Answer to Question #100075 in Algebra for Tanay

Question

In a meeting a government officer wanted to distribute certain number of apples equally
among the physically handicapped children. If the number of children was 2 less, each of
them would have got one apple more. Again when the ten officers present in the day are
also included, then each of them will get 3 apples less. Find the number of the
handicapped children and the number of apples

Answer 1

Let a - amount of apples, c - amount of children. Will compose a system of equations describing these two conditions.

$\quad\begin{cases} \cfrac{a}{c-2} - \cfrac{a}{c} = 1 \\ \cfrac{a}{c} - \cfrac{a}{c+10} = 3 \end{cases}$

Will carry out the elementary transformations

$\quad\begin{cases} \cfrac{ac-ac+2a}{c(c-2)} = 1 \\ \cfrac{ac+10a-ac}{c(c+10)} = 3 \end{cases}$

$\quad \begin{cases} 2a = c(c-2) \qquad\qquad\qquad \small\text{(1)} \\ 10a = 3c(c+10) \qquad\qquad\space \small\text{ (2)} \end{cases}$

Will multiply the first equation by 5, equate the right sides of the equations and determine the $c$ .

$\quad 5c(c-2) = 3c(c+10)\\ \quad5c-3c=30+10\\ \quad c=\cfrac{40}2\\ \quad\underline{c=20}$

Will substitute $c$ in equation (1) and define the $a$

$\quad2a = 20*(20 - 2)\\ \quad a = \cfrac12*20*18\\ \quad\underline{a = 180}$

Answer: 20 children, 180 apples.