1. Understanding Reciprocal Formulas

In work problems, we often deal with rates, which is how much work is done in a specific time. The key formula to remember is:

1/A + 1/B = 1/T

Where:

  • A = time taken by worker A to complete the job
  • B = time taken by worker B to complete the job
  • T = time taken by both workers together to complete the job

For example, if Worker A can complete a task in 3 hours and Worker B in 6 hours, their combined rate can be calculated as follows:

1/3 + 1/6 = 1/T

Finding a common denominator (which is 6), we get:

2/6 + 1/6 = 1/T

3/6 = 1/T

This simplifies to:

T = 2 hours

2. Product-Over-Sum Shortcut

When only two workers are involved, you can use the product-over-sum shortcut formula:

T = (A * B) / (A + B)

This is particularly useful for quickly calculating how long it will take for two workers to complete a task together. Using the previous example:

T = (3 * 6) / (3 + 6) = 18 / 9 = 2 hours

3. Sample Practice Questions & Rationalizations

Question 1: Worker A can finish a job in 4 hours, while Worker B can do it in 6 hours. How long will it take for both to finish the job together?
A) 2.4 hours
B) 3.0 hours
C) 2.5 hours
D) 3.5 hours
Answer: Using the reciprocal formula: 1/4 + 1/6 = 1/T, T = 2.4 hours. Correct answer: A.
Question 2: If Worker A can complete a task in 5 hours and Worker B in 10 hours, how long will it take them to finish the task when working together?
A) 7 hours
B) 3.33 hours
C) 6 hours
D) 4 hours
Answer: Using product-over-sum: T = (5*10)/(5+10) = 3.33 hours. Correct answer: B.
Question 3: If a leak in a tank can empty it in 12 hours, how long will it take to fill the tank if it is being filled at a rate of 1/4 tank per hour?
A) 12 hours
B) 16 hours
C) 24 hours
D) 8 hours
Answer: This is a leak problem. Effective rate = 1/4 - 1/12. Solving gives T = 8 hours. Correct answer: D.
Question 4: Worker A can do a job in 8 hours, but leaves after 2 hours. Worker B finishes the remaining work in 4 hours. How long would Worker B take to complete the entire job alone?
A) 8 hours
B) 6 hours
C) 10 hours
D) 4 hours
Answer: Worker A completes 1/4 of the job in 2 hours. Worker B does the remaining 3/4 in 4 hours, so T = (8 * 4) / (8 + 4) = 3.2 hours alone. Correct answer: A.

4. Key Traps to Avoid & Test Day Tips

Be mindful of common traps such as miscalculating the total time when workers leave midway. Always verify the rates and ensure you understand whether the problem involves filling or emptying. On test day, manage your time wisely and practice mental math shortcuts to speed up calculations.

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