Averages , Ratio and Proportion Question 7 - CAT 2021
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:
Step-by-Step Explanation
Let the original prices of the cups be:
Small = 2x
Medium = 5x
Large = y
- Original Product:
2x × 5x × y = 800
10x²y = 800
x²y = 80 --- (Equation 1)
- Analyze the updated state:
The prices of the small and medium cups increase by ₹6, while the large cup's price remains unchanged:
New Product = (2x + 6)(5x + 6)y = 3200 --- (Equation 2)
Divide Equation 2 by Equation 1 to cancel y:
[(2x + 6)(5x + 6)y]/(10x²y) = 3200/800
[(2x + 6)(5x + 6)]/(10x²) = 4
(10x² + 42x + 36) = 40x²
30x² − 42x − 36 = 0
Divide the quadratic equation by 6:
5x² − 7x − 6 = 0
Factorizing:
(5x + 3)(x − 2) = 0
Since the price multiplier must be positive: x = 2
- Find the original values:
Small price = 2(2) = 4
Medium price = 5(2) = 10
From Equation 1:
(2²)y = 80
4y = 80
y = 20
Large price = 20
Sum of original prices = 4 + 10 + 20 = 34
Correct Answer: 34