Fractions and decimals are two different ways to represent parts of a whole. Understanding how to move between these representations is a fundamental skill in mathematics and everyday life. Whether you’re working on homework, splitting a bill, or adjusting a recipe, knowing how to convert a fraction to a decimal can be incredibly useful. This guide will walk you through the process step-by-step, making it clear and easy to understand.
Understanding Fractions and Decimals
Before diving into the conversion process, let’s briefly define what fractions and decimals are.
A fraction represents a part of a whole and is written as one number over another, separated by a line. The number on top is called the numerator, and it represents how many parts you have. The number on the bottom is the denominator, and it represents the total number of equal parts the whole is divided into. For example, in the fraction 1/2, 1 is the numerator and 2 is the denominator.
A decimal is another way to represent parts of a whole, using a base-ten system. Decimals use a decimal point to separate the whole number part from the fractional part. Numbers to the right of the decimal point represent values less than one, like tenths, hundredths, thousandths, and so on. For example, 0.5 represents five-tenths, which is equivalent to the fraction 1/2.
The Basic Method: Division
The most straightforward method to convert a fraction to a decimal is through division. Remember that a fraction bar essentially means “divided by.” Therefore, to convert a fraction to a decimal, you simply divide the numerator (the top number) by the denominator (the bottom number).
Let’s look at some examples:
Example 1: Convert 1/2 to a decimal
To convert 1/2 to a decimal, divide 1 by 2:
1 ÷ 2 = 0.5
Therefore, 1/2 is equal to 0.5.
Example 2: Convert 3/4 to a decimal
To convert 3/4 to a decimal, divide 3 by 4:
3 ÷ 4 = 0.75
Therefore, 3/4 is equal to 0.75.
Example 3: Convert 5/8 to a decimal
To convert 5/8 to a decimal, divide 5 by 8:
5 ÷ 8 = 0.625
Therefore, 5/8 is equal to 0.625.
Converting Fractions to Terminating Decimals
In the examples above, the division resulted in decimals that end, or “terminate.” These are called terminating decimals. A fraction will convert to a terminating decimal if the prime factors of its denominator are only 2 and/or 5.
Let’s consider why this is the case. Decimals are based on powers of 10 (tenths, hundredths, thousandths, etc.). Since 10 is the product of 2 and 5, any denominator that is composed of only factors of 2 and 5 can be converted into a power of 10 by multiplying the numerator and denominator by appropriate factors.
For example, with 5/8, the denominator 8 is 2 x 2 x 2 (23). We can convert this to a power of 10 by multiplying by factors of 5:
8 x 5 x 5 x 5 = 8 x 125 = 1000 (103)
So, to convert 5/8 to a decimal, we can multiply both the numerator and the denominator by 125:
(5 x 125) / (8 x 125) = 625 / 1000
And 625/1000 is easily represented as the decimal 0.625.
Converting Fractions to Repeating Decimals
Not all fractions convert to terminating decimals. Some fractions, when converted to decimals, result in a pattern of digits that repeats endlessly. These are called repeating decimals or recurring decimals.
A fraction will convert to a repeating decimal if its denominator has prime factors other than 2 and 5.
Example 4: Convert 1/3 to a decimal
Divide 1 by 3:
1 ÷ 3 = 0.3333…
The digit 3 repeats endlessly. We represent this repeating decimal by writing a bar over the repeating digit: 0.$overline{3}$.
Example 5: Convert 2/3 to a decimal
Divide 2 by 3:
2 ÷ 3 = 0.6666…
The digit 6 repeats endlessly. We represent this as 0.$overline{6}$.
Example 6: Convert 1/7 to a decimal
Divide 1 by 7:
1 ÷ 7 = 0.142857142857…
Here, the sequence of digits “142857” repeats. We represent this as 0.$overline{142857}$.
Using Long Division for Fraction to Decimal Conversion
For more complex fractions, especially those that might result in repeating decimals, long division is a helpful method. It allows you to perform the division process manually and identify the repeating pattern if one exists.
Let’s demonstrate with an example:
Example 7: Convert 5/7 to a decimal using long division
- Set up the long division: Write 5 inside the division symbol and 7 outside. Since 5 is smaller than 7, we’ll start by adding a decimal point and zeros to the dividend (5).
_______
7 | 5.0000- Divide: 7 goes into 50 seven times (7 x 7 = 49). Write 7 above the 0 after the decimal point, and subtract 49 from 50, leaving a remainder of 1. Bring down the next 0.
0.7____
7 | 5.0000
- 4 9
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10- Continue dividing: 7 goes into 10 once (7 x 1 = 7). Write 1 next to 7 above, and subtract 7 from 10, leaving a remainder of 3. Bring down the next 0.
0.71___
7 | 5.0000
- 4 9
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10
- 7
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30- Keep going: 7 goes into 30 four times (7 x 4 = 28). Write 4 next to 71 above, and subtract 28 from 30, leaving a remainder of 2. Bring down the next 0.
0.714__
7 | 5.0000
- 4 9
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10
- 7
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30
- 28
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20- Repeat and identify the pattern: Continue the division process. You’ll notice remainders will start repeating, which indicates a repeating decimal. For 5/7, you will find the decimal representation is approximately 0.7142857… with the sequence “714285” repeating. Therefore, 5/7 = 0.$overline{714285}$.
Conclusion
Converting a fraction to a decimal is a fundamental mathematical operation that is simpler than it might initially seem. By understanding the basic principle of division and practicing with different types of fractions, you can easily master this skill. Whether you are dealing with terminating or repeating decimals, the process of dividing the numerator by the denominator remains the core method. So next time you encounter a fraction and need its decimal equivalent, remember this guide and confidently perform the conversion!
