosen fractions you could have said 1 - 1/3 = 2/3 but that is 0.666 which is a bit less and makes calculations less accurate)

E.g. 50% increase from 1 = 1.5

E.g. 300% increase from 1 = 4

Question 44

Terminating Decimals:

Answer 44

Terminating Decimals are decimals that when turned into fraction have nominators that are integers and denominators that are integers and only factor into 2’s and 5’s. I.e. any integers divided by a power of 2 or 5 will result in a terminating decimal. Terminating decimal means that the decimal is not infinite but ends at some point.

E.g. 0.03 = 3/100 = 3/2x2x5x5

E.g. 1/8 = 1/2x2x2

Question 45

Repeating Decimals:

Answer 45

Repeating decimals go on forever.

E.g. 7.3333333….
6.454545…

7.3333 as a fraction could be written as:

7 1/3

Question 46

Yearly Percent Increase:

Answer 46

If a number increases per year by 25% e.g. an easy way to make the calculation is to multiply that number by 1.25 (decimal value of 125%) because ultimately you are increasing the number by 125%. So if over the course of 3 year the number is increased by 25% of the previous year each year you can calculate:

End of year one: 1.25n (n is number items at beginning)
End of year two: 1.25 (1.25n)
End of year three: 1.25 (1.25 (1.25n)

At end of year three: 1.25^3n

Question 47

Five Raised to Power:

Answer 47

Memorize:

5^2 = 25
5^3 = 125
5^4 = 625

Question 48

Factors in Products of Numbers:

Answer 48

If you are asked:

What is the greatest integer k for which 3^k is a factor of p, where p is the product of integers from 1-30 inclusive, they essentially want to know how many factors of 3 there are in the product of 1 to 30 inclusive. So write down all the multiples of 3 between 1 and 30 inclusive, then count the numbers of factor 3 in each one.

Answer is: 3^14

in the multiples of 3 between 1 and 30 inclusive, the factor 3 appears 14 times.

This meant that that 3^14 is the greatest possible factor of the product build by multiplying integers 1 through 30 inclusive.

Question 49

Remember for Ratio:

Answer 49

If ratio is 2:3:4

Then you know real numbers are in ratio 2x:3x:4x
and the total number is: 9x

Question 50

Percents Expressed as Fractions:

Answer 50

Remember if A is 25% less than B then you can’s make B 125 and A 100 because 25%, or 1/4, of 125 is not 25 but more like 31.

If A is 25% less than B rather make B 100 and A 75 because 1/4 of 100 is 25.

If A is 40% more than B you can make B 100 and A 140.

Question 51

Backsolving in Questions With Ratios:

Answer 51

E.g. There are male and female students at a college party. The ratio of male to female students is 3 to 5. If 5 male students were to leave the ratio would change to 1 to 2. How many total students are at the party?

2 approaches:

1. Algebraically:

m/f = 3/5
m = 3/5 f

m - 5/f = 1/2

substitute m = 3/5 f into the second equation:

3/5f - 5/f = 1/2 
3/5f/f - 5/f = 1/2 
3/5 - 5/f = 1/2
-5/f = 1/2 -3/5
-5/f = 5/10 - 6/10
-5/f = -1/10
-5 = -1/10 f
50 = f 

and m = 3/5 x 50 = 30

so altogether there are 80 students at the party.

2. Backsolving:

Answer choices:
A) 24 B) 30 C) 48 D) 80 E) 90

total number of students based on ratio is 3 + 5 = 8. That tells you that the total number has to be a multiple of 8 otherwise you wouldn’t get integers in the calculations. So you can eliminate E and B.

Test D: 80

If there are 80 Students there are 3/8 x 80 = 30 males and 5/8 x 80 = 50 females. Another way of expressing this same calculation is to see that if there are 80 students total the ratio multiplier would be 10 (because 10x8 = 80) and so you have to calculate 3x10 to get 30 male students and 5x10 to get 50 female students. Next, if 5 males leave, 25 males remain and the ratio of males to females changes to 25/50 = 1/2. That works! CORRECT!
You can stop the calculation here as you have found the answer.

Question 52

Converting Between Fractions, Decimals, Percents:

Answer 52

To convert into percents try to get denominator to equal 100.

3/8 = 3/8 x 12.5/12.5 = 37.5/100 = 0.375 = 37.5%

0. 375 = 375/1000 = 15/40 = 3/8
1. 5% = 37.5/100 = 3/8 (because denominator and numerator were divided by 12.5)

Question 53

Fractions, Decimals, Percents: When to Use What?

Answer 53

Decimals and percents work well with subtraction and addition, as well as for estimating and comparing numbers.

Fractions work well with multiplication and division.

