Sweets

Patricia loves Indian sweets. During her visit to the Indian Sweets Shop, she purchased three different types of sweets – Rasgulla, Gulab Jamun and Barfi.

She purchased as many pieces of each type of sweets as its price in dollar. Each of the three types of sweet cost a different amount per piece. She spent less than $100.

Rasgulla cost the most, Gulab Jamun cost less and Barfi cost the least per piece. Compared to Rasgulla, one of the other two types of sweet costs $ 5 less per piece.

If she paid an average of $ 7 for each piece of sweet, how many of each type did she purchase?

Patricia purchased 9 Rasgulla, 4 Gulab Jamun and 1 Barfi. She paid $ 98.

Assume that she purchased R pieces of Rasgulla. Also assume that P and Q are the number of pieces of other sweets.

It is given that - she purchased as many pieces of each type of sweets as its price in dollar and she paid an average of $ 7 for each piece of sweet. Hence,
7 * (R + P + Q) = R*R + P*P + Q*Q

Since the average price per piece is $ 7 and Rasgulla cost the most, per piece cost of Rasgulla should be more than $ 7.

Also, it is given that - Compared to Rasgulla, one of the other two types of sweet costs $ 5 less per piece.

Consider R = 8 and hence, P = 3
7 * (8 + 3 + Q) = 8*8 + 3*3 + Q*Q
7 * (11 + Q) = 64 + 9 + Q*Q
Q*Q – 7*Q – 4 = 0
For Q = 1, 2, 4, 5, 6, 7, above equation is not valid.

Consider R = 9 and hence, P = 4
7 * (9 + 4 + Q) = 9*9 + 4*4 + Q*Q
7 * (13 + Q) = 81 + 16 + Q*Q
Q*Q – 7*Q + 6 = 0
Q = 6 or Q = 1

Q can not be 6 as total cost (81 + 36 + 16 = 133) would be more than $ 100. Hence, Q=1.

For all other values of R, the total cost would be more than $ 100.

Thus, Patricia purchased 9 Rasgulla, 4 Gulab Jamun and 1 Barfi. She paid $ 98.

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