Problems Based On Recurring Decimals As Rational Numbers

We know that recurring decimal numbers are those which are non- terminating but have repeating digits after the decimal point. These numbers are never ending. They go on till infinity.

For example: 1.23232323… is an example of recurring decimal number as 23 are the repeating digits in the number.

In this topic of rational number we will learn to solve different types of problems based on conversions of recurring decimals into rational fractions. Let us have look at some steps which we need to follow while converting a recurring decimal number into a rational fraction:

Step I:Assume ‘x’ to be a recurring number whose rational fraction we need to find.

Step II: Have a careful observation on the repeating digits of the decimal number.

Step III: Now place repeating digits to the left of the decimal point.

Step IV: After step 3 put the repeating digits on the right side of the decimal point.

Step V: After doing so subtract both sides of the equation as such to maintain the equality of the equations. Make sure that after subtraction difference of both sides are positive.

Now let us have a look at following examples:

1. Convert 1.333… into rational fraction.

Solution:

Step I: Let x = 1.333

Step II: Repeating digit is ‘3’

Step III: Placing repeating digit on the left side of the decimal point can be done by multiplying the original number by 10, i.e.,

10x = 13.333

Step IV: By placing repeating digit to the right of the decimal point it becomes the original number. Technically this can be done by multiplying original number by 1, i.e.,

x = 1.333

Step V: So, our two equations are:

10x = 13.333

⟹ x = 1.333

On subtracting both sides of the equation, we get:

10x – x = 13.333 – 1.333

⟹ 9x = 12

⟹ x = $\frac{12}{9}$

⟹ x = $\frac{4}{3}$

Hence, the required rational fraction is $\frac{4}{3}$.

2. Convert 12.3454545… into rational fraction.

Solution:

Step I: Let x = 12.34545…

Step II: The repeating digits of the given decimal fraction are ‘45’.

Step III: Now we need to transfer repeating digits to the left of the decimal point. To do so, we need to multiply the original number by 1000. So,

1000x = 12345.4545

Step IV: Now we have to shift the repeating digits to the right of the decimal point. To do so we have to multiply the original number by 10. So,

10x = 123.4545

Step V: Two equations are as:

1000x = 12345.4545, and

⟹ 10x = 123.4545

Now we have to perform the subtraction on both sides of the equation to maintain the equality.

1000x – 10x = 12345.4545 – 123.4545

⟹ 990x = 12222

⟹ x = $\frac{12222}{990}$

⟹ x = $\frac{1358}{110}$

⟹ x = $\frac{679}{55}$

Hence, the required rational fraction is $\frac{679}{55}$.

3. Convert 134.45757… into the rational fraction.

Solution:

Step I: Let x = 134.45757.

Step II: The repeating digits of the given decimal number are ‘57’.

Step III: Now we need to transfer the repeating digits of the decimal number to the left side of the decimal point. To do so, we need to multiply the given number with 1000. So,

1000x = 134457.5757

Step IV: Now we need to transfer the repeating digits of the decimal number to the right side of the decimal point. To do so, we need to multiply the original number by 10. So,

10x = 1344.5757

Step V: Two equations are as follows:

1000x = 134457.5757, and

⟹ 10x = 1344.5757

Now we have to perform subtraction on both sides of the equations so as to maintain the equality.

1000x - 10x = 134457.5757 - 1344.5757

⟹ 990x = 133113

⟹ x = $\frac{133113}{990}$

⟹ x = $\frac{44371}{330}$

Hence, the required rational fraction is $\frac{44371}{330}$.

All the conversion of recurring decimal numbers to rational fractions can be done by following the above mentioned steps.

Rational Numbers