Time and Work

Time and Work deals with the relationship among:

• The amount of work

• The time taken

• The rate or efficiency of doing work

It’s widely applicable in questions involving people, machines, pipes, and tanks.

🔑 Key Concepts

1. Work: A job or task, often considered as 1 whole unit.

2. Time: The duration to complete the work.

3. Rate/Efficiency: Work done per unit of time.

🧮 Formulas

1. Work = Rate × Time

2. Rate = Work / Time

3. Time = Work / Rate

4. If A can do a work in ‘x’ days, then A’s 1 day work = 1/x

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🧠 Important Shortcuts & Tricks

✅ Shortcut 1: Work done = 1/Time taken

If A can finish a work in 10 days → A’s 1 day work = 1/10

✅ Shortcut 2: Combined work

If A does a work in x days and B in y days:
A + B together can do the work in:

$$\frac{xy}{x+y}\text{ days}$$

Example: A = 6 days, B = 12 days
→ Together = (6×12)/(6+12) = 72/18 = 4 days

✅ Shortcut 3: Efficiency Ratio

If A is twice as efficient as B:
→ A finishes in half the time of B
→ Their efficiency ratio = 2:1
→ Their time ratio = 1:2 (Inverse)

✅ Shortcut 4: Worker and Time Proportion (Inverse Relationship)

More men ⇒ less time
If 4 men complete work in 6 days, then to finish it in 3 days:

$$4\times 6=x\times 3\to x=8\text{ men}$$

✅ Shortcut 5: Pipes and Cisterns

Treat inflow as a positive rate, outflow as a negative rate.

If a pipe can fill a tank in x hours, then 1 hour of work = 1/x
If another empties in y hours → 1 hour work = -1/y

Combined rate = 1/x – 1/y

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⚡ Important Note: Use the LCM Method

Let the total work be the LCM of individual time values to make calculations easier.

Example: A can do a work in 10 days, B in 15 days
→ LCM = 30 units
→ A's 1 day work = 3 units, B's = 2 units
→ Together = 5 units/day
→ Total time = 30 / 5 = 6 days

📌 Examples

🔹 Example 1

A can do a job in 5 days → 1 day work = 1/5

🔹 Example 2

A = 6 days, B = 12 days → Together = (6×12)/(6+12) = 72/18 = 4 days

🔹 Example 3

A and B together = 1/8 work/day, A alone = 1/12
→ B = 1/8 - 1/12 = (3 - 2)/24 = 1/24
→ B alone = 24 days

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🧠 MCQs on Time and Work

1. A can do a job in 10 days. How much of the job is done in 1 day?

A) 10
B) 1/10
C) 5
D) 0

✅ Answer: B) 1/10

2. A and B can do the work in 6 and 12 days, respectively. How long together?

A) 4 days
B) 3 days
C) 5 days
D) 6 days

✅ Answer: A) 4 days
➤ (6×12)/(6+12) = 72/18 = 4

3. A is twice as efficient as B. Together, they do a job in 12 days. Time taken by B alone?

A) 18
B) 24
C) 36
D) 30

✅ Answer: C) 36 days
➤ Let B = x days → A = x/2 →
Combined = (1/x + 2/x) = 3/x = 1/12 → x = 36

4. If 6 men can complete a task in 8 days, how many men are needed to do it in 4 days?

A) 3
B) 12
C) 9
D) 24

✅ Answer: B) 12
➤ 6×8 = x×4 → x = 12

5. A and B can do the work in 10 days. B alone can do it in 15 days. In how many days will A alone complete the work?

A) 30
B) 24
C) 40
D) 20

✅ Answer: D) 20
➤ A = 1/10 – 1/15 = (3 – 2)/30 = 1/30 → A alone = 30 days

Oops! That’s wrong!
Correcting:

Combined = 1/10, B = 1/15
→ A = 1/10 – 1/15 = (3 - 2)/30 = 1/30
✅ Answer: A) 30 days