Question

A sum of money was divided equally among a certain number of persons had there been three persons more, each would have received $Rs.1$ less and there had been two persons less, each would have received $Rs.1$$\left(1\right)$more than he did. Find the number of persons and the amount each received.

Answer 1

Step 1: Represent the unknown quantities by two variables

Let , The number of persons be $x$and Total sum of money be$y$.

Money received by each person$=y/x$

Step 2: Two equations using assumed variables

If there $3$ more persons added $x$ becomes $x+3$ and each person received

$\frac{y}{x}-1Rs$.

Therefore the equation we get

$\frac{y}{x}+3=\frac{y}{x}-1...............\left(1\right)$

Step 3: Simplify the equation $\left(1\right)$

we get

$\frac{y}{x+3}=\frac{y-x}{x}(take\text{'}x\text{'}asL.C.M)$

Cross multiplying it becomes

$$\begin{gathered} y\times x=(x+3)(y-x) \\ xy=xy-x^2+3y-3x \\ xy-xy=-x^2+3y-x^2 \\ 0=3y-3x-x^2 \\ {x}^2+3x=3y..................\left(2\right) \end{gathered}$$

If there two persons less $x$ becomes $(x-2)$ and each person received $\frac{y}{x}+1$ Rs.

and the equation we get

$\frac{y}{x-2}=\frac{y}{x}+1$

Simplifying the equation

$\frac{y}{x-2}=\frac{y+x}{x}[take\text{'}x\text{'}asL.C.M]$

Cross multiplying we get

$$\begin{gathered} xy=(x-2)(y+x) \\ xy=xy+x^2-2y-2x \\ xy-xy=x^2-2y-2x \\ 0=x^2-2y-2x \\ {x}^2-2y=2y.........................\left(3\right) \\ \Rightarrow \frac{x^2-2y}{2}=y......................\left(4\right) \end{gathered}$$

Step 4: Using substitution method

Substitute the value of $y=\frac{x^2-2x}{2}$ in $\left(2\right)$

we get

$x^2+3x=3\left[\frac{x^{2-}2x}{2}\right]$

Cross multiplying we get

$$\begin{gathered} 2(x^2+3x)=3x^2-6x \\ \Rightarrow 2x^2+6x=3x^2+6x=0 \\ \Rightarrow 6x+6x=3x^2-2x^2 \\ 12x=x^2 \\ 12=\frac{x^2}{x}=x \\ x=12 \end{gathered}$$

Substitute the value of $x=12$ in equation $\left(4\right)$

we get

$$\begin{gathered} y=\frac{{12}^2-2\times 12}{2} \\ =\frac{144-24}{2}=\frac{120}{2}=60 \\ x=12\&y=60 \end{gathered}$$

Hence the number of persons $x=12$ and the sum of money $y=60$ Rs.