Chapter 3.6, Problem 45PPS

To determine

To calculate: have to calculate the difference of the given fraction.

Answer to Problem 45PPS

The difference of the given fraction is $\frac{3300909}{90000}$ .

Explanation of Solution

Given data: $24.\overline{56}-\left(-12.\overline{1}\right)$

Some simple steps have to follow to add or subtract the, unlike fractions.

Sept-I:Convert Recurring numbers into the fractions.

Since, $24.\overline{56}$ is a recurring number.

Suppose the $x=24.56565656\mathrm{...}\left(\text{i}\right)$

Now, multiply the equation (i) by 10 both side, we get

  $10x=245.65656\mathrm{...}\left(\text{ii}\right)$

Again, multiply the equation (ii) by 10 both side, we get

  $100x=2456.565656565\mathrm{...}\left(\text{ii}\right)$

Subtract the equation (ii) to equation (iii), we get

  $\begin{array}{l}\Rightarrow 90x=2210.909\\ \Rightarrow x=\frac{2210909}{90000}\end{array}$

Since, $12.\overline{1}$ is a recurring number.

Suppose the $x=12.111111\mathrm{...}\left(\text{i}\right)$

Now, multiply the equation (i) by 10 both side, we get

  $10x=121.11111\mathrm{...}\left(\text{ii}\right)$

Subtract the equation (i) to equation (ii), we get

  $\begin{array}{l}\Rightarrow 9x=109\\ \Rightarrow x=\frac{109}{9}\end{array}$

Step-II: Convert the mixed fraction into an improper fraction (if any).

  $=\frac{2210909}{90000}-\left(-\frac{109}{9}\right)$

  $=\frac{2210909}{90000}+\frac{109}{9}$ [Unlike fraction]

Step-III: Calculate the ‘Least Common Multiple’ (LCM) of the denominator of the given fraction.

Now, Least Common Multiple (LCM) $=10000$

Step-IV: Convert each fraction into an equivalent fraction by multiplies and divide of its denominator and numerator to the equal to the ‘Least Common Multiple’ (LCM) obtained in step -II.

  $\begin{array}{l}=\frac{2210909}{90000}+\frac{109}{9}\\ =\frac{2210909}{90000}+\frac{109\times 10000}{9\times 10000}\end{array}$

  $=\frac{2210909}{90000}+\frac{1090000}{90000}$ [Like fraction]

Step-V: Add or subtract the like fraction to convert into a simple form.

  $\begin{array}{l}=\frac{2210909}{90000}+\frac{1090000}{90000}\\ =\frac{2210909+1090000}{90000}\\ =\frac{3300909}{90000}\end{array}$