Question

Ramesh, Suresh, and Mahesh started a business with the investment in the ratio 5:8:10 respectively. After 1 year Mahesh withdraws 50% of his capital and Ramesh increased his capital by 80% of his investment. After 2 years in what ratio should the earned profit be distributed among Ramesh, Suresh, and Mahesh?

Answer 1

Hint: Ramesh, Suresh, and Mahesh started a business with the investment in the ratio 5:8:10 respectively. After 1 year Mahesh withdrew 50% of his capital and Ramesh increased his capital by 80% of his investment. After 2 years in what ratio should the earned profit be distributed among Ramesh, Suresh, and Mahesh?

Complete step-by-step solution
We know that when two or more two people invest money in a business they are called partners and the amount of investment is called capital. The share of the profit varies directly with the capital and time period of investment. If P is the amount of profit for a particular partner, t is the time period of investment and $c$ is the capital then we have;
\[P\propto t,P\propto c\Rightarrow P\propto t\times c\]
We are given in the question that Ramesh, Suresh, and Mahesh started a business with the investment in the ratio 5:8:10 respectively. Let us assume the amount of capital invested by in first year Ramesh, Suresh, and Mahesh are ${{R}_{1}}=5x,{{S}_{1}}=8x,{{M}_{1}}=10x$ respectively. \[\]
We are further given that after 1 year Mahesh withdrew 50% of his capital and Ramesh increased his capital by 80% of his investment. \[\]
So the amount of capital invested by Mahesh in the second year is $100-50=50$% of original investment which in amount is ${{M}_{2}}=10x\times \dfrac{50}{100}=5x$. \[\]
The amount of capital invested by Ramesh is original investment plus 80% of original investment which in amount is ${{R}_{2}}=5x+5x\times \dfrac{80}{100}=5x+4x=9x$.\[\]
The investment of Suresh remains the same $8x$ for the first and second year which means${{S}_{2}}={{S}_{1}}=8x$.
The time periods are ${{t}_{1}}=1$year and then ${{t}_{2}}=1$year. So by rule of partnership sharing the ratio of shared amount among Ramesh, Suresh and Mahesh is
\[\begin{align}
  & \text{Ramesh:Suresh:Mahesh} \\ \
&={{R}_{1}}{{t}_{1}}+{{R}_{2}}{{t}_{2}}:{{S}_{1}}{{t}_{1}}+{{S}_{2}}{{t}_{2}}:{{M}_{1}}{{t}_{1}}+{{M}_{2}}{{t}_{2}} \\
 & =5x\times 1+9x\times 1:8x\times 1+8x\times 1:10x\times 1+5x\times 1 \\
 & =14x:16x:15x \\
 & =14:16:15 \\
\end{align}\]

Note: We note that if the time period would have been the same we call the partnership a simple partnership and if the time period is different for different amounts of investments then we call the partnership compound partnership like in this problem. We know that the continued proportion $a:b:c$ is the same as $ka:kb:kc$ for some non-negative integer $k$. We also remember that when we say $p$% of $a$ means $\dfrac{p}{100}\times a$.