Question

# The total income of $A$ and $B$ is $Rs\text{ }6000$. $A$ spends $60%$ of his income and $B$ spends $80%$ of his income if their savings are equal then what is the income of $A$?A) $2000$B) $2400$C) $2600$D) $2800$

Hint:In case of unknown always assume any variable as the reference value. So, here we will first assume $A's$ income. Convert given word statements in the form of mathematical expressions. Find the amount of savings and accordingly form the equation. Compare both the savings amount and simplify.

Complete step-by-step solution:
Given: Total income of $A$ and $B$$=\text{ }6000$, $A$spend $60%$ of his income and $B$ spends $80%$ of his income and both have equal savings.
Let income of$A$is $x$
Then income of $B$will be $6000-x$[ As, $income\text{ }of\text{ }A\text{ }+\text{ }income\text{ }of\text{ }B\text{ }=6000$]
$A$spends $60%$of his income.
So, $A's$ Saving is $x\text{ }-\text{ }60%\text{ }of\text{ }x$
Convert it in the mathematical form and simplify.
\begin{align} & =x-\dfrac{60}{100}of\text{ }x \\ & =x-\dfrac{6x}{10} \\ & =\dfrac{10x-6x}{10} \\ & =\dfrac{4x}{10}.............(1) \\ \end{align}
Similarly Now, $B$ spends $80%$ of his income
So, $B's$ saving is $\left( 6000\text{ }-\text{ }x \right)\text{ }-\text{ }80%\text{ }of\text{ }\left( 6000\text{ }\text{ }x \right)$
Simplify the above equations -
\begin{align} & \Rightarrow \left( 6000-x \right)-\dfrac{80}{100}of(6000-x) \\ & =\left( 6000-x \right)-\dfrac{8(6000-x)}{10} \\ & =\dfrac{10\left( 6000-x \right)-8(6000-x)}{10} \\ & =\dfrac{60000-10x-48000+8x}{10} \\ & =\dfrac{12000-2x}{10}.................(2) \\ \end{align}
According to question eq(1) is equal to eq(2) as both $A$ and $B$ has equal saving.
$\dfrac{4x}{10}=\dfrac{12000-2x}{10}$
[Same denominator from both sides of the equations cancels each other]
\begin{align} & \Rightarrow 4x=12000-2x \\ & \Rightarrow 4x+2x=12000 \\ & \Rightarrow 6x=12000 \\ & \therefore x=\dfrac{12000}{6} \\ & \therefore x=2000 \\ \end{align}
Therefore, the required solution – The income of A is $=Rs.\ \text{2000}$

Hence from the given multiple choices, option A is the correct answer.

Note: In these types of problems we first assume the unknown quantity that relates most quantities of the problem. Then form the equation for all the required conditions. Then compare the final two equations to find the required value. Always double check the conversion of word statements to the mathematical form rest goes well and just simplify for the required result.