Time and Work - Mathematics (SSC CGL CHSL)
Let us now explore a very useful and flexible formula used to solve Time and Work problems quickly:
\( M_1 \times W_1 \times T_1 = M_2 \times W_2 \times T_2 \)
Where:
- M = number of workers (men, women, or machines)
- W = amount of work (can be assumed or given)
- T = time taken
Why does this work?
The concept is simple: Work = Men × Time.
If more people work, the task is completed faster. If fewer people work, more time is needed. The relation assumes equal efficiency unless otherwise stated.
Example 1: Simple Case with Same Work
Q: 6 men can complete a task in 12 days. How many days will 9 men take to complete the same task?
Step 1: Use the formula:
\[ 6 \times 1 \times 12 = 9 \times 1 \times T_2 \]
Step 2: Solve for \( T_2 \):
\[ T_2 = \frac{6 \times 12}{9} = 8 \text{ days} \]
Answer: 9 men will finish the work in 8 days.
Example 2: Work Gets Doubled
Q: 4 men can complete a work in 15 days. How many days will 5 men take to complete double the work?
Step 1: Set up the formula:
\[ 4 \times 1 \times 15 = 5 \times 2 \times T_2 \]
Step 2: Solve for \( T_2 \):
\[ T_2 = \frac{4 \times 15}{5 \times 2} = \frac{60}{10} = 6 \text{ days} \]
Answer: 5 men will take 6 days to complete double the work.
Example 3: Work Reduced, People Reduced
Q: 10 workers can finish 80 units of work in 20 days. If only 5 workers are available, how many days will they take to finish 60 units of work?
Step 1: Use the full formula (as work is different):
\[ 10 \times 80 \times 20 = 5 \times 60 \times T_2 \]
Step 2: Solve for \( T_2 \):
\[ T_2 = \frac{10 \times 80 \times 20}{5 \times 60} = \frac{16000}{300} = \frac{160}{3} \approx 53.33 \text{ days} \]
Answer: 5 workers will take 53⅓ days to complete 60 units of work.
Example 4: Total Work Unknown (Relative Comparison)
Q: A team of 8 men can build a wall in 25 days. How many men are required to build the wall in 10 days?
Step 1: Use the formula assuming same work:
\[ 8 \times 1 \times 25 = M_2 \times 1 \times 10 \]
Step 2: Solve for \( M_2 \):
\[ M_2 = \frac{8 \times 25}{10} = 20 \]
Answer: 20 men are needed to finish the same work in 10 days.
Example 5: Mix of Men and Women (Equal Efficiency)
Q: 6 men or 9 women can complete a task in 12 days. How many men and women working together (3 men and 3 women) can finish the same task in how many days?
Step 1: First, find total work in units:
6 men × 12 days = 72 man-days = Total work
Step 2: Convert women to men:
If 6 men ≡ 9 women, then 1 man ≡ 1.5 women
So 3 women ≡ 2 men ⇒ 3 men + 2 men = 5 men equivalent
Step 3: Time = Total work / Effective men per day = \( \frac{72}{5} = 14.4 \) days
Answer: Together, 3 men and 3 women can complete the work in 14.4 days.
Mixed Group Work (Men, Women, and Children)
In many practical problems, workers of different types — men, women, and children — work together. Their work capacities (efficiencies) may vary. The key is to assign a common base unit of work to compare and calculate total work done.
Let’s understand this with a detailed example:
Example 5 : 6 men or 9 women can complete a work in 12 days. In how many days will 3 men and 3 women together finish the same work?
Step 1: Find total work
Let’s calculate the total work in “man-days”: \[ 6 \text{ men} \times 12 \text{ days} = 72 \text{ man-days} \] So, total work = 72 units (assuming 1 man works 1 unit per day)
Step 2: Convert women's efficiency to men
Given: 6 men ≡ 9 women ⇒ 1 man = 1.5 women ⇒ 1 woman = \( \frac{2}{3} \) of a man
Step 3: Find total efficiency per day of 3 men + 3 women
3 men = 3 units/day 3 women = \( 3 \times \frac{2}{3} = 2 \) units/day Total = 3 + 2 = 5 units/day
Step 4: Time = Total Work / Daily Efficiency
\[ \text{Time} = \frac{72}{5} = 14.4 \text{ days} \]
Answer: They will complete the work in 14.4 days.
Example 6: Man, Woman, and Child Working Together
Q: 4 men, 6 women, and 3 children can complete a task in 10 days. One man = 2 women = 4 children in efficiency. In how many days can 2 men, 4 women, and 5 children do the same task?
Step 1: Find total work using first group
Let’s convert everything into "children units":
- 1 man = 4 children ⇒ 4 men = 16 children
- 6 women = 6 × 2 = 12 children
- 3 children = 3 children
Step 2: Efficiency of second group
- 2 men = 8 children
- 4 women = 8 children
- 5 children = 5 children
Step 3: Time = Work / Efficiency
\[ \text{Time} = \frac{310}{21} \approx 14.76 \text{ days} \]
Answer: About 14.76 days (or exactly \( \frac{310}{21} \))
Example 7: People Leaving the Work Midway
Q: 8 men can complete a task in 12 days. They work together for 6 days, and then 2 men leave. In how many more days will the remaining work be completed?
Step 1: Total work = 8 × 12 = 96 man-days
Step 2: Work done in first 6 days by 8 men:
8 × 6 = 48 man-days Remaining = 96 − 48 = 48 man-days
Step 3: Now 6 men are working
Time = \( \frac{48}{6} = 8 \) days
Answer: Remaining work will take 8 more days
Example 8: Workers Join in Later
Q: A job can be completed by 5 men in 20 days. After 10 days, 5 more men join. In how many more days will the job be completed?
Step 1: Total work = 5 × 20 = 100 man-days
Step 2: Work done in first 10 days:
5 × 10 = 50 man-days Remaining = 100 − 50 = 50 man-days
Step 3: New team = 10 men
Time = \( \frac{50}{10} = 5 \) more days
Answer: Total time = 10 + 5 = 15 days
📝 Summary Table: What to Remember
| Type | Key Point |
|---|---|
| Men/Women/Children | Convert all to one unit (usually man or child) |
| Different Efficiencies | Use ratios: 1 man = 2 women = 4 children, etc. |
| Leaving Midway | Calculate done and remaining work, adjust manpower |
| Joining Midway | Split total work into parts before/after joining |
These problem can be solved by ratio! If your are having diffulty in the conversion of workers in single worker. We can use ratio method to find the value of each worker in ratios M1W1T1 = M2W2T2 using ratio method.
Summary Notes:
- Always check if the work is same or changes (e.g., doubled, halved).
- Cancel work units when identical on both sides.
- If time is unknown, isolate it and solve. If men or work is unknown, do the same.
- This is the fastest method for many SSC questions — practice using it mentally!
Quick Tip:
This formula also applies to problems involving machines, pipes, or even groups of animals — just treat them as "workers" with consistent efficiency.
💬 Let us remember:
These types of questions are all about units of work. Stick to one unit (man-days, woman-days, etc.), and apply the same principle: \[ \text{Total Work} = \text{Efficiency per day} \times \text{Time} \] Once you master this, you’ll solve these questions faster than ever!