Explainer: Converting Fractions to Recurring Decimals

In this explainer, we will learn how to convert a fraction into a decimal, noting that the decimal expansion of a rational number either terminates or eventually repeats.

A fraction, in everyday language, often means a tiny part. In math, a fraction compares a part to a whole and describes what we call a proportion. The denominator of the fraction is the number of equal shares (or portions) the whole is split into, while the numerator is the number of these shares that make the part we are considering.

But a fraction can also be understood as a quotient: it is simply dividing one quantity by another. Any rational number can be expressed in this way, and in this case a “fraction” is actually a number.

Let us take 34 as an example.

Fraction: If 34 is a fraction, it means that we are comparing a part made of 3 equal shares to the whole made of 4 such equal shares.

If we consider the whole to be the unit (1), then the part is 0.75.

Number: If 34 is a number obtained by dividing 3 by 4, then it is 0.75.

We see that the rational number 34 (3 divided by 4) is the same as 34 (three-quarters) of 1. They can both be expressed as a decimal, 0.75.

We are going to learn with some examples how to express a fraction as a decimal. As we have just seen, a general method is to divide the numerator by the denominator. For this, we may have to use long division.

Let us look at a couple of examples and recall what happens when the division ends.

Example 1: Converting a Fraction to a Decimal

Write 1325 as a decimal.

Answer

Here, we notice that the denominator of the fraction, 25, is a divisor of 100. Therefore, it is easy to rewrite this fraction with 100 as the denominator, namely, by multiplying both the numerator and the denominator by 4.

We find that 1325=13×425×4=52100.

Now, it is easy to divide 52 by 100; we find that it is 0.52.

We can also carry out a long division to find the result of 13 divided by 25. We would then find that the division ends with a remainder of 0. The result of the division is of course 0.52.

Example 2: Converting a Unitary Fraction to a Decimal

Convert 17 to a decimal.

Answer

To convert 17 to a decimal, we need to divide 1 by 7. For this, we carry out a long division.

In the next example, we are asked to round a fraction to the nearest tenth.

Example 3: Converting a Fraction to a Decimal Rounded to the Nearest Tenth

23 to the nearest 110.

Answer

We are asked here to write 23 as a decimal rounded to the nearest 110. Let us divide 2 by 3.

First Step: We find there is 0 times 3 in 2, with a remainder of 2.

Second Step: We find there is 6 times 3 in 20, with a remainder of 2.

The remainder, 2, is repeated, so the loop “there is 6 times 3 in 20, with a remainder of 2” will be repeated forever. Hence, 23=0.66666=0.6. To round this to the nearest tenth, we need to look at the digit in the second decimal place: 6. It is greater than 5; therefore, we need to round up. The answer is, therefore, 230.7.

We are going to look at two word problems where we need to convert a fraction to a decimal in order to compare two numbers.

Example 4: Converting Fractions to Decimals in a Real-World Context

Daniel was told he could have 13 or 0.34 of the total jam inside a jar. Convert 13 to a decimal, and determine which option would give Daniel more jam.

Answer

As said in the question, we need first to convert 13 to a decimal. Let us divide 1 by 3.

First Step: We find there is 0 times 3 in 1, with a remainder of 1.

Second Step: We find there is 3 times 3 in 10, with a remainder of 1.

The remainder, 1, is repeated, so the loop “there is 3 times 3 in 10, with a remainder of 2” will be repeated forever. Hence, 13=0.3333=0.3.

Now, we want to compare 0.3=0.333 with 0.34. Since 4 is greater than 3, 0.34 is greater than 0.3. Hence, Daniel will have more with 0.34 of the total jam.

Example 5: Identifying Two Decimals That Lie between Two Fractions

When two students were asked to find two decimals that lie between 3740 and 293200, the first student picked 0.935 and 1.03, whereas the second student picked 0.44 and 1.28. Which of the two is correct?

Answer

We are given two fractions, 3740 and 293200, and we need to check whether the decimals 0.935, 1.03, 0.44, and 1.28 lie between them or not.

To do this, we are going to convert the fractions to decimals. Let us start with 3740. We could carry out the long division of 37 by 40. However, given that the denominator is a multiple of 40, there is an easier way by first dividing both the dividend and the divisor by 4: 3740=37÷440÷4=9.2510=0.925.

Similarly, we can first divide both the dividend and the divisor by 2 in 293200 to convert it more easily to a decimal: 293200=293÷2200÷2=146.5100=1.465.

We can now check whether the numbers given lie between 0.925 and 1.465 or not.

We find that 0.925<0.935<1.465,0.925<1.03<1.465,0.44<0.925,0.925<1.28<1.465.

The number 0.44 does not lie between 0.925 and 1.465. Hence, the first student is correct.

Key Points

  1. A fraction can be understood as a quotient; it is simply dividing one quantity by another. Any rational number can be expressed in this way, and in this case, a “fraction” is actually a number. If 34 is a number obtained by dividing 3 by 4, then it is 0.75.
  2. We see that the rational number 34 (3 divided by 4) is the same as 34 (three quarters) of 1. They both can be expressed as a decimal, 0.75.
  3. Converting a fraction to a decimal is simply carrying out the division of the numerator of the fraction by its denominator.
  4. There are two possible situations when carrying out a long division of whole numbers:
    1. We get at some point a remainder of zero, so the result of this division is a terminating decimal. For instance, 38=0.375.
    2. We get at some point a remainder that we have already had, so the steps between the first time we got this remainder and just before we got this remainder again will be repeated forever; the result of the division is a repeating decimal. For instance, 17=0.142857.

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