#### Percent

#### Percent & Conversions

The word percent means per 100. It is a fraction whose denominator is 100. For example, 26% is equivalent to the fraction \dfrac{26}{100}.

A percentage is used to compare values. It can describe the relationship of a part to a whole, or it can describe a magnitude of change.

**Decimal to Percent**

To change a decimal number to a percentage, multiply by 100; this moves the decimal point 2 places to the right. Then add the percent symbol.

0.32 = 32%

0.07 = 7%

1.75 = 175%

**Percent to Decimal**

To change a percentage to a decimal, divide by 100; this moves the decimal point 2 places to the left. Then remove the percent symbol.

25% = 0.25

90% = 0.90 = 0.9

32.1% = 0.321

100% = 1

**Fraction to Decimal**

You can convert a fraction to a decimal by dividing the numerator by the denominator. Sometimes a quicker method is to change the fraction so it has 100 as the denominator.

{21}{100} = 0.21

\dfrac{43}{50} = \dfrac{86}{100} = 0.86

\dfrac{13}{16} = \dfrac{13}{16} = 0.8125

**Decimal to Fraction**

To convert decimal to a fraction, move the decimal point 2 places to the right and use 100 as the denominator. Then simplify the fraction. If there is still a decimal in the numerator, move the decimal point to the right as needed, and add a 0 to the denominator for each place value.

0.55 = \dfrac{55}{100} = \dfrac{11}{20}

2.18 = \dfrac{218}{100} = \dfrac{109}{50} = 2 \,\dfrac{9}{50} **or ** 2.18 = 2

\dfrac{18}{100} = 2\dfrac{9}{50}

0.123 = \dfrac{12.3}{100} = \dfrac{123}{1000}

Note: There is a chart at the end of this section with common fraction and decimal conversion values.

**Fraction and Percent **

To change a percentage to a fraction, change the percentage to a decimal, then the decimal to a fraction.

34% = 0.34 = \dfrac{34}{100} = \dfrac{17}{50}

Correspondingly, to change a fraction to a percentage, change the fraction to a decimal, then the decimal to percent.

\dfrac{18}{90} = \dfrac{2}{10} = 0.2 = 20%

### Solution

To convert 4% into a decimal, move the decimal point two places to the left. This requires adding a leading 0.

4% = 0.04

To express 4% as a fraction, put it over a denominator of 100, and then simplify the fraction.

\dfrac{4}{100} = \dfrac{4}{4} × 25 = \dfrac{1}{25}

#### Multiplying by Percent

Multiplying by a percentage is a way to find a “part” of the original. Multiplying by a percentage is the same as multiplying by the decimal equivalent.

15% of 40 = 15% × 40 = 0.15 × 40 = 6

8% of 25 = 8% × 25 = 0.08 × 25 = 2

*Quick trick:* Multiplying by 10%

To get 10% of anything, simply slide the decimal point one place to the left.

3456 × 10% = 345.6

For 20%, slide the decimal point one place to the left, then multiply by 2.

3456 × 20% = 345.6 × 2 = 691.2

The same pattern holds true for 30%, 40%, 60%, 70%, 80% and 90%. But for 50%, simply divide by 2.

### Solution

You are trying to find the “part” of 210 that is equal to 30%.

30% of 210 = 210 × 30% Change to multiplication.

= 21 × 3 Slide the decimal point one place to the left, then multiply by 3.

= 63

30% of 210 is 63.

You may also be given two numbers (a part and a whole) and asked to find the percentage of the whole represented by the part. One way to do this is to use a proportion, which sets 2 fractions equal to each other. Change the given numbers to a fraction and set it equal to a fraction with denominator 100. Solve for the numerator and that will be the percentage.

### Solution

You are trying to find the percent that matches the given fraction.

\dfrac{part}{whole} = \dfrac{percent}{100} so \dfrac{9}{24} =\dfrac{\textit{x}}{100}

*Method 1*

900 = 24*x* Use cross multiplication.

*x* = \dfrac{900}{24} = 3 × 3 × 4 × \dfrac{25}{2} × 3 × 4 = 3 × \dfrac{25}{2} = \dfrac{75}{2} = 37.5

*Method 2 *

Start by simplifying the fraction.

