Time and Work - Mathematics (SSC CGL CHSL)
Using Ratio Method — M₁W₁T₁ = M₂W₂T₂
This equation is based on the principle that: Total Work Done = Number of Workers × Work Units per Day × Time
In competitive exams like SSC CGL and CHSL, this formula is super useful when we deal with two scenarios or two different teams working under different conditions.
M₁W₁T₁ = M₂W₂T₂
Where:
M = Number of Men (or any worker)
W = Work per Day (Efficiency per worker)
T = Time in Days
Example 1: Direct Application (Same Work Type)
Q: 5 men can complete a work in 12 days. How many days will 6 men take to complete the same work?
Solution:
Using ratio method: \[ M₁T₁ = M₂T₂ \Rightarrow 5 \times 12 = 6 \times T₂ \Rightarrow T₂ = \frac{60}{6} = 10 \text{ days} \]Answer: 10 days
Example 2: Mixed Workers Without Converting
Q: 4 men and 6 women can do a work in 8 days. How many days will 6 men and 4 women take to do the same work, if their efficiencies are in ratio 3:2?
Step 1: Use ratio instead of converting
Let man’s efficiency = 3, woman’s = 2
Step 2: Apply formula using weighted efficiencies
Left side = (4×3 + 6×2) × 8 = (12 + 12) × 8 = 24 × 8 = 192 Right side = (6×3 + 4×2) × T₂ = (18 + 8) × T₂ = 26 × T₂
\[ 192 = 26 × T₂ \Rightarrow T₂ = \frac{192}{26} = 7.38 \text{ days (approx)} \]Answer: 7.38 days
Example 3: Men, Women, and Children Together
Q: 3 men, 5 women, and 2 children can complete a work in 10 days. In how many days will 4 men, 4 women, and 4 children do the same work? Efficiencies are: Man : Woman : Child = 6 : 4 : 2
Step 1: Weighted workers
Total work by first group per day = 3×6 + 5×4 + 2×2 = 18 + 20 + 4 = 42 units Total work = 42 × 10 = 420 units
Step 2: Efficiency of second group
4×6 + 4×4 + 4×2 = 24 + 16 + 8 = 48 units/day Time = 420 / 48 = 8.75 days
Answer: 8.75 days
Example 4: Ratio Method When Some Leave the Job
Q: 8 men can do a job in 15 days. They work for 6 days, and then 2 men leave. In how many more days will the remaining work be completed?
Step 1: Total Work = 8×15 = 120 man-days
Step 2: Work done in first 6 days
8 × 6 = 48 man-days Remaining = 120 − 48 = 72 man-days
Step 3: Remaining work done by 6 men
T = 72 / 6 = 12 days
Answer: 12 more days
Example 5: Ratio Method in Reverse — Find Number of Workers
Q: 6 men can do a work in 16 days. How many men are needed to do it in 12 days?
Apply ratio:
\[ M₁T₁ = M₂T₂ \Rightarrow 6 × 16 = M₂ × 12 \Rightarrow M₂ = \frac{96}{12} = 8 \]Answer: 8 men
Tips for Ratio Method
- Don’t convert into one worker — use their efficiency ratio directly.
- Calculate total efficiency of each group (multiply number × efficiency).
- Then apply: Total Work = Group Efficiency × Time
- Or set: (E₁ × T₁) = (E₂ × T₂) when work is constant
| Scenario | Approach |
|---|---|
| Different groups, same work | Use M₁W₁T₁ = M₂W₂T₂ |
| Mixed worker types | Assign ratios, use group efficiency |
| Joining/leaving midway | Break into parts before and after |