Question

A and B are partners in a business . A contributes $\left( {\dfrac{1}{4}} \right)$ of the capital for 15 months and B received $\left( {\dfrac{2}{3}} \right)$ of the profit. Find for how long B’s money was used?

Answer 1

Hint: In order to solve the given question, we should know the important concepts related to the question that is Ratio and Proportion. We can say that the comparison or simplified form of two quantities of the same kind is referred to as ratio. This relation gives us how many times one quantity is equal to the other quantity. The sign used to denote a ratio is ‘:’. Also the Proportion is an equation which defines that the two given ratios are equivalent to each other. In other words, the proportion states the equality of the two fractions or the ratios. When two ratios are equal in value, then they are said to be in proportion. In simple words, it compares two ratios. Proportions are denoted by the symbol ‘::’ or ‘=’. Using these concepts we can get our required solution.

Complete step by step solution:
We are given the following information in the question
A’s contribution of capital = $\left( {\dfrac{1}{4}} \right)$
Therefore, B’s contribution of capital = $\left( {\dfrac{3}{4}} \right)$
Profit received by B = $\left( {\dfrac{2}{3}} \right)$
Therefore, Profit received by A = $1 - \dfrac{2}{3} = \dfrac{1}{3}$
Now, we will see the proportion of the Capital’s ratio to Profit’s sharing ratio that is
Let B’s money was used for $x$ months.
We will now calculate –
Capital’s ratio :: Profit’s sharing ratio
We can represent it as –
$
   \Rightarrow \dfrac{1}{4} \times 15:x \times \dfrac{3}{4}::\dfrac{1}{3}:\dfrac{2}{3} \\
   \Rightarrow \dfrac{{15}}{4}:\dfrac{{3x}}{4}::\dfrac{1}{3}:\dfrac{2}{3} \\
   \Rightarrow \dfrac{{15}}{4}:\dfrac{{3x}}{4}::\dfrac{1}{3}:\dfrac{2}{3} \\
   \Rightarrow 15:3x::1:2 \\
 $
This can also be written as –
\[
   \Rightarrow \dfrac{{15}}{{3x}} = \dfrac{1}{2} \\
   \Rightarrow 3x = 30 \\
   \Rightarrow x = 10 \\
 \]
Therefore, the B’s money was used for \[x = 10\] months.

Note:
> The ratio should exist between the quantities of the same kind .
> Ratio and Proportion are explained majorly based on fractions. When a fraction is represented in the form of a:b, then it is a ratio whereas a proportion states that two ratios are equal.
> While comparing two things, the units should be similar
> There should be significant order of terms
> The comparison of two ratios can be performed, if the ratios are equivalent like the fractions.