\dfrac{9}{24} = \dfrac{3}{8}

Since you know \dfrac{1}{8} = 0.125, you can quickly calculate \dfrac{3}{8} = 3(0.125) = 0.375 = 37.5%

**Percent Multipliers**

#### Video Quiz

#### Percent

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8 questions with video explanations

100 seconds per question

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https://www.youtube.com/watch?v=NIuATVe8OfY

#### Percent Increase and Decrease

Percentages can be used to describe the magnitude of change. To find the percentage of change, you compare the new amount to the original amount.

**Percent increase:** If the price of a $30 item increases by 10%, the new price is the original $30 plus 10% of the $30 original, $33. This is 110% or 1.1 times the original price. “Increase” may go by other names such as the “markup” from “wholesale” (cost from the factory) to “retail” (cost to the public).

**Percent decrease:** If the price of an item decreases by 10%, the new price is the original $30 minus 10% of the $30 original, or $27. This is 100% – 10% = 90% or 0.9 times the original price. “Decrease” may go by other names such as “sale” or “discount.”

#### How much is a 100% increase?

An increase of 100% is the original plus 100% of the original, which is the same as doubling or multiplying by 2. So 30 increased by 100% is 60 because you are adding 100% of 30 to 30.

If the price of a stock drops from $60 to $45, what is the percentage decrease?

### Solution

You are trying to find the percent decrease based on the original value and the new value.

\dfrac{change}{original} = \dfrac{percent}{100}

*so*

new – \dfrac{original}{original} = \dfrac{percent}{100}

45 – \dfrac{60}{60} = \dfrac{\textit{x}}{100}

\\[2ex]\dfrac{-15}{60} = \dfrac{x}{100}

The negative tells you this is a decrease. You can leave it out of the rest of the calculations as long as you remember to use the correct terminology (e.g., “decreased” or “dropped”).

\dfrac{15}{60} = \dfrac{1}{4} = \dfrac{25}{100} = 25%

For this fraction, it is easier to simplify than cross-multiply.

The price dropped by 25%.

### Solution

The original price here is the introductory price.

\dfrac{change}{original} = \dfrac{percent}{100}

*which is*

new – \dfrac{original}{original} = \dfrac{percent}{100}

50.99 – \dfrac{39.99}{39.99} = \dfrac{\textit{x}}{100}

\\[2ex]\dfrac{11}{39.99} = \dfrac{\textit{x}}{100}

39.99x = 1100 (Round and use 40 instead of 39.99)

*x* = 27.5

The price increased by 27.5%.

#### Discounts and Markups

A percentage can be used to apply the same proportion of change to multiple values.

Discount is the decrease in price of an item when the price is decreased by a certain percentage.

Markup is the increase in price when the cost of an item is increased by a certain percentage. The following examples illustrate this concept.

For markups and discounts, calculate:

new – \dfrac{original}{original} = \dfrac{percent}{100}

If the value is negative, it is a discount. If the value is positive, it is a markup.

### Solution

new – \dfrac{original}{original} = \dfrac{percent}{100}

69 – \dfrac{115}{115} = \dfrac{\textit{x}}{100}

\\[2ex]\dfrac{-46}{115} = \dfrac{\textit{x}}{100}

The negative tells you this is a decrease. You can leave it out of the rest of the calculations.

\dfrac{46}{115} = 2 × \dfrac{23}{5} × 23 = \dfrac{2}{5} = \dfrac{40}{100} = \dfrac{\textit{x}}{100}\\[2ex]*x* = 40

The discount is 40%.

### Solution

new – \dfrac{original}{original} = \dfrac{percent}{100}

115 – \dfrac{69}{69} = \dfrac{\textit{x}}{100}\\[2ex]

\dfrac{46}{69} = \dfrac{\textit{x}}{100}

Notice the original price here is $69, not $115.

\dfrac{46}{69} = 2 × \dfrac{23}{3} × 23 = \dfrac{2}{3} = \dfrac{67}{100} = \dfrac{\textit{x}}{100}\\[2ex]

*x* = 67

The markup is 67%.

### Solution

Notice that the price is being marked up *by* 120%, not *to* 120%.

The amount of the markup is 120% of $110 so it’s 1.2 × 110 = $132.

The selling price is then the original price plus the markup, so $110 + $132 = $242.

