## Learning Math: Number and Operations

# Fractions and Decimals Part B: Decimals to Fractions (30 minutes)

In Part A of this session, you learned that the decimal representation for every rational number was either a terminating or a repeating decimal. You also learned how to find the decimal representation for any rational number. Is the converse of that statement true? That is, is every terminating or repeating decimal a rational number? The answer is yes. And any non-terminating, non-repeating decimal cannot be a rational number. So, for instance, is an irrational number, as is .

So how do we find the fractional representation of a decimal? See Note 2 below. If the decimal is terminating, it’s already a fraction; you just can’t see the denominator. For example, 0.25 means 25/100, which reduces to 1/4. However, if the fraction is repeating, the process isn’t quite so simple. To find the fractional representation for 0.232323…, for example, here’s what you need to do.

First, choose a letter to represent the fraction you are looking for; let’s say, F. This fraction, F, represents your repeating decimal; that is, F = 0.232323…. Now we need to think of a way to get rid of those repeating parts. To do this, multiply F by 10^{n}, where n equals the size of the period. In this case, the period is two, so multiply F by 10^{2}, or 100. Finally, subtract F. The problem looks like this:

Since 99F = 23, F = 23/99.

This worked out nicely, didn’t it? But it does raise some questions:

**• **Why can we do this? We can do this because we subtracted equal quantities from both sides of an equation.

**• **How did we know to multiply by 100? The period of this decimal is two, so if we multiply by 10^{2}, the repeating part will “move over” two places and the repeating parts then “line up” under each other. In other words, if the period is p, we can multiply by 10^{p}.

**• **What if the decimal doesn’t repeat right away? Then we need to modify the process. Let’s look at another decimal number, 0.45545454…. We know that F represents the repeating decimal number; that is, F = 0.45545454…. Once again, we need to think of a way to get rid of those repeating parts. To do this, we again find 100 times F (because the repeating part has a period of two) and then subtract F:

So, since 99F = 45.09, F = 45.09/99.

Notice that, unlike in the previous example, the first couple of digits didn’t “line up,” which resulted in having a terminating decimal number in the numerator. To simplify this fraction that contains a decimal point, multiply both top and bottom by 100, which gives us F = 4,509/9,900 = 501/1,100.

**Problem B1
**

**a.**Find the fraction equivalent for 0.125.

**b.**Find the fraction equivalent for 0.125125125….

**Problem B2
**

**a.**Find the fraction equivalent for 0.5436.

**b.**Find the fraction equivalent for 0.543654365436….

**Problem B3
a. **Find the fraction equivalent for 0.236.

**Find the fraction equivalent for 0.2363636….**

**b.**Look carefully at which digits repeat. This is not the same type of decimal as the ones used in Problems B1 and B2.

**Video Segment
**In this video segment, Donna and Tom convert into a fraction the type of decimal number where not all the digits repeat. Watch this segment after you’ve completed Problem B3.

Think about why, when using this method for converting decimals, the fraction’s denominator is always in the form of 9 multiplied by some power of 10.

You can find this segment on the session video approximately 18 minutes and 1 second after the Annenberg Media logo.

**Problem B4
**Find the fraction equivalent for 0.111111111….

In this session, you’ve learned that in our base ten system a decimal will terminate if all the factors of its denominator are factors of some power of 10. In other words, a decimal will terminate if the prime factorization of the denominator can be reduced to powers of the prime numbers 2 and 5.

Watch this video segment to find out how the people of ancient Babylonia used a base sixty number system, which allowed them to have more terminating decimals (simply because, in a base sixty system, 60 comprises more prime factors).

You can find this segment on the session video approximately 22 minutes and 22 seconds after the Annenberg Media logo.

### Notes

**Note 2
**Why would we want to change decimals to fractions or fractions to decimals? Sometimes computations are easier with decimals, and sometimes they’re easier with fractions. For example, it might be easier to multiply by 3/4 than 0.75. On the other hand, it may be easier to divide by 2 than to multiply by 0.5.

### Solutions

**Problem B1
a. **Since this is a terminating decimal, 0.125 = 125/1,000, which can be reduced to 1/8.

**b.**Use the method of multiplication. If F = 0.125125125…, then 1,000F = 125.125125125… Subtracting F from both sides gives you 999F = 125, so F = 125/999.

**Problem B2**

**a. **Since this is a terminating decimal, 0.5436 = 5,436/10,000 = 1,359/2,500.

**b. **F = 0.54365436…. Multiply by 10,000 to get 10,000F = 5,436.54365436…. Subtracting F from both sides gives you 9,999F = 5,436, so F = 5,436/9,999, which can be reduced to 604/1,111.

**Problem B3**

**a. **0.236 = 236/1,000 = 59/250.

**b. **Note the difference; only the 36 is repeated! In this case, F = 0.2363636…. Multiply by 100 to get 100F = 23.6363636…. Subtracting F from both sides gives you 99F = 23.4, so F = 23.4/99, or 234/990, which can be reduced to 13/55.

**Problem B4
**Multiplying by 10 yields 10F = 1.111111…. Subtracting F from both sides gives you 9F = 1. Thus, F = 1/9. (It may seem counterintuitive at first that 0.11111111… = 1/9 since there are no 9s in the decimal!)