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CAT 2018 Slot 1 — QA Question 9

MixturesEasy

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 

Answer & solution

Correct answer: 19 : 24

  • A

    17 : 25

  • B

    18 : 25

  • C

    21 : 25

  • 19 : 24

Solution

Easy

Let costs per kg be aa for A and bb for B. A 3:23:2 mix sells at Rs. 4040/kg for 10%10\% profit; a 2:32:3 mix at the same Rs. 4040/kg gives 5%5\% profit. Over 55 kg the selling value is 200200 in both cases. Equate cost ×\times (profit factor) for the two mixes to get the ratio a:ba:b.

1

Mix 1 (3:23:2, 10%10\% profit). Cost of 55 kg is 3a+2b3a+2b; selling value of 55 kg is 5×40=2005\times40 = 200.

(3a+2b)×1.1=200\begin{aligned} &(3a + 2b)\times 1.1 = 200 \end{aligned}
2

Mix 2 (2:32:3, 5%5\% profit). Same selling value 200200.

(2a+3b)×1.05=200\begin{aligned} &(2a + 3b)\times 1.05 = 200 \end{aligned}
3

Equate and simplify. Both equal 200200, so set the left sides equal.

(3a+2b)(1.1)=(2a+3b)(1.05)⇒ 3.3a+2.2b=2.1a+3.15b⇒ 1.2a=0.95b(collect terms)\begin{aligned} &(3a+2b)(1.1) = (2a+3b)(1.05)\\ &\Rightarrow\ 3.3a + 2.2b = 2.1a + 3.15b\\ &\Rightarrow\ 1.2a = 0.95b \quad\text{(collect terms)} \end{aligned}
4

Form the ratio.

ab=0.951.2=1924\begin{aligned} &\frac{a}{b} = \frac{0.95}{1.2} = \frac{19}{24} \end{aligned}
a:b=19:24  ⇒  option (d)\boxed{a : b = 19 : 24 \;\Rightarrow\; \text{option (d)}}

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CAT 2018 Slot 1 QA Q9: 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 — Solution | TheCATExam