Question
A, B and C invested their capitals in the ratio 3 : 4 : 6. However their share of profit are equal. The duration of their investments must be in the ratio ?
Answer: Option A
Let their investments be Rs. 3x for p months
Rs. 4x for q months and Rs. 6x for r months respectively
Then,
$$\eqalign{
& 3xp:4xq:6xr = 1:1:1 \cr
& \Rightarrow 3p:4q:6r = 1:1:1 \cr
& {\text{So,}}3p = 4q \cr
& \Leftrightarrow q = \frac{{3p}}{4} \cr
& {\text{And, }}4q = 6r \cr
& \Leftrightarrow r = \frac{{2q}}{3} = \left( {\frac{2}{3} \times \frac{3}{4}p} \right) = \frac{p}{2} \cr
& \therefore p:q:r \cr
& = p:\frac{{3p}}{4}:\frac{p}{2} \cr
& = 4:3:2 \cr} $$
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Let their investments be Rs. 3x for p months
Rs. 4x for q months and Rs. 6x for r months respectively
Then,
$$\eqalign{
& 3xp:4xq:6xr = 1:1:1 \cr
& \Rightarrow 3p:4q:6r = 1:1:1 \cr
& {\text{So,}}3p = 4q \cr
& \Leftrightarrow q = \frac{{3p}}{4} \cr
& {\text{And, }}4q = 6r \cr
& \Leftrightarrow r = \frac{{2q}}{3} = \left( {\frac{2}{3} \times \frac{3}{4}p} \right) = \frac{p}{2} \cr
& \therefore p:q:r \cr
& = p:\frac{{3p}}{4}:\frac{p}{2} \cr
& = 4:3:2 \cr} $$
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