#### 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 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

43/50 = 86/100 = 0.86

13/16 = 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 = 55/100 = 11/20

2.18 = 218/100 = 109/50 = 2\dfrac{9}{50} **or ** 2.18 = 2\dfrac{18}{100} = 2\dfrac{9}{50}

0.123 = 12.3/100 = 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 = 34/100 = 17/50

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

18/90 = 2/10 = 0.2 = 20%

## Example

Convert 4% into a decimal and a fraction in lowest terms.

### 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.

4/100 = 4/4 × 25 = 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.

## Example

What is 30% of 210?

### 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.

## Example

What percent of 24 is 9?

### Solution

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

part/whole = percent/100 so 9/24 = *x*/100

*Method 1*

900 = 24*x* Use cross multiplication.

*x* = 900/24 = 3 × 3 × 4 × 25/2 × 3 × 4 = 3 × 25/2 = 75/2 = 37.5

*Method 2 *

Start by simplifying the fraction.

9/24 = 3/8

Since you know 1/8 = 0.125, you can quickly calculate 3/8 = 3(0.125) = 0.375 = 37.5%

**Percent Multipliers**

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#### 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.

## Example

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.

change/original = percent/100

*so*

new – original/original = percent/100

45 – 60/60 = x/100

-15/60 = 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”).

15/60 = 1/4 = 25/100 = 25%

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

The price dropped by 25%.

## Example

The monthly cost of cable internet service went from the introductory price of $39.99 a month to $50.99 a month. What was the percentage increase from the introductory price?

### Solution

The original price here is the introductory price.

change/original = percent/100

*which is*

new – original/original = percent/100

50.99 – 39.99/39.99 = *x*/100

11/39.99 = *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 – original/original = percent/100

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

## Example

A pair of aerobic shoes was priced $115 and is now discounted to $69. What is the percentage discount?

### Solution

new – original/original = percent/100

69 – 115/115 = *x*/100

-46/115 = *x*/100

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

46/115 = 2 × 23/5 × 23 = 2/5 = 40/100 = *x*/100

*x* = 40

The discount is 40%.

## Example

A pair of aerobic shoes is purchased at wholesale for $69 and sold by the store for $115. What is the percentage markup?

### Solution

new – original/original= percent/100

115 – 69/69 = *x*/100

46/69 =* x*/100

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

46/69 = 2 × 23/3 × 23 = 2/3 = 67/100 = *x*/100

*x* = 67

The markup is 67%.

## Example

An employee is to mark up the price of a piece of jewelry by 120%. If its wholesale cost was $110, what will be its selling price?

### 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

## Example

A college bookstore purchases trade books on a 30% margin, i.e., it purchases a trade book for 30% less than its retail price. What is the percentage markup from the wholesale price?

### 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 – original/original = percent/100

retail – wholesale/wholesale = markup/100

100 – 70/70 = markup/100

30/70 = 3/7 = 42.8/100 = markup/100

So the markup from wholesale to retail is 43%.

## Example

Find the number of residents in a city if 20% of them, or 6200 people, ride bicycles.

### 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

(2/10)*R *= 6200

To divide by a fraction, multiply by the reciprocal.

*R* = 6200 × (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.

## Example

Kathy buys a bike for $240 after a 40% markdown. What was the original 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* = 240/0.6 = 240/6/10 = 240 × 10/6 = 400

The original price was $400.

## Example

A sweater, originally $80, is on sale for 25% off. What is the sale price?

### 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 = 3/4 × 80 = 60

The sale price is $60.

## Example

Kent pays 20% tax on income between $10,000 and $20,000 and 30% on income over $20,000. The first $10,000 is tax-free. If he paid $14,000 in taxes, what was his income?

### 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 × (10/3) = 60,000

Kent’s income was $60,000.

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#### Mixture Problems

## Example

How many gallons of pure water must be added to 100 gallons of a 4% saline solution to produce a 1% saline solution?

### 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* = 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.

## Example

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

## Example

Joe’s portfolio lost 80% of its value, then gained back 10% of its value. What was Joe’s final percentage loss?

### 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.

## Example

A backpack was marked for sale at 30% off, and an additional 20% was taken off at the register. If the original price was $50, what was the final purchase price?

### 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 1/8 = 0.125, then 3/8 = 3 × 0.125 = .375

1⁄6 = 16.66%

5⁄6 = 83.66%

1⁄7 = 14.28%

2⁄7 = 28.56%

3⁄7 = 42.85%

4⁄7 = 57.14%

5⁄7 = 71.42%

6⁄7 = 85.71%

1⁄8 = 0.125 = 12.5%

3⁄8 = 0.375 = 37.5%

5⁄8 = 0.625 = 62.5%

7⁄8 = 0.875 = 87.5%

1⁄9 = 11.1%

2⁄9 = 22.2%

4⁄9 = 44.4%

5⁄9 = 55.6%

7⁄9 = 77.7%

8⁄9 = 88.8%

1⁄11 = 9.09%

2⁄11 = 18.18%

3⁄11 = 27.27%

4⁄11 = 36.36%

5⁄11 = 45.45%

6⁄11 = 54.54%

7⁄11 = 36.36%

8⁄11 = 72.72%

9⁄12 = 81.81%

10⁄11 = 90.90%

1⁄12 = 8.3%

5⁄12 = 41.7%

7⁄12 = 58.3%

11⁄12 = 91.7%

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