Question

A man earns x% on the first Rs.2000 and y% on the rest of his income. If he earns Rs.700 from Rs.4000 and Rs.900 from Rs.5000 of income, find x.

Answer 1

Hint: Analyze both cases separately. That is, take Rs.4000 as income in 1st case and Rs.5000 as income in 2nd case. Now, as per case take Rs.700 as earning in 1st case and Rs.900 as earning in 2nd case.

Complete step by step answer:
Let the total income of man be Z.
Earning on first Rs.2000= x% of 2000
=\[\dfrac{x}{100}\times 2000=20x\]
Earning on the remaining amount of income=y% of (Z-2000)
=\[\dfrac{y(Z-2000)}{100}\]
Total Earning=
\[\text{20}x+\dfrac{y(Z-2000)}{100}\] ……….Eq (i)
Now, according to the question, man earns Rs.700 from Rs.4000.
Putting Z=4000 in eq(i) we can get the total earning.
\[\begin{align}
  & 20x+\dfrac{y(4000-2000)}{100}=20x+20y \\
 & \Rightarrow 20x+20y=700 \\
 & \Rightarrow 2x+2y=70........eq(ii) \\
\end{align}\]
In case 2nd man earns Rs.900 from Rs.5000.
Putting Z=5000 in eq(i), we can get the total earning.
\[\begin{align}
  & 20x+\dfrac{y(5000-2000)}{100}=20x+30y \\
 & \Rightarrow 20x+30y=900 \\
 & \Rightarrow 2x+3y=90..........eq(iii) \\
\end{align}\]
Subtracting eq(ii) from eq(iii),we get y=20.
Putting y=20 in eq(ii),we get
\[\begin{align}
  & 2x+2(20)=70 \\
 & \Rightarrow 2x=70-40 \\
 & \Rightarrow 2x=30 \\
 & \Rightarrow x=15 \\
\end{align}\]

Hence, the value of x is \[15%\] .

Note: In this type of question, one can make mistakes in percentage. It means one may forget to divide the percentage term by 100 during calculations. Like, x percent of 2000 should be written as \[\dfrac{x}{100}\times 2000\] not as \[x\times 2000\] . So, it should be remembered that we divide by 100 in percentage terms for removing the percentage sign.