Question

Two friends P and Q started a business investing in the ratio of 5:6. R joined them after six months investing an amount equal to Q's. At the end of the year, 20% of profit was earned which was equal to Rs.98000. What was the amount invested by R?
\[\begin{align}
  & A.Rs.105000 \\
 & B.Rs.175000 \\
 & C.Rs.210000 \\
 & \text{D}.\text{None of these} \\
\end{align}\]

Answer 1

Hint: For solving this question, we will first suppose the total investment as Rs.x and then we will find 20% profit of x which will be equal to Rs.98000. Using this, we will get total investment by all three. Now, P and Q invested for 12 months whereas R invested for 6, so we will find their shares in terms of t which will be common ratio. At last, we will find the value of t using total investment which will help us in finding R's investment.

Complete step by step answer:
Here let us suppose that, total investment by all three members was Rs.x
Now we are given that 20% of the investment became profit which is also equal to Rs.98000. So we can say that 20% of x is equal to Rs.98000.
Writing it in mathematical terms we get: $\dfrac{20}{100}\times x=98000$.
Simplifying we get: $\dfrac{1}{5}\times x=98000$.
Taking 5 on the other side we get: $x=Rs.98000\times 5=Rs.490000$.
Hence total investment by all three was Rs.490000.
Now we are given that, P and Q invested in the ratio of 5:6. Let the common ratio be t. Therefore, P's one month investment will be 5t and Q's one month investment will be 6t.
Since R's investment was equal to Q's investment, so R's one month investment will be 6t.
Now, P invested for a whole year. Therefore, P's share will be $5t\times 12=60t$. (12 months in a year).
Q also invested for a whole year. Therefore, Q's share will be $6t\times 12=72t$.
Now, R came in after 6 months, so he invested for six months only. Therefore, his share will be $6t\times 6=36t$.
Now their total investment will be equal to the sum of all their shares.
Therefore, total investment will be $Rs.\left( 60t+72t+36t \right)=Rs.168t$.
Since we found monthly investment of three of them as Rs.490000, so their investment for the whole year will be $Rs.\left( 490000\times 12 \right)$.
Hence we can say that,
$Rs.168t=Rs.\left( 490000\times 12 \right)\Rightarrow 168t=490000\times 12$.
Dividing both sides by 168 we get:
$t=\dfrac{490000\times 12}{168}$.
Now simplifying it we get:
$t=Rs.35000$.
So the common ratio is Rs.35000, since R's investment was equal to Rs.6x.
So we get R's investment as $Rs.\left( 6\times 35000 \right)=Rs.210000$.

So, the correct answer is “Option C”.

Note: Students must not forget to multiply their investment by 12 as we are required to make an annual investment here. Students should do all calculations carefully as big numbers are involved here. Students should make sure that R invested 6 months later, so his share will be multiplied by 6 only. We are asked for a monthly investment here so we have found a value of 6x rather than 36x.