Question 54

Estimation on Percent-Fraction Problems

Answer 54

If the values in the answer choices are far apart, estimating is a good strategy to save time on GMAT.

So if you are for instance asked what is 65% of something, you know that 2/3 is 66/7% and much easier to do multiplications as needed in percentage computations with. So, if answer choices are far enough apart. Just do 2/3 of the number and keep in mind that you rounded up so the actual number will be a little less than your result.

Question 55

Expressing Fractions in Percents:

Answer 55

1000/10 = 100 and in order to express 100 as a percent multiply the number by 100 and set 100 as the denominator of the fraction:

10,000/100 = 10,000%

25/8 = 25 x 12.5/8 x 12.5 = 315.5/100 = 312.5%

Or another way of calculating is using fractions:

25/8 = 3 1/8 you should memorize that 1/8 is 0.125
so 3 1/8 is 3.125 and expressed as a percent that is 312.5/100 = 312.5%

Question 56

Non-Integers and Digits

Answer 56

Non-integers are not generally classified by the number of digits they contain, since you can always add any number of zeros on the right side of the decimal point.

E.g. 9.1 = 9.10 = 9.10000…

Question 57

Powers of 10 Expressed in Fractions and Decimals:

Answer 57

10^3 = 1000 = 1000.0
10^2 = 100 = 100.0
10^1 = 10 = 10,0
10^-1 = 1/10 = 0.1
10^-2 = 1/100 = 0.01
10^-3 = 1/1000 = 0.001

When multiplying by a power of 10 move the decimal point to the right by the value of the power.

E.g. 3.985 x 10^3 = 3985

2.57 x 10^6 = 2,570,000

When dividing by a power of 10 move the decimal point to the left by the value of the power.

E.g. 398.5/100 = 3.985

14.29/10^5 = 0.0001429

Note: multiplying by a negative power of 10 shifts the decimal point to the left, making the number smaller. Dividing by a negative power of 10 shifts the decimal point to the right making the number larger.

E.g. 6,782.01 x 10^-3 = 6,782.01/1000 = 6.78201

53.0447/10^-2 = 53.0447/1/100 = 53.0447 x 100 = 5304.47

Question 58

Multiplying Large Numbers With Decimals:

Answer 58

Trick to simplify calculation: Move decimals the same number of places in opposite direction. This is like trading decimal places in one number for decimal places in the other. The multiplication is simplified but the values stay the same.

0.0003 x 40,000 = 3 x 4 = 12

Question 59

Division with Decimals:

Answer 59

If the decimal is in the numerator only you can divide normally:

12.42 ÷ 3 = 12÷3 + 0.42÷3 = 4 + 0.14 = 4.14

If the decimal is in the denominator, change the numbers by moving the decimal point until there is no decimal in the denominator:

12.42÷0.3 = 124.2÷3 = 120÷3 + 4.2÷3 = 40 + 1.4 = 41.4

or 1242÷30 = 1200÷30 + 42÷30 = 40 + 1.4 = 41.4

Question 60

Units Digits and Tens Digits:

Answer 60

E.g. if x is a positive integer, what is the units digit of x?

(1) The units digit of x/10 is 4.
(2) The tens digit of 10x is 5.

This question is best solved by testing cases.

(1) e.g. x = 40 then 40/10 = 4 which fulfills the statement and the units digit of x would be 0. But if 44 then 44/10 = 4.4 which fulfills the statement but units digit of 4 is 4. Since there are at least two different possibilities for the units digit this is NOT SUFFICIENT.
(2) e.g. x = 5 then 10x5 = 50 which fulfills the statement and units digit of x is 5. If x = 55 then 10x55 = 550 which fulfills the statement and the units digit of x is 5 again. It turns out the units digit of x can only be 5 according to the statement because if you multiply it by 10 that units digit shifts to become the tens digit and since that is supposed to be 5, the units digit of x has to be 5. SUFFICIENT!

Question 61

Adding Numbers to Numerator and Denominator:

Answer 61

Adding the same number to the denominator and numerator brings the fraction closer to 1.

1/3
1+1/3+1 = 2/4 = 1/2
1+1/2+1 = 2/3 
2+1/3+1 = 3/4 
etc. 

1/3

Question 62

Building Percents.

Answer 62

For some percents problems it’s fastest to build percents.

E.g. what is 40% of 60?

You first calculate 10% which you can do in your head, 6, and then multiply that by 4. This is an easy and quick computation that you can do in your head with many of these problems.

REMEMBER: But with 75% it’s easier to calculate with fractions, 3/4 of a number.

Question 63

Population Increase and Decrease Problems:

Answer 63

In population problems it’s crucial to decide the starting point, i.e. the original.

E.g. if problems asks how much smaller population was in 1980 than in 1990, then the population of 1990 is the starting point, or original.

