# Fractions

### Changing Between Fractions and Decimals

#### Expressing Decimals as Fractions in Lowest Terms

Sometimes it is easier to work with fractions than to work with decimals. It is therefore important to learn how to change decimals into fractions. The secret to this task lies in understanding the meaning of place value for decimals. When, in reference to decimals, we talk about "a 4 in the tenths place" (as in the section on place value in Decimals, what we mean is 4 tenths, or 4/10. Similarly, a 9 in the hundredths place is equal to 9/100.

Because our system is base ten, a value of 10 in one place is equal to a value of 1 in the place to the left, or a value of 1 in one place is equal to a value of 10 in the place to the right. Thus, a "4" in the tenths place is equal to a "40" in the hundredths place (if it were possible to have a double-digit number in a single place). Therefore, we know that 4/10 = 40/100 . We further know that 0.49 = 40/100 + 9/100 = 49/100 . Similarly, 0.876 = 800/1, 000 + 70/1, 000 + 6/1, 000 = 876/1, 000 .

We also know that 0.876 = 876/1, 000 because 0.876 = 0.876/1.000 . Moving the decimal point three spaces to the right in both the numerator and the denominator does not change the fraction, so 0.876/1.000 = 876/1, 000 .

In general, to convert a decimal into a fraction, write the decimal as a fraction over 1. Add as many zeros after the "1." as there are places after the decimal in the original number. Then, move the decimal point in the numerator and the denominator to the right the same number of places until there is a whole number in the numerator. The result will be a numerator that is the original number with the decimal point removed, over a denominator that is a 1 followed by the same number of zeros as there were decimal places in the original number.
Note: It may sometimes be necessary to reduce this fraction--see Reducing Fractions for information on how to reduce fractions.

Example 1. Convert 0.437 into a fraction.

1. 0.437 = 0.437/1 (writing as a fraction over 1)
2. 0.437/1 = 0.437/1.000 (adding 3 zeros after the decimal point)
3. 0.437/1.000 = 437/1, 000 (moving the decimal point)
4. 437/1,000 cannot be reduced.

Thus, 0.437 = 437/1, 000 . Note that there were 3 decimal places in the original number, and there are now 3 zeros in the denominator.

Example 2. Convert 2.45 into a fraction.

1. 2.45 = 2.45/1
2. 2.45/1 = 2.45/1.00
3. 2.45/1.00 = 245/100 . Note that there were 2 decimal places in the original number, and there are now 3 zeros in the denominator.
4. Reduction. 245/100 = 49/20 .

#### Expressing Fractions as Decimals

Sometimes we will want to work with decimals instead of fractions. To convert fractions to decimals, simply divide the numerator by the denominator, either using a calculator or by hand using the usual method of long division. When using long division, don't leave the remainder as a whole number. Instead, treat the numerator as a decimal with zeroes after the decimal point (for example, treat 56 as 56.000). Continue to divide, making sure the decimal point is in the same place in the numerator and the answer.

Example 3. Convert 7/4 into a decimal.

By calculator: 7/4 = 1.75
By long division: 7/4 = 7.00/4 = 1.75

#### Terminating and Repeating Decimals

Try to change 1/3 into a decimal on a calculator. The answer should be 0.3333333... . Similarly, 5/12 = 0.416666666... and 13/99 = 0.13131313...

Decimals that endlessly repeat one number or a group of numbers are called repeating decimals. The part that repeats is usually written as a single number (or group of numbers) with a line over it. For example, 0.3333333... , 0.4166666... , and 0.13131313... are written as 0. , 0.41 , 0. ,
Decimals that stop after a certain place are called terminating decimals. Every fraction can be written as either a terminating decimal or a repeating decimal.

Example 4. Do the following fractions convert to terminating or repeating decimals? 4/6, 9/6, 109/99, 5/4.

4/6. Repeating
9/6. Terminating
109/99. Repeating
5/4. Terminating

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