Another way to calculate the selling price is 120% + 100% = 220%, so 2.2 × $110 = $242

### Solution

The wholesale price is the retail price minus 30% of the retail price.

wholesale = retail – (30% of retail) = (100% – 30%) × retail = 70% × retail

So the wholesale price is 70% of the retail price.

But don’t stop there. It is a common GMAT trick to require you to use the result of one part of a question to get the final answer.

This question is asking for the markup from wholesale to retail.

To make calculations easier, use $100 rather than a variable as the retail price of a trade book. Then the wholesale is 70% × retail = 70% × $100 = $70.

new – \dfrac{original}{original} = \dfrac{percent}{100}

retail – \dfrac{wholesale}{wholesale} = \dfrac{markup}{100}

100 – \dfrac{70}{70} = \dfrac{markup}{100}

\\[2ex]\dfrac{30}{70} = \dfrac{3}{7} = \dfrac{42.8}{100} = \dfrac{markup}{100}

So the markup from wholesale to retail is 43%.

### Solution

Let *R* be the number of residents. Translate the words into an equation.

20% of *R* is 6200, so 0.2*R* = 6200

Change the decimal to a fraction.

0.2*R* = 6200

(\dfrac{2}{10})*R *= 6200

To divide by a fraction, multiply by the reciprocal.

*R* = 6200 × (\dfrac{10}{2}) = 62,000 = 31,000

The city has 31,000 residents.

#### Sale Prices

In determining sale prices, be careful not to mix up the amount taken off the original price with the new, sale price.

### Solution

Since the markdown is 40%, the purchase price is 60% of the original price.

100% – 40% = 60%

purchase price = 60% of original price

Let *x* be the original price.

0.6*x* = 240

*x* = \dfrac{240}{0.6} = \dfrac{\dfrac{240}{6}}{10} = 240 × \dfrac{10}{6} = 400

The original price was $400.

### Solution

Since the markdown is 25%, the purchase price is 75% of the original price.

100% – 25% = 75%

sale price = 75% of original price

Let *s* be the sale price.

0.75(80) = *s*

*s* = 0.75 × 80 = \dfrac{3}{4} × 80 = 60

The sale price is $60.

### Solution

Let Kent’s income be *k*. Divide his income into the tax brackets:

- For $0 to $10,000, there is no tax.
- The $10,000 between $10,000 to $20,000 is taxed at 20%.
- Income beyond $20,000 is taxed at 30%.

So, the equation is:

10,000 (0%) + 10,000 (20%) + (*k* – 20,000)(30%) = 14,000

0 + 2000 + 0.3(*k* – 20,000) = 14,000

2000 + 0.3*k* – 6000 = 14,000

0.3*k* = 18,000

*k* = 18,000 × (\dfrac{10}{3}) = 60,000

Kent’s income was $60,000.

#### Video Quiz

#### Percent

Best viewed in landscape mode

9 questions with video explanations

100 seconds per question

# Are you sure you want to refresh the question?

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https://www.youtube.com/watch?v=Elm2C1LXfUI

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https://www.youtube.com/watch?v=ZqIv50jT6Fc

https://www.youtube.com/watch?v=aKx7rMfwvtw

#### Mixture Problems

### Solution

Let *x* be the gallons of pure water to be added.

In 100 gallons of a 4% saline solution, there are:

0.04(100) = 4 gallons of salt.

In the 1% solution, the total number of gallons will be 100 + *x*. The amount of salt will remain constant at 4 gallons.

0.01(100 + *x*) = 4

1 + 0.01*x* = 4

0.01*x* = 3

*x* = \dfrac{3}{0.01} = 300

300 gallons of water need to be added to produce a 1% saline solution.

#### Taking a Percentage of a Percentage

What happens if you take a percentage of a number and then take a percentage of that new value? You are just multiplying the new total by a second percentage. This is a common GMAT trick.

If the price of a stock starts at $100, increases by 10% during the first year, then increases again by another 20% in the second year, what is the stock’s final price?

### Solution

In the first year, $100 × 110% = $110.

For the second year, $110 × 120% = $132.

The stock price would be $132.

Notice that this is not the same price change as 10% + 20% = 30%.

A 30% increase would have been resulted in a $130 price.

