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Question:

A tea shop offers tea in cups of three different sizes. The product of the prices, in INR, of three different sizes is equal to 800. The prices of the smallest size and the medium size are in the ratio 2 : 5. If the shop owner decides to increase the prices of the smallest and the medium ones by INR 6 keeping the price of the largest size unchanged, the product then changes to 3200. The sum of the original prices of three different sizes, in INR, is

Explanation:

Let the price of small, medium and large cups be Rs. 2x, 5x and p respectively.

⇒ 2x × 5x × p = 800   …(1)

Also, (2x + 6) × (5x + 6) × p = 3200   …(2)

(2) = (1) × 4

⇒ (2x + 6) × (5x + 6) × p = 2x × 5x × p × 4

⇒ 10x² + 42x + 36 = 40x²

⇒ 30x² - 42x - 36 = 0

⇒ 5x² – 7x – 6 = 0

⇒ 5x² – 10x + 3x – 6 = 0

⇒ (5x + 3)(x - 2) = 0

⇒ x = 2

∴ Price of small cup = 2x = 4
Price of medium cup = 5x = 10
Price of large cup = 800/(4 × 10) = 20

⇒ Sum of the prices of three cups = 4 + 10 + 20 = 34.

Hence, 34.