Always think of the original as 100%!

Question 64

Converting Percent Increases and Decreases:

Answer 64

It’s easier and saves steps to find the new percent of the original (instead of finding the value of the percentage and then adding or subtracting):

10% increase = 110% of original = 1.1x
10% greater than = 110% of original = 1.1x

45% decrease = 55% of original = 0.55x
45% less than = 55% of original = 0.55x

50% greater than 60 = 150% of 60 = 1.5x

Question 65

Successive Percent Change:

Answer 65

In successive percent change problems calculate the pieces separately.

E.g. a plane ticket is first increased by 25% and then goes on sale for 20% less than the new price. what is the overall percent change?

Let’s say original price is 100

New price: 125

Then goes on sale for 20% less:

125x20/100 = 5x5 = 25

So 125-25 = 100 is end price. The 20% reduction totally offsets the 25% increase in the beginning. There has been no percent change at all because:

100-100/100 = 0/100 = 0

Question 66

Multiple Ratios: Make a Common Term

Answer 66

If a problem contains ratio numbers for three different things but the numbers are given in separate two-part ratios where each ratio contains a common element but different numbers.

E.g. ratio of something is:
C:A = 3/2 and C:L = 5:4

You cant combine these two ratios to get a three-part ratio because the numbers for C, 3 and 5, are different numbers. To solve this issue, you can multiply each ratio by the other number for C, i.e. first ratio by 5 and second ratio by 3 to get C to the same number in both ratios.

C:A = 3/2 x 5/5 = 15/10
C:L = 5/4 x 3/3 = 15/12

So three-part ratio is: C:A:L = 15:10:12

The real number of things is a multiple of those numbers, 15, 10, and 12. The smallest possible number would be all multiplied by 1, meaning the multiplier would be one in which case the actual numbers would be 15, 10, and 12.

Question 67

Three-part Ratios:

Answer 67

If you have want to find the ratio of three-part ratios of the form x:y:z you don’t need actual values because you are asked for a ratio, not the actual numbers. You only need to find out the relationship between the variables, in other words you need to know the ratio of x/y AND the ratio of y:z. Once you know that, you can set up the ratio.

Question 68

Raising Decimals to Higher Powers

Answer 68

To raise decimals to powers higher than 3 there’s an easier way than do the actual calculation: Rewrite the decimal as the product of an integer and a power of 10.

E.g. 0.5 = 5 x 10^-1

0. 5^4 = (5 x 10^-1)^4
1. 5^4 = 5^4 x 10^-4
2. 5^4 = 625 x 1/10000
3. 5^4 = 0.0625

Question 69

Solving Roots of Decimals:

Answer 69

Recall that roots of decimals means the decimal raised to a fractional power. E.g. square root of a decimal means raised to power of 1/3, cube root means raised to power of 1/3. You can use that to solve problems with roots of decimals. First rewrite the number as a product of an integer that you can take the root from easy and a power of 10.

E.g. 3√0.000027

Since 0.000027 = 27 x 10^-6 you can rewrite as:

= 27^1/3 x 10^-6^1/3
= 27^1/3 x 10^-2
= 3 x 10^-2
= 0.03

Once you understand the principle you can make these calculations much quicker without writing out each step of the calculation. REMEMBER this: The number of decimal places in a cubed decimal is 3 times the number of decimal places in the original number.

0.04^3 = 0.000064 because through the cube we get 2x3 decimal places of which 4^3 = 64 are the last two digits.

Likewise the number of decimal places in a cube root is 1/3 the number of decimal places in the original number.

E.g. 3√0.000000008= 0.002 because through the square root you make out of 9 decimal places 9x1/3 = 3 decimal places and the cube root of 8 is 2 which is the last of the three decimal places.

Question 70

Dividing by 9, 99, 999, etc.

Answer 70

The number will be a repeating number, which never ends and you can tell the never-ending number from the number of the denominator:

23/99 = 0.232323...
3/11 = 27/99 = 0.272727...

Question 71

Terminating Decimals:

Answer 71

There are numbers like √2 = 1.4… and π that never end and never repeat themselves. And then there are terminating decimals, like 0.2 or 0.47. If GMAT asks about properties of certain terminating decimals remember that they can be rewritten as fractions.

E.g. 0.2 = 2/10 = 1/5

0. 47 = 47/100
1. 375 = 375/1000 = 15/40 = 3/8

REMEMBER: Positive powers of 10 are composed of only 2’s and 5’s as prime factors. Terminating decimals share the same characteristic. The rule is that if after reducing the fraction built to re-express a decimal there are only 2’s and 5’s as prime factors IN THE DENOMINATOR then the decimal will terminate. If there are any other prime factors int he denominator, then it is not terminating.