Question

On her birthday, Seema decided to donate some money to children of an orphanage home. If there were $8$ children less everyone would have got Rs.$10$ more, however if there were $16$ children more everyone would have got Rs.$10$ less. Using matrix method, find the number of children and the amount distributed by Seema. What values are reflected by the Seema?

Answer 1

according to the question,

Let number of children = $x$

Amount received by one children = $y$

total amount received = $xy$ .....(1)

According to first condition :

If no.of children = ($x$ - 8)

Amount received by one = ($y$ + 10)

total amount= ($x$- 8)($y$ + 10)

implies $xy$ + 10$x$ - 8$y$ - 80 ......(2)

put (1) = (2) {both are total amount}

$xy$ = $xy$ 10$x$ - $y$ + 80

implies 10$x$ - 8$y = 80

implies 5$x$ - 4$y$ = 40 ....(3)

According to the second condition:

If No. of children = ($x$ + 16)

Amount received by one = ($y$ - 10)

total amount = ($x$ +16)($y$- 10)

implies $xy$ - 10$x$ + 16$y$ -160 .....(4)

put (1) = (4) {both are total amount}

implies $xy$= =$xy$ -10$x$ + 16$y$

- 160

implies -10$y$ + 16$y$= 160

implies -5$y$ + 8$y$ = 80 ......(5)

We can express (3) and (5) in Matrix as follow :

A = 5-4-58, B = 4080, $x$ = $xy$

As we know,

AX = B

implies X = $A^{-1}$ B

To find $A^{-1}$ , We need coffector of A

$A_{11}$ = 8

$A_{21}=5$

$A_{12}=4$

$A_{22}=5$

Adjoint of A = 8455

$A^{-1}$ = 1 AAdjoint of A

impliesA-1 = 1208455

impliesA-1 B = 120320 + 320200 + 400

implies it A-1 B = 3230 ....(7)

From (6) and (7), We get

$xy$ = 3230

So No of children = 32

Amount got by one children = 30

Total Amount = 960