Time and Work Problems: Tricks, Formulas, Mock Tests

The Basics: Understanding the Concept

In time and work problems, we deal with three main variables:
Work = Efficiency × Time.

Trick 1: Assume Total Work as LCM

Instead of dealing with fractions, assume the total work as the LCM of the time taken by individuals. This simplifies calculations.

Example

If A can complete a task in 10 days and B in 15 days, what is the total time taken if they work together?

1. Assume total work = LCM of 10 and 15 = 30 units.
2. A’s efficiency = 30/10 = 3 units/day.
3. B’s efficiency = 30/15 = 2 units/day.
4. Combined efficiency = 3 + 2 = 5 units/day.
5. Total time = Total work / Combined efficiency = 30/5 = 6 days.

Trick 2: Inverse Proportionality for More Workers

If more people join, the time taken decreases. Use inverse proportionality to solve such problems.

Example

If 4 workers can build a wall in 8 days, how long will 8 workers take?

1. Workers × Time = Constant.
2. 4 workers × 8 days = 32 worker-days.
3. 8 workers × Time = 32 worker-days.
4. Time = 32/8 = 4 days.

Trick 3: Efficiency Ratios

When efficiencies are given in ratios, use them directly to find work rates.

Example

A is twice as efficient as B. If B can complete a task in 12 days, how long will A take?

1. Efficiency ratio (A:B) = 2:1.
2. Time ratio (A:B) = 1:2 (since time is inversely proportional to efficiency).
3. If B takes 12 days, A takes 6 days.

Trick 4: Work Done in Stages

Sometimes, workers join or leave midway. Break the problem into stages and calculate work done in each stage.

Example

A can complete a task in 20 days. After 5 days, B joins, and together they finish the remaining work in 6 days. How long would B take to complete the task alone?

1. Total work = 20 units (A’s efficiency = 1 unit/day).
2. Work done by A in 5 days = 5 units.
3. Remaining work = 20 - 5 = 15 units.
4. A and B together complete 15 units in 6 days.
5. Combined efficiency = 15/6 = 2.5 units/day.
6. B’s efficiency = Combined efficiency - A’s efficiency = 2.5 - 1 = 1.5 units/day.
7. Time taken by B alone = Total work / B’s efficiency = 20/1.5 = 13.33 days.

Trick 5: Vedic Math Shortcut for Combined Work

Use the formula: Time taken by A and B together = (A × B) / (A + B).

Example

If A can do a task in 12 days and B in 18 days, how long will they take together?

1. Time taken together = (12 × 18) / (12 + 18) = 216/30 = 7.2 days.

Real-Life Example: Teamwork in Office

Imagine you and your colleague are working on a project. You can finish it in 10 hours, and your colleague takes 15 hours. If you both work together, how long will it take?

1. Assume total work = LCM of 10 and 15 = 30 units.
2. Your efficiency = 30/10 = 3 units/hour.
3. Colleague’s efficiency = 30/15 = 2 units/hour.
4. Combined efficiency = 3 + 2 = 5 units/hour.
5. Total time = 30/5 = 6 hours.

Quick Recap: Key Takeaways

• Assume total work as LCM for easy calculations.
• Use inverse proportionality for more/less workers.
• Efficiency ratios help compare work rates.
• Break problems into stages for complex scenarios.
• Use Vedic math shortcuts for combined work.

Practice Problem

A can complete a task in 12 days, and B can do it in 18 days. They work together for 4 days, and then A leaves. How long will B take to finish the remaining work?

Hint: Use the LCM method and break the problem into stages.

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