Question

Ram’s income is \[40\%\] more than Raj’s income. What percent is Raj’s income less than that of Ram’s?

Answer 1

Hint: We are given that Ram’s income is \[40\%\] more than Raj’s income and we are asked to tell by what percent Raj's income is less than that of Ram’s. We will first let the Raj’s income be Rs.x, then the income of Ram will be x more than \[40\%\] of x, that is, \[x+40\%\times x\]. Then, we will find the difference between the incomes of Ram and Raj and find the percentage with respect to Ram’s income. Hence, we will have the required percentage.

Complete step by step answer:
According to the given question, we are given that Ram’s income is \[40\%\] more than Raj’s income and we are asked to tell by what percent is Raj's income less than that of Ram’s income.
Let the income of Raj be Rs.x
So as per the question, the income of Ram will be x more than \[40\%\] of x, that is,
\[x+40\%\times x\]
We will simplify the above expression further and we get,
\[\Rightarrow x+\dfrac{40}{100}x\]
Taking the LCM now, we get the expression as,
\[\Rightarrow \dfrac{100x}{100}+\dfrac{40x}{100}\]
\[\Rightarrow \dfrac{100x+40x}{100}\]
\[\Rightarrow \dfrac{140x}{100}=\dfrac{14x}{10}\]
That is, the income of Ram is Rs. \[\dfrac{14x}{10}\]
The difference of the Ram and Raj incomes is,
\[=\dfrac{14x}{10}-x\]
Taking the LCM and we get the expression as,
\[\Rightarrow \dfrac{14x-10x}{10}\]
\[\Rightarrow \dfrac{4x}{10}\]
In order to find the percentage by how much is Raj’s income is less than Ram’s is by the ratio of the difference of Ram and Raj’s income and Ram’s income and we will multiply this ratio by 100 and we get,
\[=\dfrac{\dfrac{4x}{10}}{\dfrac{14x}{10}}\times 100\]
Simplifying the expression, we get the value as,
\[\Rightarrow \dfrac{4x}{14x}\times 100\]
Cancelling out the common terms, we get the expression as,
\[\Rightarrow \dfrac{2}{7}\times 100\]
\[\Rightarrow 0.2857\times 100\]
\[\Rightarrow 28.57\%\]
Therefore, the percentage decrease in the income of Raj is \[28.57\%\] than Ram’s income.

Note: The interpretation of the question while finding the value of the income of Ram should be done correctly. Also, while finding the difference in the income, always the bigger income comes first and then the smaller income else, we will get a negative value. Also, we are calculating the percentage decrease in Raj’s income with respect to Ram’s income, that is why we took Ram ‘s income in the denominator.