Question

Seven friends planned a tea party.The expenses per boy in rupees is numerically 1 less than the number of girls and the expenses per girl in rupees is numerically 1 less than the number of boys. If the ratio of the total expenses of the boys and the girls is 8:9,then what is the expenditure of each boy?

Answer 1

Answer :

Let,

No of Girls = x
No. of Boys = y
From the question , we get

x + y = 7

y = 7 - x --------------------1

The expenses per boy = x - 1

The expenses per girl = y - 1

Total expense by the boys = y$\times$ ( x - 1 )

Total expense by the girls = x $\times$( y - 1 )

Given
Ratio of the Total expense of the boys to the girl = 8 : 9
Hence
$\Rightarrow$ $\frac{y\left(x-1\right)}{x\left(y-1\right)}=\frac{8}{9}$
$\Rightarrow$9( xy - y ) = 8( xy - x )

$\Rightarrow$ 9xy - 9y = 8xy - 8x

$\Rightarrow$8x - 9y + xy = 0

Substitute value from equation 1 , And get

$\Rightarrow$8x - 9( 7 - x ) + x( 7 - x )​ = 0

$\Rightarrow$8x - 63 +9 x + 7x - x² = 0

$\Rightarrow$- x² + 24x - 63 = 0

Multiply by -1 on both hand side , we get

$\Rightarrow$x² - 24x + 63 = 0

Using splitting middle term method we get

$\Rightarrow$x² - 21x - 3x + 63 = 0

$\Rightarrow$x ( x - 21 ) - 3 (x - 21 )=0

$\Rightarrow$( x - 21 )(x - 3 ) = 0

$\Rightarrow$ x = 21 , 3

x can’t be equal to 21 as given x + y = 7
So,
we take x = 3
So total girls = 3
hence expenditure of each boy = x - 1 = 3 - 1 = 2 (Ans)