# Fraction to decimal

There are a number of ways to convert fractions to decimals. A fraction can be read as a division problem where the numerator is divided by the denominator; can be read as "four divided by five," where normally it would be read as "four fifths", or maybe "four over five."

The simplest way to convert a fraction to a decimal would be using a calculator if it is available and entering the division problem. Otherwise, we can perform long division, or in some cases use strategies involving equivalent fractions and estimation.

### Using long division

Since fractions are essentially division problems, long division can be used to convert fractions to decimals. In a fraction, the numerator is the dividend and the denominator is the divisor.

Example

Convert to decimal form:

This is a relatively simple example where the fraction is a terminating decimal. The process remains the same for other fractions, but some may be significantly more tedious, or may exhibit a repeating pattern.

Unfortunately, there are no particularly easy manual methods of division that can be applied to all fractions, and for fractions that don't quite work out as nicely as the one above, converting the fraction to a decimal by hand is very tedious. In some cases though, the method below can be used to quickly convert fractions to decimals.

### Using equivalent fractions and estimation

One way to convert fractions to decimals is to express the fraction in terms of an equivalent fraction where the denominator is a power of 10. The reason for this is because decimals are a numeration system based on powers of 10. If we can express a fraction with a denominator that is a power of 10, it makes it relatively easy for us to convert the fraction to a decimal.

In fractions,

is equivalent to

0.1, 0.01, 0.001, 0.0001...

in decimals, respectively. These are read as one tenth, one hundredth, one thousandth, one ten-thousandth, and so on.

So, to convert a fraction to a decimal:

- Multiply the numerator and denominator of the fraction by a number that makes the denominator a power of 10
- Count the number of 0s in the power of 10 in the denominator; the number of 0s will be referred to as n
- Count n decimal spaces from the right-most digit of the numerator, where each digit represents one decimal place, then write a decimal point; if n is greater than the number of digits in the numerator, write a 0 in the empty decimal place

Examples

1. Convert to its decimal form:

Step 1:

Step 2:

There is one 0 in 10, so n = 1

Step 3:

2. Convert to its decimal form:

Step 1:

Step 2:

There are four 0s in 10000, so n = 4

Step 3:

In the examples above, the fractions have equivalent fractions that have denominators exactly equal to powers of 10. There are many fractions where this is not the case, and for these fractions, the above method of conversion cannot be used. In some cases though, we may be able to estimate a decimal value using this method depending how exact we need the answer to be. We can do this by forming an equivalent fraction where the denominator is close to a power of 10.

Example

Estimate in decimal form:

327 cannot be multiplied by an integer to equal a power of 10, but 327 × 3 = 981, which is relatively close to 1000, so we can estimate the fraction as:

Since 981 < 1000, this estimate is going to be slightly smaller than the actual value. The actual value to three decimal places is 0.171 and the percent error between this value and our estimate is: