Kelly and Mallory are comparing the number of boxes of erasers that each one of them has sold. Kelly works in the store one day and Mallory works in the store the next day. They decide to write a riddle about their sales and give it to Trevor to figure out. When Trevor comes in the store on the day after Mallory has worked he finds this riddle.

Trevor is puzzled. How is he going to solve this riddle?

In this concept, you will learn to find equivalent forms of rational numbers.

### Rational Numbers as Fractions

A **rational number** is a number that can be written in fraction form. In other words, if a number can be written as a fraction, it is called a rational number. Rational numbers can also be positive or negative. For example, \begin{align*}-\frac{1}{2}\end{align*} and \begin{align*}\frac{1}{2}\end{align*} are both rational numbers.

Integers are also rational numbers. An **integer** is a whole number and can be positive or negative. But even whole numbers can be fractions; they just have 1 as the denominator. For example, 5 can be written as \begin{align*}\frac{5}{1}\end{align*} as can -4 be written as \begin{align*}-\frac{4}{1}\end{align*}. So integers are rational numbers.

Decimals can also be written as fractions if the decimal is terminating or repeating. When you understand how to convert a decimal to a fraction and a fraction to a decimal, you will be able to determine whether or not the decimal is a rational number.

Let's look at an example.

Is 0.34 a rational number?

First, look at the decimal to determine its value. Its value is 34 hundredths.

Next, you can write this decimal as a fraction with a denominator of 100.

\begin{align*}0.34 = \frac{34}{100}\end{align*}

This is a rational number since it can be written as a fraction.

Here is another example.

Is 0.343434 a rational number?

First, look at this decimal. It repeats but also terminates (ends).

Next, you know that \begin{align*}0.34=\frac{34}{100}\end{align*} so 0.343434 must be slightly more than \begin{align*}\frac{34}{100}\end{align*}. Place the numbers in the decimal over the appropriate place value. Count the decimal places. There are six decimal places, so there will be six zeros in the denominator.

\begin{align*}\frac{343434}{1000000}\end{align*}

Then, simplify.

\begin{align*}\frac{343434}{1000000} = \frac{34}{99}\end{align*}

This is a rational number since it can be written as a fraction.

Here is one more example.

Is \begin{align*}0.4526753 \ldots\end{align*} a rational number?

First, look at the number. The decimal does not terminate (notice the … just after the 3) nor does it repeat.

Next, since the decimal does not end, you cannot make it into a fraction.

The number is not a rational number. It is an irrational number. The most common example of an irrational number is Pi. Most times we use 3.14 for pi but the actual irrational number, given the name pi, has over 200 million known digits, and it does not repeat nor end.

Consider this example.

What decimal is equivalent to the fraction \begin{align*}\frac{7}{8}\end{align*}?

Divide the numerator by the denominator to find the equivalent decimal.

\begin{align*}7 \div 8 = 0.875\end{align*}

The decimal 0.875 is equivalent to the fraction \begin{align*}\frac{7}{8}\end{align*}.

To convert from a decimal to a fraction, place the numbers in the decimal over the appropriate place value.

Convert 0.875 back to a fraction.

First, count the decimal places. There are three decimal places, so there will be three zeros in the denominator.

\begin{align*}0.875 = \frac{875}{1000}\end{align*}

Next, simplify.

\begin{align*}\begin{array}{rcl} \frac{875}{1000} &=& \frac{35}{40} \\ \frac{35}{40} &=& \frac{7}{8} \end{array}\end{align*}

A percent also represents a part of a whole; percent can also be a rational number. You can convert a percent to a decimal and to a fraction. Remember that **percent** means out of 100.

Let's look at another example.

What is 30% as a fraction? As a decimal?

First, remember that percent means out of 100.

\begin{align*}30\% = \frac{30}{100}\end{align*}

Next, simplify.

\begin{align*}\frac{30}{100} = \frac{3}{10}\end{align*}

Then, write the decimal.

\begin{align*}\frac{3}{10} = 0.30\end{align*}

Percentages that have been converted to decimals and fractions can also be considered rational numbers.

