40 men can finish a piece of work in 26 days. How many men will be needed to finish it in 16 days?

Answer

To solve the problem step by step, we can use the concept of inverse proportion. Here's how we can find the number of men needed to finish the work in 16 days:

### Step 1: Understand the relationship
We know that if more men are working, the number of days required to finish the work decreases. This indicates an inverse relationship between the number of men and the number of days.

### Step 2: Set up the known values
From the problem, we have:
- \( x_1 = 40 \) men
- \( y_1 = 26 \) days
- \( y_2 = 16 \) days (the new number of days we want to finish the work)
- \( x_2 = ? \) (the number of men we need to find)

### Step 3: Use the inverse proportion formula
According to the inverse proportion rule:
\[ x_1 \times y_1 = x_2 \times y_2 \]
Substituting the known values into the equation:
\[ 40 \times 26 = x_2 \times 16 \]

### Step 4: Solve for \( x_2 \)
First, calculate \( 40 \times 26 \):
\[ 40 \times 26 = 1040 \]
Now we have:
\[ 1040 = x_2 \times 16 \]
To find \( x_2 \), divide both sides by 16:
\[ x_2 = \frac{1040}{16} \]

### Step 5: Perform the division
Calculating \( \frac{1040}{16} \):
\[ x_2 = 65 \]

### Conclusion
Therefore, the number of men needed to finish the work in 16 days is **65 men**.

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