Time & Work

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Quick recall

• Convert completion times into work rates before adding.
• Worker-hours comparisons assume equal, constant efficiency.

On this page

Formula Cards

1. Individual work rate

$$R_A=\frac{1}{a}$$

a = days to complete one whole job.

2. Combined rate

$$\frac{1}{T}=\frac{1}{a}+\frac{1}{b}$$

Add another reciprocal for each additional worker.

3. Efficiency and time

$$\text{Efficiency}\propto\frac{1}{\text{Time}}$$

Twice the efficiency means half the time for the same job.

4. Worker-hours for a fixed job

$$M_1D_1H_1=M_2D_2H_2$$

Assumes equal, constant worker efficiency. In general, work is proportional to workers × days × hours × efficiency.

Worked Examples

Example 1

Question: A finishes a job in 10 days and B in 15 days. Find their time together.

Step 1: Add daily work rates.

$$R=\frac{1}{10}+\frac{1}{15}=\frac{1}{6}$$

Step 2: Invert the combined rate.

$$T=6\text{ days}$$

Answer: 6 days.

Example 2

Question: A, B, C finish a job in 10, 15, 30 days. Find their combined time.

Step 1: Add all three work rates.

$$R=\frac{1}{10}+\frac{1}{15}+\frac{1}{30}=\frac{1}{5}$$

Step 2: Invert the rate.

$$T=5\text{ days}$$

Answer: 5 days.

Example 3

Question: A is twice as efficient as B. Together they finish in 12 days. Find A's time alone.

Step 1: Let B's daily rate be x and A's rate be 2x.

$$2x+x=\frac{1}{12}$$

Step 2: Solve for A's rate.

$$x=\frac{1}{36},\qquad2x=\frac{1}{18}$$

Step 3: Invert A's rate.

$$T_A=18\text{ days}$$

Answer: 18 days.

Example 4

Question: Ten equally efficient workers finish a job in 12 days. How long for 15 workers?

Step 1: Keep worker-days constant.

$$10\times12=15D$$

Step 2: Solve.

$$D=\frac{120}{15}=8$$

Answer: 8 days.

Drill

Try each question before opening its answer.

Question 1

A=8 days, B=12 days. Together?

• Show answer

  4.8 days

Question 2

A does a job in 20 days, B in 25. They work together for 5 days, then A leaves. How many more days for B?

• Show answer

  11.25 days

Question 3

12 men build a wall in 15 days. How many men to finish in 9 days?

• Show answer

  20 men

Question 4

A is thrice as efficient as B. If B takes 24 days, find combined time.

• Show answer

  6 days

Question 5

A, B, C finish a job in 6, 8, 12 days respectively. Working together, time taken?

• Show answer

  2.67 days

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