If a : b is 32 : 35 and b : c is 21 : 32, then what is the simplified ratio a : c?

Introduction / Context:
This question checks understanding of how to combine two given ratios involving three variables a, b and c. Such problems are common in ratio and proportion chapters of aptitude tests. We are given a : b and b : c, and we need to find the relationship between a and c directly as a : c.

Given Data / Assumptions:

• a : b = 32 : 35.
• b : c = 21 : 32.
• All quantities are positive and we want the simplified ratio a : c.

Concept / Approach:
From the two ratios, we can write a/b and b/c as fractions. The ratio a : c is obtained by multiplying a/b with b/c, because a/c = (a/b) * (b/c). After we compute this product, we simplify the resulting fraction to lowest terms to get the final ratio in simplest form.

Step-by-Step Solution:

Verification / Alternative check:
We can construct actual numbers that satisfy both ratios. Let a = 32k and b = 35k from the first ratio. From the second ratio, let b = 21m and c = 32m. Equate b: 35k = 21m. Choose k = 3 and m = 5 so that 35 * 3 = 105 and 21 * 5 = 105. Then a = 32 * 3 = 96 and c = 32 * 5 = 160. The ratio 96 : 160 simplifies by dividing both parts by 32, giving 3 : 5, which matches our earlier result.

Why Other Options Are Wrong:

• 5 : 3 would correspond to a/c = 5/3, which is the reciprocal of the correct ratio.
• 1 : 1 would only occur if a and c were equal, which they are not.
• 5 : 7 does not agree with the product of the given ratios and is simply a distractor.

Common Pitfalls:
Some students mistakenly add the ratios or average them instead of multiplying a/b by b/c. Others attempt to directly compare 32 : 35 and 21 : 32 without forming a fraction, leading them away from the correct method. Remember that to link a and c when b is common, we rely on multiplication of the corresponding fractions and then simplify.

Final Answer:
The required ratio is 3 : 5.