Question

The total of the daily wages of Ram, Shyam and Mohan is Rs.450. If they respectively spend 25%, 20% and 50% of their income then ratio of their savings become 9:8:4. What is the income of Ram?
\[\begin{align}
  & A.180 \\
 & B.190 \\
 & C.200 \\
 & D.175 \\
\end{align}\]

Answer 1

Hint: For solving this question, we will first suppose the income of Ram, Shyam and Mohan as x, y and z respectively. Then we will find their spent income using the given percentage in form of variables. Using them, we will find their savings in the form of variables. After that, we will compare the ratio of their savings with the given ratio and take a common value of all ratios as k. Using total wages, we will find the value of k and then values of x, y and z which are required income.

Complete step-by-step answer:
Let us suppose that, the income of Ram is Rs.x, the income of Shyam is Rs.y and the income of Mohan is Rs.z. Here we are given that, the sum of their wages is Rs.450. Therefore, we have:
$x+y+z=450\cdots \cdots \cdots \left( 1 \right)$.
Now let us find the spending and saving income of all three men.
We are given that Ram spent 25% of his income.
Therefore, Ram spent 25% of x $\Rightarrow \dfrac{25}{100}x=\dfrac{1}{4}x$.
Now total income of Ram is Rs.x and spent income is $\dfrac{1}{4}x$. So, savings of Ram will be $Rs.x-Rs.\dfrac{x}{4}=Rs.\dfrac{3x}{4}$.
We are given that Shyam spends 20% of his income, therefore, he spends 20% of y $\Rightarrow \dfrac{20}{100}y=\dfrac{y}{5}$.
Total income of Shyam is Rs.y and spent income is $\dfrac{y}{5}$. So, his savings will be equal to $Rs.y-Rs.\dfrac{y}{5}=Rs.\dfrac{4}{5}y$.
We are given that Mohan spends 50% of his income, therefore, he spends 50% of z $\Rightarrow \dfrac{50}{100}z=\dfrac{z}{2}$.
Total income of Mohan is Rs.z and his spent income is $\dfrac{z}{2}$. So, his savings will be equal to $Rs.z-Rs.\dfrac{z}{2}=Rs.\dfrac{z}{2}$.
We are given the ratio of savings of Ram, Shyam and Mohan as 9:8:4.
Let us suppose that, k is the common ratio.
Therefore, savings of Ram will be 9k.
Savings of Shyam will be 8k and
Savings of Mohan will be 4k.
So taking both found savings equal we get:
$\dfrac{3x}{4}=9k,\dfrac{4y}{5}=8k\text{ and }\dfrac{z}{2}=4k$.
Let us find values of x, y, z in terms of k, we get: $\dfrac{3x}{4}=9k$.
Taking $\dfrac{3}{4}$ to the other side, we get:
$x=9\times \dfrac{4}{3}k\Rightarrow x=12k$.
\[\begin{align}
  & \dfrac{4y}{5}=8k \\
 & \Rightarrow y=8\times \dfrac{5}{4}k\Rightarrow y=10k \\
\end{align}\]
$\begin{align}
  & \dfrac{z}{2}=4k \\
 & \Rightarrow z=8k \\
\end{align}$
Now putting these values in (1) we get:
$\begin{align}
  & 12k+10k+8k=450 \\
 & \Rightarrow 30k=450 \\
\end{align}$.
Dividing both sides by 30, we get:
$k=\dfrac{450}{30}=15$.
Hence the common ratio is 15.
So income of Ram $x=12k=12\times 15=Rs.180$.
Hence Rs.180 is the required answer.

So, the correct answer is “Option A”.

Note: Students always forget to take common ratio and put $9=\dfrac{3x}{4},8=\dfrac{4y}{5}\text{ and }4=\dfrac{z}{2}$. Students should note that we are given a percentage of spending income and ratio of savings. Take care of signs while solving the equations. We can find income of Shyam and Mohan also.