Rates and time are inversely proportional: If the working rates of A and B are in the ratio 3 : 4, then the numbers of days taken by A and B to finish the same job (working individually) will be in what ratio?

Introduction / Context:
Time-and-Work questions often pivot on the inverse relation between rate and time for a fixed amount of work. If one worker is faster, they take less time, and the ratios invert perfectly when comparing identical jobs.

Given Data / Assumptions:

• rate_A : rate_B = 3 : 4.
• They work individually on the same one-job task.

Concept / Approach:
For one complete job, time = 1 / rate. Therefore, the ratio of times is the reciprocal of the ratio of rates. If rate_A : rate_B = 3 : 4, then time_A : time_B = 1/3 : 1/4 = 4 : 3 after clearing denominators.

Step-by-Step Solution:
rate_A : rate_B = 3 : 4time_A : time_B = (1/3) : (1/4)Multiply both terms by 12 to clear denominators ⇒ 4 : 3

Verification / Alternative check:
Pick concrete rates: let rate_A = 3 units/day and rate_B = 4 units/day. For 12 units of work: time_A = 12/3 = 4 days; time_B = 12/4 = 3 days; thus time_A : time_B = 4 : 3, confirming the inverse relationship.

Why Other Options Are Wrong:

• 3 : 4 is the rate ratio, not the time ratio.
• 9 : 16 is the square of the rate ratio; time does not scale with the square of rates here.
• None of these: incorrect because a valid ratio (4 : 3) is available.

Common Pitfalls:

• Confusing the rate ratio with the time ratio.
• Forgetting to invert the ratio when moving from rates to times.

Final Answer:
4 : 3