Question

Some students planned a picnic. The total budget for hiring a bus was Rs.1440. Later on, eight of these refused to go and instead paid their total share of money towards the fee of one economically weak student, and thus the cost of each member, who went for picnic increased by $Rs.30$
How many students attended the picnic and how much money was paid towards the fees?

Answer 1

Total budget for hiring the bus $=Rs.1440$

No. of students who were supposed to attend $=x$

No. of students who attended $=x-8$

The fees paid by each member in case $1$ (when x would attend) $=\frac{1440}{x}$

The fees paid by each member in case $2$ $=\frac{1440}{x-8}$

According to the problem, we have

Final per head amount $=$ Intial per head amount $+$ 30 [$\because$ Each one has to pay $Rs.30$ extra when $8$ person is not going]

$\Rightarrow \frac{1440}{x-8}=\frac{1440}{x}+30$

$\Rightarrow \frac{1440}{x-8}-\frac{1440}{x}=30$

$\Rightarrow 1440(\frac{1}{x-8}-\frac{1}{x})=30$

$\Rightarrow 1440\left[\frac{x-(x-8)}{(x-8)x}\right]=30$

$\Rightarrow 1440\left[\frac{8}{(x^2-8x)}\right]=30$

$\Rightarrow 1440\times 8=30(x^2-8x)$

$\Rightarrow \frac{1440}{30}\times 8=(x^2-8x)$

$\Rightarrow 48\times 8=(x^2-8x)$

$\Rightarrow x^2-8x-384=0$

$\Rightarrow x^2-24x+16x-384=0$

$\Rightarrow x(x-24)+16(x-24)=0$

$\Rightarrow (x-24)(x+16)=0$

$\Rightarrow (x-24)=0or,(x+16)=0$

$\Rightarrow x=24or,x=-16$

Therefore, $x=24$ [$\because$ students cannot be negative]

No. of students who attended the picnic $=24-8=16$

Money paid by$8$ students towards the fees $=\frac{1440}{24}\times 8=Rs.480$