## 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 *percent*means “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*.