Rani decided to distribute some amount to poor students for their books. If there are 8 students less, everyone will get ₹ 10 more. If there are 16 students more, everyone will get ₹ 10 less. What is the number of students and how much does each get? Also, find the total amount distributed?

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No. of students- 60, Amount per student- ₹ 40, Total amount- ₹ 800

No. of students- 22, Amount per student- ₹ 20, Total amount- ₹ 360

No. of students- 32, Amount per student- ₹ 30, Total amount- ₹ 960

No. of students- 72, Amount per student- ₹ 50, Total amount- ₹ 670

answer is A.

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Detailed Solution

It is given that if the count of students is 8 less then everyone will get ₹ 10 more and if the count of students is 16 more then everyone gets ₹ 10 less.
Let the amount given to one person be ₹ x.
Let the number of students be y.
So,
Total amount to be distributed =

yx .

Now, according to the first condition,

(y - 8) (x + 10) = yx

According to the second condition,

(y + 16) (x - 10) = yx

Therefore, we can say that,

(y − 8) (x + 10) = (y + 16) (x − 10)

Simplifying the above equation,

⇒ y x − 8 x + 10 y − 80 = y x + 16 x − 10 y − 160 ⇒ y x − y x + 160 − 80 = 16 x + 8 x − 10 y − 10 y ⇒ 80 = 24 x − 20 y … … (1)

Also, since the total amount to be distributed is

xy

in both the case,

⇒ y x = (y − 8) (x + 10) ⇒ y x = y x − 8 x + 10 y − 80 ⇒ y x − y x + 80 = 10 y − 8 x ⇒ 80 = 10 y − 8 x … … … (2)

Equating the RHS values from equation 1 and 2,

⇒ 10 y − 8 x = 24 x − 20 y ⇒ 10 y + 20 y = 24 x + 8 x ⇒ 30 y = 32 x ⇒ y = 3230 x .... (3)

Putting values of equation 3 in equation 1,

⇒ 80 = 24 x − 20 (3230 x) ⇒ 80 = 24 x − 643 x ⇒ 80 = 72 − 64 3 x ⇒ 80 = 83 x ⇒ x = 30

Putting the values of x in equation 3,

⇒ y = 32 x 30 ⇒ y = 32 (30) 30 ⇒ y = 32

Therefore, the number of students is y = 32 and the amount per student is x = ₹ 30.
Thus, the total amount will be
⇒32 × 30 = ₹ 960
Hence, option 1 is correct.

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