Question

Anuj decided to donate some money for books for the children living in an orphanage. If there are 8 children less, everyone will get Rs. 20 more. If there are 7 children more everyone will get Rs. 10 less. What is the number of children and how much does each get? What is the total amount distributed? Why did Anuj decide to distribute money for books?

Answer 1

Hint: First we take a variable as the number of children and another for the amount given to each child.
Then applying the two conditions we will get two equations.
By solving the equations we will get the number of children and the amount is given to each child.
By multiplying these two we can derive at the total amount he spent.

Complete step-by-step answer:
It is given that Anuj donated some money for books for the children living in an orphanage.
Let, the number of children be \[x\] and the amount given to each child be \[y\]
Then the total amount anuj donated is \[xy\].
It is given that when there are 8 children less, then the number of children are\[\;x - 8\]
Also, everyone will get Rs. 20 more.
That is each child will get the amount Rs.\[y + 20\]
Then the total amount is Rs.\[\left( {x - 8} \right)\left( {y + 20} \right)\].
Since the total amount is fixed we get, \[\left( {x - 8} \right)\left( {y + 20} \right) = xy\]
Let us simplify the above equation and name it as equation (1).
\[xy + 20x - 8y - 160 = xy\]
\[10x - 4y = 80....(1)\]
Also it is given that when there are 7 more children, then the number of children are\[\;x + 7\]
Also, everyone will get Rs. 10 less.
That is each child will get the amount Rs.\[y - 10\]
Then the total amount is Rs.\[\left( {x + 7} \right)\left( {y - 10} \right)\]
Since the total amount is fixed we get,\[\;\left( {x + 7} \right)\left( {y - 10} \right) = xy\]
Let us simplify the above equation and name it as equation (2).
\[xy - 10x + 7y - 70 = xy\]
\[ - 10x + 7y = 70...(2)\]
Let us add (1) and (2) we get,
\[ - 4y + 7y = 80 + 70\]
Let us simplify the above equation to get\[y\],
\[3y = 150\]
\[y = \dfrac{{150}}{3} = 50\]
Let's put the value of y in (1), so that we can find x,
\[10x - 4 \times 50 = 80\]
\[10x = 80 + 200\]
On solving the above equation, we get,
\[x = \dfrac{{280}}{{10}} = 28\]
Hence the number of children is 28.
The amount given to each child is Rs. 50.
The total amount distributed is \[28 \times 50 = 1400\] .
No conclusion can be drawn about why Anuj decided to distribute money for books.

Note: Whatever we do with the number of students or the amount given to each student making more or less the total amount is fixed. This plays a major role in finding the total amount distributed.