Arithmetic

Profit & Loss

Medium

Two types of tea, $A$ and $B$ , are mixed and then sold at Rs. $40$ per kg. The profit is $10 \%$ if A and B are mixed in the ratio $3: 2$ , and $5 \%$ if this ratio is $2: 3$ . The cost prices, per kg , of $A$ and $B$ are in the ratio

$17:25$

$21:25$

$18:25$

$19:24$

✅ Correct Option: 4

When dealing with mixture problems involving profit percentages, we need to connect three key concepts:

Cost Price (CP): What we pay to buy the tea

Selling Price (SP): What we sell the tea for

Profit Percentage: The extra money we make as a percentage of cost price

If profit is 10%, then SP = CP × (1 + 10/100) = CP × 1.1

Let's call the cost prices of tea A and B as Ca and Cb respectively.

Given Information:

Selling price of mixture = Rs. 40 per kg (constant in both cases)

Case 1: A:B = 3:2 ratio gives 10% profit

Case 2: A:B = 2:3 ratio gives 5% profit

When we mix 3 kg of A with 2 kg of B, we get 5 kg of mixture.

Cost of this mixture = 3Ca + 2Cb

Average cost per kg = (3Ca + 2Cb)/5

Since we make 10% profit:

40 = [(3Ca + 2Cb)/5] × 1.1

40/1.1 = (3Ca + 2Cb)/5 ---------(1)

When we mix 2 kg of A with 3 kg of B, we get 5 kg of mixture.

Cost of this mixture = 2Ca + 3Cb

Average cost per kg = (2Ca + 3Cb)/5

Since we make 5% profit:

40 = [(2Ca + 3Cb)/5] × 1.05

40/1.05 = (2Ca + 3Cb)/5 ---------(2)

Instead of solving each equation individually, let's use a smart approach by dividing equation (1) by equation (2):

$\frac{40/1.1}{40/1.05} = \frac{(3Ca + 2Cb)/5}{(2Ca + 3Cb)/5}$

The 40s and 5s cancel out:

$\frac{1.05}{1.1} = \frac{3Ca + 2Cb}{2Ca + 3Cb}$

Converting to fractions (since 1.05 = 21/20 and 1.1 = 22/20):

$\frac{21/20}{22/20} = \frac{21}{22}$

So: $\frac{21}{22} = \frac{3Ca + 2Cb}{2Ca + 3Cb}$

Cross-multiplying:

$21(2Ca + 3Cb) = 22(3Ca + 2Cb)$

Expanding both sides:

$42Ca + 63Cb = 66Ca + 44Cb$

Collecting like terms:

$63Cb - 44Cb = 66Ca - 42Ca$

$19Cb = 24Ca$

Therefore: $\frac{Ca}{Cb} = \frac{19}{24}$

The cost prices of tea A and B are in the ratio 19:24.

This makes sense because tea A is cheaper than tea B, so when we use more of A (3:2 ratio), we get higher profit (10%) compared to when we use more of B (2:3 ratio gives only 5% profit).

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CAT Past Year Questions (Topic-Wise)

Two types of tea, $A$ and $B$ , are mixed and then sold at Rs. $40$ per kg. The profit is $10 \%$ if A and B are mixed in the ratio $3: 2$ , and $5 \%$ if this ratio is $2: 3$ . The cost prices, per kg , of $A$ and $B$ are in the ratio