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# Ishan’s income is 10 % more than Sophia’s. By what percent is Sophia’s income less than Ishan’s income?

Last updated date: 16th Jun 2024
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. To solve this question, we will, first of all, assume a variable for Sophia’s income and then try to calculate Ishan’s income in terms of that variable. Finally, we will calculate the difference of the income and then calculate the percentage difference by using the formula $\text{Percentage}=\dfrac{\text{Difference in income}}{\text{Ishan }\!\!'\!\!\text{ s Income}}\times 100.$

Complete step-by-step solution
Ishan’s income is 10 % more than Sophia’s, so we get,
$\text{Ishan }\!\!'\!\!\text{ s Income}=x+10\text{ Percent of x}$
$\Rightarrow \text{Ishan }\!\!'\!\!\text{ s Income}=x+\dfrac{10}{100}x$
$\Rightarrow \text{Ishan }\!\!'\!\!\text{ s Income}=x+0.1x\text{ or }\dfrac{100x+10x}{100}$
$\Rightarrow \text{Ishan }\!\!'\!\!\text{ s Income}=1.1x\text{ or }\dfrac{110x}{100}$
Now, we have to calculate that by how much percentage of Sophia’s income is less than Ishan’s income. So, we need the difference between their income. Therefore, the difference between their income is
Ishan’s Income – Sophia’s Income
$\Rightarrow \text{Difference}=\dfrac{110x}{100}-x$
$\Rightarrow \text{Difference}=\dfrac{110x-100x}{100}$
$\Rightarrow \text{Difference}=\dfrac{10x}{100}.......\left( i \right)$
Now, finally, the percentage would be calculated by using
$\text{Percentage}=\dfrac{\text{Difference}}{\text{Ishan }\!\!'\!\!\text{ s Income}}\times 100$
Therefore, the difference is obtained by equation (i). Substituting the values, we get,
$\Rightarrow \text{Percentage}=\dfrac{\dfrac{10x}{100}}{\dfrac{110x}{100}}\times 100$
$\Rightarrow \text{Required Percentage}=\dfrac{10x}{110x}\times 100$
$\Rightarrow \text{Required Percentage}=\dfrac{10}{110}\times 100$
$\Rightarrow \text{Required Percentage}=\dfrac{10\times 10}{11}$
$\Rightarrow \text{Required Percentage}=\dfrac{100}{11}$
$\Rightarrow \text{Required Percentage}=9.09\%$
Therefore, Sophia’s income is less than Ishan’s income by $9.09\%.$

Note: The possibility of mistake can be directly subtracting 10 % from 100 % to get 90 % as the answer. But this is wrong because we are given the relative value of the income of Ishan and Sophia and that too in percentage. So, if we need to relative income again then we first need to find their exact income in terms of some variable and then proceed to get the result.