CAT 2017

Slot 2

QA

Question & Solution

1. Concept Used

• Topic: Ratio and Proportion — Combining Ratios
• Formula: $$ a : b : c = \frac{a}{b} \times \frac{b}{c} \text{ (chain ratio method)} $$

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2. Calculation

We are given two separate ratios: $$a : b = 3 : 4$$ and $$b : c = 2 : 1$$.

To combine these into a single ratio $$a : b : c$$, we need the value of $$b$$ to be consistent (i.e., the same number) in both ratios.

In the first ratio, $$b = 4$$. In the second ratio, $$b = 2$$. The LCM of 4 and 2 is 4, so we scale the second ratio by 2:

$$b : c = 2 : 1 \implies b : c = 4 : 2$$

Now both ratios share the same value for $$b$$:

$$a : b : c = 3 : 4 : 2$$

So we can write $$a = 3x,\ b = 4x,\ c = 2x$$ for some positive integer $$x$$.

Therefore:

$$a + b + c = 3x + 4x + 2x = 9x$$

This means $$(a + b + c)$$ must always be a multiple of 9.

Now let's check each option:

$$201 \div 9 = 22.\overline{3} \quad \text{(Not a multiple of 9)}$$

$$205 \div 9 = 22.\overline{7} \quad \text{(Not a multiple of 9)}$$

$$207 \div 9 = 23 \quad \text{(Multiple of 9 ✅)}$$

$$210 \div 9 = 23.\overline{3} \quad \text{(Not a multiple of 9)}$$

Only 207 is divisible by 9, corresponding to $$x = 23$$, giving $$a = 69,\ b = 92,\ c = 46$$.

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3. Solution

Answer = Option C ✅

The final calculated value is 207, which is the only multiple of 9 among the given options.