Question

A and B entered into partnership with capitals in the ratio $4:5$. After $3$ months, A withdraws $\dfrac{1}{4}$ of his capital and B withdraws $\dfrac{1}{5}$ of his capital. The gain at the end of $10$ months was $Rs.760$. A’s share in this profit is
(A) $Rs.330$
(B) $Rs.360$
(C) $Rs.380$
(D) $Rs.430$

Answer 1

Hint: We are given that the ratio of the capitals, when A and B entered into partnership, is $4:5$.
So, we will let the capital of A as $4x$ and B as $5x$ when they both enter into partnership.
Now, we are given that after $3$ months, A withdraws $\dfrac{1}{4}$ of his capital and B withdraws $\dfrac{1}{5}$ of his capital.
So, we will multiply the initial capitals of A and B with $3$ to find out the profit in these $3$ months when none of them withdraw any of their capital.
Lastly, we will use the fact that the gain at the end of the $10$ months was $Rs.760$ to find the ratio of profit of A and B.


Complete step by step solution:
We are given that the ratio of the capitals, when A and B entered into partnership, is $4:5$.
So, let the capital of A be $4x$ and let the capital of B be $5x$.
Now, we are given that after $3$ months, A withdraws $\dfrac{1}{4}$ of his capital and B withdraws $\dfrac{1}{5}$ of his capital.
So, the profit of A and B in these $3$ months when none of them withdrew their money will be $(4x \times 3)$ and $(5x \times 3)$ respectively.
Now, since after $3$ months, A withdraws $\dfrac{1}{4}$ of his capital and B withdraws $\dfrac{1}{5}$ of his capital, so the remaining capital of A and B are:
$1 - \dfrac{1}{4} = \dfrac{3}{4}$
and $1 - \dfrac{1}{5} = \dfrac{4}{5}$ , respectively.
We are also given the gain at the end of $10$ months.
Since we have already calculated the profit for the initial $3$ months, we need to calculate the profit of the remaining $7$ months because $10 - 3 = 7$.
The profit of A in these $7$ months will be the remaining capital of A multiplied by the initial capital of A multiplied by $7$, that is, profit of A in these $7$ months is $\dfrac{3}{4} \times 4x \times 7$.
Similarly, the profit of B in these $7$ months will be the remaining capital of B multiplied by the initial capital of B multiplied by $7$, that is, profit of $B$ in these $7$ months is $\dfrac{4}{5} \times 5x \times 7$.
Now, adding both the profits, that is, of the initial $3$ months and the after $7$ months of A and B, we get the total profit of A and B as $\left( {4x \times 3} \right) + \left( {\dfrac{3}{4} \times 4x \times 7} \right)$ and $\left( {5x \times 3} \right) + \left( {\dfrac{4}{5} \times 5x \times 7} \right)$ respectively.
So, the ratio of the profit of A and B is as follows:
$\left( {4x \times 3} \right) + \left( {\dfrac{3}{4} \times 4x \times 7} \right):\left( {5x \times 3} \right) + \left( {\dfrac{4}{5} \times 5x \times 7} \right)$
Multiplying the terms in the brackets separately, we get the above ratio as:
$\left( {12x + 21x} \right):\left( {15x + 28x} \right)$.
Adding the terms in the brackets, we get the ratio as:
$33x:43x$.
Cancelling out $x$ in the ratio, we get the final ratio as:
$33:43$.
Since, the gain at the end of $10$ months was $Rs.760$,
therefore, A’s share in this profit will be:
$\dfrac{{33}}{{33 + 43}} \times 760$.
Simplifying the numerator and denominator separately, we get:
$\dfrac{{33 \times 760}}{{33 + 43}} = \dfrac{{25080}}{{76}}$.
Dividing $25080$ with $76$, we get $330$.
Thus, A’s share in the profit is $Rs.330$.
Hence, option (A) $Rs.330$ is correct.

Note:
In this question, don’t forget to add the profits in the end and also, while finding A’s share in the profit, we added both the terms of the ratio, that is, $33$ and $43$ in the denominator and took A’s share of the total profit as the numerator, that is, $33$ and then multiplied the whole term by $760$, which was the total gain of A and B.