﻿ ﻿Percents - Easy Mathematics Step-by-Step

## Easy Mathematics Step-by-Step (2012)

### Chapter 7. Percents

In this chapter, you learn about percents and their relationship to fractions and decimals.

Percent Concepts

A percent is a special type of fraction with a denominator of 100. There are three ways of writing a percent: as a fraction, as a decimal, or with the special percent (%) symbol. For example, . The word percentmeans “per hundred.” The % symbol is a shorthand way of writing or 0.01. You use the % symbol in writing, but calculations require percents to be in decimal or fraction form.

The word rate is sometimes used for percent.

To remember that % means , think of the slash in the % symbol as a fraction bar and the two zeros as part of the number 100.

Changing Percents to Decimal Form

You can change a percent to decimal form by substituting 0.01 for the % symbol and multiplying. For example, 25% = 25 (0.01) = 0.25. A shortcut for this process is as follows.

Changing a Percent to Decimal Form

To change a percent to decimal form, move the decimal point two places to the left and drop the percent symbol.

The decimal point in a whole number is to the immediate right of the rightmost digit.

Problem Change the percent to decimal form.

a. 75%

b. 35.5%

c. 6%

d. 300%

Solution

a. 75%

Step 1. Move the decimal point two places to the left and drop the percent symbol.

Step 2. State the main result.

b. 35.5%

Step 1. Move the decimal point two places to the left and drop the percent symbol.

Step 2. State the main result.

c. 6%

Step 1. Move the decimal point two places to the left and drop the percent symbol.

You may have to insert zeros in order to move two decimal places to the left.

Step 2. State the main result.

Be sure to move the decimal point two places to the left when changing from a percent to a decimal number.

d. 300%

Step 1. Move the decimal point two places to the left and drop the percent symbol.

Step 2. State the main result.

Step 1. Move the decimal point two places to the left and drop the percent symbol.

A fraction in a decimal number does not occupy its own decimal place. It occupies the same decimal place as the digit to which it is attached.

Step 2. State the main result.

Changing Decimals to Percent Form

You can reverse the process just presented to change decimals to percent form as follows.

Changing a Decimal to Percent Form

To change a decimal number to percent form, move the decimal point two places to the right and attach a % symbol.

The % symbol is equivalent to two decimal places, so when your number “gives up” two decimal places, you replace them with a % symbol.

Problem Change the decimal number to percent form.

a. 0.34

b. 0.425

c. 0.09

d. 2

e. 0.0025

Solution

a. 0.34

Step 1. Move the decimal point two places to the right and attach a % symbol.

Step 2. State the main result.

b. 0.425

Step 1. Move the decimal point two places to the right and attach a % symbol.

Step 2. State the main result.

c. 0.09

Step 1. Move the decimal point two places to the right and attach a % symbol.

Step 2. State the main result.

d. 2

Step 1. Move the decimal point two places to the right and attach a % symbol.

You may have to insert zeros in order to move the decimal point two places to the right.

Step 2. State the main result.

e. 0.0025

Step 1. Move the decimal point two places to the right and attach a % symbol.

Step 2. State the main result.

Changing Percents to Fraction Form

To change a percent to fraction form, you substitute for the % symbol and multiply.

Changing a Percent to Fraction Form

To change a percent to fraction form, substitute for the % symbol and then multiply. Simplify the resulting fraction to lowest terms, if possible.

Problem Change the percent to lowest fraction form.

a. 5%

b. 25%

c. 50%

d. 100%

e. 125%

Solution

a. 5%

Step 1. Substitute for the % symbol and then multiply.

Step 2. Reduce to lowest form.

Step 3. State the main results.

b. 25%

Step 1. Substitute for the % symbol and multiply.

Step 2. Reduce to lowest form.

Step 3. State the main results.

c. 50%

Step 1. Substitute for the % symbol and multiply.

Step 2. Reduce to lowest form.

Step 3. State the main results.

d. 100%

Step 1. Substitute for the % symbol and multiply.

Step 2. Reduce to lowest form.

Step 3. State the main results.

e. 125%

Step 1. Substitute for the % symbol and multiply.

Step 2. Reduce to lowest form.

Step 3. State the main results.

Step 1. Substitute for the % symbol and multiply.

Step 2. State the main results.

In changing a percent to a fraction, if the percent contains a mixed fraction, change the mixed fraction to an improper fraction as an initial step.

Problem Change the percent to lowest fraction form.

Solution

Step 1. Change to an improper fraction.

Step 2. Substitute for the % symbol and multiply.

Step 3. Reduce to lowest form.

Step 4. State the main results.

Step 1. Change to an improper fraction.

Step 2. Substitute for the % symbol and multiply.

Step 3. Reduce to lowest form.

Step 4. State the main results.

In changing a percent to a fraction, if the percent contains a decimal fraction, before reducing to lowest form, multiply the numerator and denominator by an appropriate power of 10 (10, 100, 1000, etc.) to remove the decimal in the numerator. Then simplify the resulting fraction, if possible.

Problem Change the percent to lowest fraction form.

a. 12.5%

b. 0.25%

Solution

a. 12.5%

Step 1. Substitute for the % symbol and multiply.

Step 2. Multiply the numerator and denominator by 10 to remove the decimal in the numerator.

Step 3. Reduce to lowest form.

Step 4. State the main results.

b. 0.25%

Step 1. Substitute for the % symbol and multiply.

Step 2. Multiply the numerator and denominator by 100 to remove the decimal in the numerator.

Step 3. Reduce to lowest form.

Step 4. State the main results.

Changing Fractions to Percent Form

An efficient way to change fractions to percent form is to first change the fractions to decimals. Here is the process.

Changing a Fraction to Percent Form

To change a fraction to percent form, first convert the fraction to a decimal number by performing the indicated division to at least two decimal places and then change the resulting decimal number to percent form. When the quotient is a repeating decimal number, write the remainder as a fraction like this: .

Note: Even if the decimal representation terminates before two decimal places, you should carry the division to at least two decimal places.

Problem Change the fraction to percent form.

Solution

Step 1. Perform the indicated division.

Step 2. Change 0.80 to percent form.

Step 3. State the main results.

Step 1. Perform the indicated division.

Step 2. Change 0.375 to percent form.

Step 3. State the main results.

Step 1. Perform the indicated division.

Step 2. Change 0.0025 to percent form.

Step 3. State the main results.

Step 1. Perform the indicated division.

Step 2. Change to percent form.

Step 3. State the main results.

. The in does not occupy a decimal place of its own.

Common Percents to Know

Here is a list of common percents and their decimal and fraction forms. Memorizing these relationships enhances your understanding of fractions, decimals, and percents.

Always remember that a % symbol means or 0.01.

Exercise 7

For 1–4, change the percent to decimal form.

1. 65%

2. 25.5%

3. 5%

4. 400%

For 5–9, change the decimal number to percent form.

5. 0.72

6. 0.325

7. 0.08

8. 10

9. 0.0075

For 10–13, change the percent to lowest fraction form.

10. 150%

12. 0.75%

For 14 and 15, change the fraction to percent form.

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