# Some students planned a picnic. The budget of the food was Rs. 480. As 8 of them failed to join the party, the cost of the food for each member increased by Rs. 10. Find how many students went for the picnic.

A. 22

B. 16

C. 24

D. 17

Last updated date: 29th Mar 2023

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Answer

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Hint: since the budget is given but neither the no, of students nor the cost per student but we can form an algebraic equation from this by equating the product of no. of students and cost per student to the budget. Using the second condition we can form another algebraic equation and then proceed further.

Complete step-by-step answer:

Let the no. of students be X

Let the cost per student be Y

∴ XY = 480

\[ \Rightarrow \]Y = \[\dfrac{{480}}{X}\]

Now, 8 students didn’t arrived, each food would have cost Rs. 10 more

\[ \Rightarrow \] (X – 8)(y + 10) = 480

\[ \Rightarrow \]\[(X - 8)\left( {\dfrac{{480}}{X} + 10} \right) = 0\]

\[ \Rightarrow \]\[{X^2} - 8X - 384 = 0\]

\[ \Rightarrow \]\[{X^2} + 16X - 24X - 320 = 0\]

\[ \Rightarrow \]\[\left( {X{\text{ }} - {\text{ }}24} \right)\left( {X{\text{ }} + {\text{ }}16} \right){\text{ }} = {\text{ }}0\]

So \[X{\text{ }} = {\text{ }}24\] or \[ - 16\]

Since no. of students cannot be negative therefore -16 is discarded’

∴ No. of students = 24.

Thus the no. of students went for picnic = 24 – 8 = 16

So, the correct option is ‘B’.

Note: The quadratic equation in this question can also be solved by using quadratic formula i.e.

\[ \Rightarrow \]\[X = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\] Where a is coefficient of \[{X^2}\], b is coefficient of \[X\] and c is the constant.

Complete step-by-step answer:

Let the no. of students be X

Let the cost per student be Y

∴ XY = 480

\[ \Rightarrow \]Y = \[\dfrac{{480}}{X}\]

Now, 8 students didn’t arrived, each food would have cost Rs. 10 more

\[ \Rightarrow \] (X – 8)(y + 10) = 480

\[ \Rightarrow \]\[(X - 8)\left( {\dfrac{{480}}{X} + 10} \right) = 0\]

\[ \Rightarrow \]\[{X^2} - 8X - 384 = 0\]

\[ \Rightarrow \]\[{X^2} + 16X - 24X - 320 = 0\]

\[ \Rightarrow \]\[\left( {X{\text{ }} - {\text{ }}24} \right)\left( {X{\text{ }} + {\text{ }}16} \right){\text{ }} = {\text{ }}0\]

So \[X{\text{ }} = {\text{ }}24\] or \[ - 16\]

Since no. of students cannot be negative therefore -16 is discarded’

∴ No. of students = 24.

Thus the no. of students went for picnic = 24 – 8 = 16

So, the correct option is ‘B’.

Note: The quadratic equation in this question can also be solved by using quadratic formula i.e.

\[ \Rightarrow \]\[X = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\] Where a is coefficient of \[{X^2}\], b is coefficient of \[X\] and c is the constant.

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