Question

Karuna and Varuna invested Rs. 2400 and Rs. x in a business. After 3 months, Karuna added Rs. 600 while Varuna withdrew Rs. 300. After a year out of a total profit of Rs. 36,920, Varuna received Rs. 17,160. Find the amount invested by Varuna at the starting of business.

Answer 1

Hint: We have Karuna invested Rs. 2400 and Varuna invested Rs. x. First, we will find the total share Karuna and Varuna have after one year. We will find the profit Varuna using the formula,
\[\text{Profit of Varuna}=\dfrac{\text{Varuna }\!\!'\!\!\text{ s shares}}{\text{Total Share}}\times \text{Total Profit}\]
To do, we will first add the share of Varuna and Karuna. Then we will apply the above formula. Once we have this equation, we will solve to find the value of x.

Complete step by step answer:
We are given that Karuna invested Rs. 2400 while Varuna invested Rs. x. First, we will find the total shares of Karuna and Varuna one by one. We have Karuna invested Rs. 2400 and after 3 months, she added 600 more. We have the profit after 1 year. So, we get that, Karuna invested Rs. 2400 for 3 months and (2400 + 600) for 9 months.
So, Karuna’s share is given as,
\[{{S}_{K}}=2400\times 3+\left( 2400+600 \right)9\]
\[\Rightarrow {{S}_{K}}=7200+3000\times 9\]
Simplifying further, we get,
\[\Rightarrow {{S}_{K}}=34200\]
Now, we will find the share of Varuna. We know that she invested Rs. x initially and after 3 months, she withdrew 300 again the profit for 1 year. So, Varun share is given as,
\[{{S}_{V}}=x\times 3+\left( x-300 \right)9\]
\[\Rightarrow {{S}_{V}}=3x+9x-2700\]
\[\Rightarrow {{S}_{V}}=12x-2700\]
The total share will be given as the sum of both of their shares.
\[{{S}_{T}}={{S}_{V}}+{{S}_{K}}\]
\[\Rightarrow {{S}_{T}}=34200+12x-2700\]
\[\Rightarrow {{S}_{T}}=31500+12x\]
Now, we have a total profit after a year as Rs. 36920. We know that the profit of Varuna is given as
\[\text{Profit of Varuna}=\dfrac{\text{Varuna }\!\!'\!\!\text{ s shares}}{\text{Total Share}}\times \text{Total Profit}\]
As Varuna’s share is 12x – 2700, the total share is 31500 + 12x and the total profit is 36920. So, we get,
\[\Rightarrow \text{Profit of Varuna}=\dfrac{12x-2700}{31500+12x}\times 36920\]
As we have given Varuna’s profit is 17160, so we get,
\[\Rightarrow \text{17160}=\dfrac{12x-2700}{31500+12x}\times 36920\]
Now, on simplifying, we get,
\[\Rightarrow \text{17160}=\left( \dfrac{4x-900}{10500+4x} \right)\times 36920\]
\[\Rightarrow \left( 10500+4x \right)429=\left( 4x-900 \right)923\]
Opening the brackets, we get,
\[\Rightarrow 10500\times 429+429\times 4x=4x\times 923-900\times 923\]
Simplifying further, we get,
\[\Rightarrow 1976x=5335200\]
\[\Rightarrow x=2700\]
So, we get that Varun invested Rs. 2700.

Note:
 While finding the total share, we should be careful about what amount of money is invested for how much time. Like Karuna started with 2400 and she added 600 after 3 months. So, Rs. 2400 is under-investment for just 3 months. After that (2400 + 600) is under-investment for (12 – 3) months = 9 months. If we take the wrong amount or the time, we will get the incorrect solution.