Question: Some students planned a picnic. The budget for the food was Rs 2,500. As five of them unable to join the picnic, the cost of the food for each students increased by Rs 25. Find how many students went for the picnic?

Solution:

Let x number of students went for the picnic and the cost of the food for each student be Rs y.

According to the question,
The budget for the food was Rs 2,500.
or, xy = 2500
or, x = \frac{2500}{y} - (i)

Also, as five of them unable to join, the cost of food for each students increased by Rs 25.
or, (x -5)(y +25) = 2500
[ Put value of x from equation (i) ]

or, (\frac{2500}{y} -5)(y +25) = 2500

or, (\dfrac{2500 -5y}{y} × (y +25) = 2500

or, (2500-5y)(y+25) = 2500y

or, 2500(y +25) -5y(y +25) = 2500y

or, 2500y + 62500 -5y² - 125y = 2500y

or, 0 = 5y² +125y +2500y -2500y -62500

or, 0 = 5y² +125y -62500

or, 0 = 5(y² +25y -12500)

or, y² +25y -12500 = 0

or, y² +(125-100)y -12500 = 0

or, y² +125y -100y -12500 = 0

or, y(y +125) - 100(y -125) = 0

or, (y-100)(y +125) = 0

Either,
y -100 = 0
\therefore y = 100

Or,y +125 = 0
\therefore y = -125

We know, rate of cost per students can never be a negative value. So, y = 100.

Put value of y in equation (i), we get,
or, x = \frac{2500}{100}
\therefore x = 25

So, (x,y) = (20,125)

And, number of students who went for the lunch is (x -5) = (25-5) = 20 students.

Therefore, the required number of students that went for the picnic is 20 students.

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