Question

If 8 (P’s Capital) = 10 (Q’s Capital) = 12 (R’s Capital), then out of the total profit of Rs 2590, R will receive?

Answer 1

Hint: Let us assume that P’s capital is p and Q’s capital is q and R’s capital is r. As it is given in the question that: 8 (P’s Capital) = 10 (Q’s Capital) = 12 (R’s Capital). So, we get a relation between p, q, and r as $8p=10q=12r$.We have two equations and three variables. So, we need to substitute the value of a variable in terms of another variable to solve the given equation to get a ratio of capital.
Now, to find the share of each one’s profit, assume their share is ‘x’. We get an equation with a single variable. Solve the equation and get the share of R.

Complete step by step answer:
As we have assumed that P’s capital is p and Q’s capital is q and R’s capital is r.
We have: 8 (P’s Capital) = 10 (Q’s Capital) = 12 (R’s Capital). So, we get a relation between p, q and r as: $8p=10q=12r$.
Further, taking 2 as common from the above equation, we get;
$4p=5q=6r......(1)$
So, we have two equations and three variables as;
$\begin{align}
  & 4p=5q......(2) \\
 & 4p=6r......(3) \\
\end{align}$
We can write q and r in terms of p from equation (2) and (3) as:
$\begin{align}
  & q=\dfrac{4p}{5}......(4) \\
 & r=\dfrac{4p}{6}=\dfrac{2p}{3}......(5) \\
\end{align}$
Hence, we get the ratio of P’s Capital, Q’s Capital and R’s Capital as:
$p:q:r=p:\dfrac{4p}{5}:\dfrac{2p}{3}$
To simplify the given ratio, multiply by 15 and divide by p, we get:
$p:q:r=15:12:10$

Hence, P’s Capital, Q’s Capital, and R’s Capital share profit in the ratio of $15:12:10$.
It is given that the total profit earned in Rs. 2590.
Let us assume that the profit earned by each is ‘x’.
So, we get:
$\begin{align}
  & \Rightarrow 15x+12x+10x=2590 \\
 & \Rightarrow 37x=2590 \\
 & \Rightarrow x=70 \\
\end{align}$

So, R’s share in total profit is:
$\begin{align}
  & =10x \\
 & =10\times 70 \\
 & =Rs.700 \\
\end{align}$

Note: Since it is given that they share the profit in a given ratio. Do not assume that they share that part of total profit. For the ratio given in the question, i.e. $8:10:12$; some might write the equation: $8x+10x+12x=2590$ which leads to the wrong answer.