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, Image. The word percentmeans “per hundred.” The % symbol is a shorthand way of writing Image 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 Image, 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.

Image

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%

Image

Solution

a. 75%

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

Image

Step 2. State the main result.

Image

b. 35.5%

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

Image

Step 2. State the main result.

Image

c. 6%

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

Image

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

Step 2. State the main result.

Image

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

d. 300%

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

Image

Step 2. State the main result.

Image

Image

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

Image

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.

Image

Changing Decimals to Percent Form

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

Image

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

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

Image

Step 2. State the main result.

Image

b. 0.425

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

Image

Step 2. State the main result.

Image

c. 0.09

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

Image

Step 2. State the main result.

Image

d. 2

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

Image

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

Step 2. State the main result.

Image

e. 0.0025

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

Image

Step 2. State the main result.

Image

Changing Percents to Fraction Form

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

Image

Changing a Percent to Fraction Form

To change a percent to fraction form, substitute Image 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%

Image

Solution

a. 5%

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

Image

Step 2. Reduce to lowest form.

Image

Step 3. State the main results.

Image

b. 25%

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

Image

Step 2. Reduce to lowest form.

Image

Step 3. State the main results.

Image

c. 50%

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

Image

Step 2. Reduce to lowest form.

Image

Step 3. State the main results.

Image

d. 100%

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

Image

Step 2. Reduce to lowest form.

Image

Step 3. State the main results.

Image

e. 125%

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

Image

Step 2. Reduce to lowest form.

Image

Step 3. State the main results.

Image

Image

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

Image

Step 2. State the main results.

Image

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.

Image

Image

Solution

Image

Image Step 1. Change Image to an improper fraction.

Image

Step 2. Substitute Image for the % symbol and multiply.

Image

Step 3. Reduce to lowest form.

Image

Step 4. State the main results.

Image

Image

Image Step 1. Change Image to an improper fraction.

Image

Step 2. Substitute Image for the % symbol and multiply.

Image

Step 3. Reduce to lowest form.

Image

Step 4. State the main results.

Image

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%

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

Image

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

Image

Step 3. Reduce to lowest form.

Image

Step 4. State the main results.

Image

b. 0.25%

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

Image

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

Image

Step 3. Reduce to lowest form.

Image

Step 4. State the main results.

Image

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.

Image

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: Image.

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.

Image

Image

Image

Image

Solution

Image

Image Step 1. Perform the indicated division.

Image

Step 2. Change 0.80 to percent form.

Image

Step 3. State the main results.

Image

Image

Image Step 1. Perform the indicated division.

Image

Step 2. Change 0.375 to percent form.

Image

Step 3. State the main results.

Image

Image

Image Step 1. Perform the indicated division.

Image

Step 2. Change 0.0025 to percent form.

Image

Step 3. State the main results.

Image

Image

Image Step 1. Perform the indicated division.

Image

Step 2. Change Image to percent form.

Image

Step 3. State the main results.

Image

Image. The Image in Image 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.

Image

Always remember that a % symbol means Image or 0.01.

Image 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%

Image

12. 0.75%

Image

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

Image

Image