Calculating The Rate Of Doing A Piece Of Work By Different Number Of Men

 

Men and Days Problems: Worker Time Rates

In math, when more people join a job, the work gets done faster (it takes fewer days). If fewer people do the job, it takes longer (more days). This is called inverse relationship.

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The "Man-Days" Formula

The total amount of work needed to finish a job can be measured in "Man-Days" (Men × Days). Because the total work stays the same, we use this simple formula:

M₁ × D₁ = M₂ × D₂

• M₁ = Number of men at first
• D₁ = Number of days the first group takes
• M₂ = Number of men in the second group
• D₂ = Number of days the second group will take

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Practice Questions and Solutions (Ascending Difficulty)

Question 1: Very Simple (Doubling the Workers)

If 5 men can complete a piece of work in 20 days, how long will it take 10 men to do the same work?

Solution:

• First Group: M₁ = 5 men, D₁ = 20 days
• Second Group: M₂ = 10 men, D₂ = ?
• Apply Formula: 5 × 20 = 10 × D₂
• Calculate: 100 = 10 × D₂
  D₂ = 100 ÷ 10 = 10 days

Answer: It will take 10 men 10 days.

Question 2: Simple (Reducing the Workers)

If 6 builders can build a wall in 4 days, how many days will it take just 2 builders to build the same wall?

Solution:

• First Group: M₁ = 6, D₁ = 4
• Second Group: M₂ = 2, D₂ = ?
• Apply Formula: 6 × 4 = 2 × D₂
• Calculate: 24 = 2 × D₂
  D₂ = 24 ÷ 2 = 12 days

Answer: It will take 2 builders 12 days.

Question 3: Medium (Finding Number of Men)

A group of 8 farmers can clear a farmland in 6 days. How many farmers are needed to clear the same farmland in exactly 4 days?

Solution:

• First Group: M₁ = 8, D₁ = 6
• Second Group: M₂ = ?, D₂ = 4
• Apply Formula: 8 × 6 = M₂ × 4
• Calculate: 48 = M₂ × 4
  M₂ = 48 ÷ 4 = 12 farmers

Answer: 12 farmers are needed.

Question 4: Hard (More Men Joining Later)

12 men are hired to dig a trench, and they can finish it in 5 days. If 3 more men join the team before they start, how many days will it take them all together?

Solution:

1. Find the new number of men: They started with 12 men, and 3 more joined. So, M₂ = 12 + 3 = 15 men.
2. Identify values: M₁ = 12, D₁ = 5, M₂ = 15, D₂ = ?
3. Apply Formula: 12 × 5 = 15 × D₂
4. Calculate: 60 = 15 × D₂
   D₂ = 60 ÷ 15 = 4 days

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