### Examples

#### Example 1

Earlier, you were given a problem about Trevor and the riddle.

Trevor knows that there has to be a variable because the number of boxes that Kelly sold is unknown. He also knows that he needs to write an equation having the rational number, three-fourths in it.

First, write an equation.

Let \begin{align*}x = \text{Kelly’s boxes}\end{align*}

Mallory sold four more boxes than three fourths of Kelly’s sold,

Therefore add 4 to \begin{align*}\frac{3}{4}x\end{align*}.

Mallory sold 13 boxes.

The equation is \begin{align*}\frac{3}{4} x + 4 = 13\end{align*}

Next, subtract 4 from both sides.

\begin{align*}\begin{array}{rcl} \frac{3}{4} x &=& 13-4 \\ \frac{3}{4} x &=& 9 \end{array}\end{align*}

Then, get \begin{align*}x\end{align*} by itself by multiplying both sides by 4 and dividing by 3. This is really just multiplying both sides by the reciprocal. Remember the Inverse Property of Multiplication that states that any number multiplied by its reciprocal is equal to 1.

\begin{align*}\begin{array}{rcl} \left( \frac{4}{3} \right) \frac{3}{4} x &=& 9 \left( \frac{4}{3} \right) \\ x &=& \frac{9 \times 4}{3} \\ x &=& \frac{36}{3} \\ x &=& 12 \end{array}\end{align*}

The answer is 12.

Kelly sold 12 boxes of erasers.

#### Example 2

Is .35678921 a rational number?

Notice that this is not a repeating decimal, but it is a terminating decimal. Remember a terminating decimal has an end to it. As long as it has an end, it is a rational number.

#### Example 3

Convert 0.67 to a percent.

First, count the decimal places. There are two decimal places, so there will be two zeros in the denominator.

\begin{align*}0.67 = \frac{67}{100}\end{align*}

Next, remember that a percent is out of 100. Since your fraction is over 100, this is your percent.

The answer is 67%.

#### Example 4

Convert 45% into a fraction.

First, remember that percent means out of 100. Therefore

\begin{align*}45\% = \frac{45}{100}\end{align*}

Next, simplify.

\begin{align*}\frac{45}{100} = \frac{9}{20}\end{align*}

The answer is \begin{align*}\frac{9}{20}\end{align*}.

#### Example 5

Convert 0.185 to a percent.

First, count the decimal places. There are three decimal places, so there will be three zeros in the denominator.

\begin{align*}0.185 = \frac{185}{1000}\end{align*}

Next, simplify.

\begin{align*}\frac{185}{1000} = \frac{37}{200}\end{align*}

Then, remember that a percent is out of 100. Since your fraction is over 200, you need to divide by 2.

\begin{align*}\frac{37}{200} = \frac{18.5}{100}\end{align*}

The answer is 18.5%.

### Review

Identify whether each is a rational number or not. Write yes or no for your answer. Then identify the form of the number: integer, decimal, repeating decimal, fraction, terminating decimal, irrational number.

1. \begin{align*}0.456\end{align*}

2. \begin{align*}\frac{2}{3}\end{align*}

3. \begin{align*}-45\end{align*}

4. \begin{align*}567\end{align*}

5. \begin{align*}-8970\end{align*}

6. \begin{align*}0.3434343434\end{align*}

7. \begin{align*}0.234 \ldots\end{align*}

8. \begin{align*}0.234567\end{align*}

9. \begin{align*}-0.876\end{align*}

10. \begin{align*}-\frac{2}{7}\end{align*}

Write each rational number as a decimal and then as a percent.

11. \begin{align*}\frac{4}{5}\end{align*}

12. \begin{align*}\frac{1}{5}\end{align*}

13. \begin{align*}\frac{14}{50}\end{align*}

14. \begin{align*}\frac{12}{100}\end{align*}

15. \begin{align*}\frac{6}{25}\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 2.15.