#### Fractions and Percentile

### Solution

Use $100 for the original value to help solve the question.

The portfolio lost 80%.

100% − 80% = 20%, so $20 was left.

That $20 gained back 10% of its value, so $20 × 110% = $22.

So 22% is the percentage of the original investment that he still has.

Subtract 100% − 22%. The final percentage loss is 78%.

Notice that this is not 100% − 80% + 10% = -70%, or a 70% loss.

### Solution

The discount is **not** 30% + 20% = 50%.

30% off means you pay 70%.

Another 20% off means paying 80% of the 70%.

$50 × 0.8 × 0.7 = $50 × 0.56 = $28

A way to check this is to calculate the dollar amount coming off the price.

30% off of $50 = 0.3 × $50 = $15. So the purchase price before getting to the register is $50 × $15 = $35.

20% off of $35 = 0.2 × $35 = $7

So the final purchase price is $35 × $7 = $28.

#### Common Conversions

1 = 100%

3⁄4 = 75%

1⁄2 = 50%

1⁄4 = 25%

1⁄10 = 10%

3⁄10 = 30%

7⁄10 = 70%

9⁄10 = 90%

1⁄5 = 2⁄10 = 20%

2⁄5 = 4⁄10 = 40%

3⁄5 = 6⁄10 = 60%

4⁄5 = 8⁄10 = 80%

1⁄3 = 33.33

2⁄3 = 66.66

For many other denominators, you can remember the first value then multiply to get other values.

For example, if you remember that \dfrac{1}{8} = 0.125, then \dfrac{3}{8} = 3 × 0.125 = .375

\dfrac{1}{6} = 16.66%

\dfrac{5}{6} = 83.66%

\dfrac{1}{7} = 14.28%

\dfrac{2}{7} = 28.56%

\dfrac{3}{7} = 42.85%

\dfrac{4}{7} = 57.14%

\dfrac{5}{7} = 71.42%

\dfrac{6}{7} = 85.71%

\dfrac{1}{8} = 0.125 = 12.5%

\dfrac{3}{8} = 0.375 = 37.5%

\dfrac{5}{8} = 0.625 = 62.5%

\dfrac{7}{8} = 0.875 = 87.5%

\dfrac{1}{9} = 11.1%

\dfrac{2}{9} = 22.2%

\dfrac{4}{9} = 44.4%

\dfrac{5}{9} = 55.6%

\dfrac{7}{9} = 77.7%

\dfrac{8}{9} = 88.8%

\dfrac{1}{11} = 9.09%

\dfrac{2}{11} = 18.18%

\dfrac{3}{11} = 27.27%

\dfrac{4}{11} = 36.36%

\dfrac{5}{11} = 45.45%

\dfrac{6}{11} = 54.54%

\dfrac{7}{11} = 36.36%

\dfrac{8}{11} = 72.72%

\dfrac{9}{12} = 81.81%

\dfrac{10}{11} = 90.90%

\dfrac{1}{12} = 8.3%

\dfrac{5}{12} = 41.7%

\dfrac{7}{12} = 58.3%

\dfrac{11}{12} = 91.7%

#### Video Quiz

#### Percent

Best viewed in landscape mode

8 questions with video explanations

100 seconds per question

# Are you sure you want to refresh the question?

https://www.youtube.com/watch?v=eaD06pD53vg

https://www.youtube.com/watch?v=LSYy7QsI8eI

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https://www.youtube.com/watch?v=WInVBeI_jk8

https://www.youtube.com/watch?v=W4LrWRjEy_4

https://www.youtube.com/watch?v=CM3y66gjqz0

https://www.youtube.com/watch?v=vAXsdJrGBCo

https://www.youtube.com/watch?v=0Y6W6iu8HTw&list=PL6F58D00ADB3C0A85&index=8

**Before attempting these problems, be sure to review this section on data sufficiency questions.**

https://www.youtube.com/watch?v=-8BM6V4jTSI

https://www.youtube.com/watch?v=60DL_O4ubfk

https://www.youtube.com/watch?v=ydTtcf9Vxd8

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https://www.youtube.com/watch?v=F4So7Lb55qM

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https://www.youtube.com/watch?v=NZ4g7TMJPJA

https://www.youtube.com/watch?v=7kR9ZfZHLO0