In a system of two meshed cogged wheels, one wheel has 32 cogs and the other wheel has 54 cogs. The wheels work into each other without slipping. If the wheel with 54 cogs makes 80 complete revolutions in three-quarters of a minute, how many complete revolutions will the 32-cog wheel make in 8 seconds?

Introduction / Context:
This aptitude problem on cogged wheels (gears) tests the concept of relative motion and how rotational speed changes when two wheels with different numbers of cogs mesh with each other. Such questions are common in quantitative aptitude exams because they combine proportional reasoning with unit conversions involving minutes and seconds.

Given Data / Assumptions:

• First wheel has 32 cogs.
• Second wheel has 54 cogs.
• The 54-cog wheel makes 80 revolutions in three-quarters of a minute, that is in 45 seconds.
• The wheels are perfectly meshed without slipping, so the linear motion of the teeth is the same for both wheels.
• We must find how many revolutions the 32-cog wheel makes in 8 seconds.

Concept / Approach:
When two cogged wheels mesh, the number of teeth passing the point of contact per unit time is the same for both wheels. Therefore, the speed of rotation is inversely proportional to the number of cogs. If one wheel has more cogs, it turns more slowly, and if it has fewer cogs, it turns more quickly. We first find the angular speed of the 54-cog wheel, then use the tooth ratio to obtain the speed of the 32-cog wheel, and finally convert that result to revolutions in 8 seconds.

Step-by-Step Solution:
Step 1: Time for 80 revolutions of the 54-cog wheel is 45 seconds, so its revolutions per second are 80 / 45.Step 2: For meshed gears, speed of 32-cog wheel / speed of 54-cog wheel = number of cogs on 54-cog wheel / number of cogs on 32-cog wheel = 54 / 32.Step 3: Revolutions per second of 32-cog wheel = (80 / 45) * (54 / 32).Step 4: Simplify: 80 * 54 / (45 * 32) = 135 / 45 = 3 revolutions per second.Step 5: In 8 seconds, revolutions of 32-cog wheel = 3 * 8 = 24.

Verification / Alternative check:
If the smaller wheel has fewer cogs, it must rotate faster than the larger wheel.The ratio of speeds should be 54 : 32, which reduces to 27 : 16, a value greater than 1, confirming that the 32-cog wheel is faster.Using 3 revolutions per second for the smaller wheel is reasonable because in 45 seconds it would complete 135 revolutions, which matches 80 revolutions of the larger wheel times 54 / 32.

Why Other Options Are Wrong:
Values like 36, 38 or 48 assume incorrect proportionality or mishandled time conversion. For example, 48 would correspond to using a wrong ratio or doubling the correct value. The option 30 is also incorrect because it ignores the exact 45 second to 8 second proportion combined with the tooth ratio.

Common Pitfalls:
Candidates often reverse the ratio of cogs and incorrectly assume that speed is directly proportional to the number of cogs. Another frequent error is to forget that three-quarters of a minute is 45 seconds, not 30 seconds, which leads to wrong revolutions per second. Some learners also multiply by the time interval in minutes rather than converting everything into seconds before calculating.

Final Answer:
The 32-cog wheel makes 24 complete revolutions in 8 seconds.