LESSON: Fractions and Decimals

READ: Write Fractions and Mixed Numbers as Repeating Decimals

Write Fractions and Mixed Numbers as Repeating Decimals

By now you’ve gotten the hang of converting fractions to decimals. So far, we have been working with what are known as terminating decimals, or decimals that have an end like 0.75 or 0.5.

One reason that we sometimes use fractions instead of decimals is because some decimals are repeating decimals, or decimals that go on forever. If you try to find a decimal for $\frac{1}{3}$ by dividing, you can divide forever because $\frac{1}{3}$ written as a decimal $= 0.3333333333 ....$ It goes on and on. That’s why we usually just simply write a line above the number that repeats. For $\frac{1}{3}$ , we write: $0.\overline{3}$ . Let’s check out some examples involving repeating decimals.

Example

Write $\frac{5}{6}$ using decimals

First, we rewrite $\frac{5}{6}$ as the division problem $5 \div 6$ . We already know that we will have to go on the right side of the decimal point, so we are going to begin by dividing 6 into 5.0.

Six goes into 5.0 .8 times, but we have the remainder of .2. Six goes into 0.2 .03 times and we have a remainder of .02. Since 6 always goes into 20 three times, $(3 \cdot 6 = 18)$ and there will always be a remainder of 2, we can see that it will never evenly divide.

If you keep dividing, you will get 0.83333333333.... forever and ever.

Our final answer is $0.8\overline{3}$ .

What about mixed numbers?

Well, there are some mixed numbers where the fraction part is a repeating decimal. Let’s look at an example.

Example

Write $2 \frac{2}{3}$ using decimals.

Just as we did with the terminating decimals, we are going to leave the whole number, 2 to the side before we are ready to add it to the final answer. So, we are simply solving for the decimal equivalent of $\frac{2}{3}$ . We write the division problem $2.0 \div 3$ . How many times does 3 go into 2.0? It goes into 2.0 0.6 times.

We have 0.20 as the remainder. How many times does 3 go into 0.20? The answer is 0.06 times.

Are you noticing a pattern here? It is obvious that there will always be a remainder whether we divide 3 into 2.0, 0.2, 0.02, 0.002, or 0.0002 and on and on. Clearly $\frac{2}{3}$ is a repeating decimal.

For our final answer we write $2.\overline{6}$ .