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3.5 | Converting Fractions to Decimals and Vice-versa |

In these revision notes for Converting Fractions to Decimals and Vice-versa, we cover the following key points:

- What are decimal fractions?
- How to turn a decimal into decimal fractions?
- What about fractions that cannot turn into decimal fractions?
- What are decimals? What is their composition?
- How to convert fractions into decimals?
- How to convert decimals into fractions?
- How to convert recurrent fractions into decimals?

**A decimal fraction is a fraction that has a denominator that is a power of ten.**

There are some fractions that do not seem to be decimal fractions, but with a few steps, they can turn into such. These steps include multiplying or dividing both numerator and denominator several times by the same number until we obtain a decimal fraction.

Not all fractions can be written as decimal fractions, as it is not always possible to write their denominator as a power of 10.

Decimals are an alternative representation of decimal mixed numbers. They are made of two parts: the left part which shows the whole part of the corresponding decimal mixed number and the right part which shows the numerator of the fractional part of the corresponding decimal mixed number. A dot known as **decimal point** separates the whole and decimal part of such numbers.

The number of zeroes in the denominator of the decimal mixed number determines the number of digits after the decimal point, i.e. the number of decimal places. Thus, if the denominator of the mixed number is 10, the corresponding decimal has one decimal place, if the denominator of the mixed number is 100, the corresponding decimal has two decimal places and so on.

In fractions that cannot be decimal ones, we apply the normal division of their numerator and denominator using the known division methods. In most cases, this division gives an infinite number of digits after the decimal place but if you look them carefully, you will detect a kind of recurrence, i.e. the pattern is repeated after a number of digits.

The conversion of a recurring decimal into fraction is impossible through the usual methods. Therefore, a completely new approach must be used for this kind of conversion. It consists of eliminating the recurring part after the decimal place and then writing the corresponding fraction obtained by subtraction. The following steps are used in this method:

**Step 1:** We denote the original number by N.

**Step 2:** Then, we multiply the original number by a factor that gives a new number with the same decimal part as the original.

**Step 3:** Then, we subtract the original number from the new one to eliminate the recurring part. The same thing is done with the factors containing N's.

**Step 4:** Now, we solve the above equation and simplify the result is needed.

Sometimes the recurrence does not begin immediately after the decimal point but more on the right. In such cases, we first multiply the original number by a factor that leaves only the recurrence after the decimal point and then, we multiply this number by another suitable factor that gives the same recurrence, in order to make possible the elimination of recurrence through subtraction.

When the period of a recurring decimal is 9, the decimal takes the value of the nearest whole number.

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