Understanding the Bridge Repair Problem

This question asks us to figure out how many days it will take to repair a bridge if more people are added to the workforce. This is a classic problem involving work and time, specifically dealing with inverse proportion.

Concept of Inverse Proportion in Work

In many work problems, the number of workers and the time taken to complete a job are inversely proportional. This means that if you increase the number of workers, the time required to finish the same amount of work decreases, assuming all workers work at the same rate.

The relationship can be expressed as:

\( \text{Number of Workers} \times \text{Time Taken} = \text{Total Work} \)

Since the total work (repairing the bridge) remains constant, we can say:

\( M_1 \times D_1 = M_2 \times D_2 \)

Where:

• \( M_1 \) = Initial number of workers
• \( D_1 \) = Initial number of days
• \( M_2 \) = Final number of workers
• \( D_2 \) = Final number of days

Analyzing the Given Information

Let's break down the information provided in the question:

In the new situation, 14 more people join the initial 56. So, the new number of persons is \( 56 + 14 = 70 \).

\( M_2 = 70 \)

Calculating the New Time Taken

Using the inverse proportion formula \( M_1 \times D_1 = M_2 \times D_2 \), we can plug in the values we know:

\( 56 \times 20 = 70 \times D_2 \)

Now, we need to solve for \( D_2 \):

1. Start with the equation: \( 56 \times 20 = 70 \times D_2 \)
2. Calculate the product on the left side: \( 1120 = 70 \times D_2 \)
3. Isolate \( D_2 \) by dividing both sides by 70: \( D_2 = \frac{1120}{70} \)
4. Simplify the fraction: \( D_2 = \frac{112}{7} \)
5. Perform the division: \( D_2 = 16 \)

So, if 14 more people join, the bridge can be repaired in 16 days.

Conclusion on Bridge Repair Time

When the number of people working on the bridge repair increases from 56 to 70, the time taken decreases from 20 days to 16 days, which is consistent with the principle of inverse proportion in work problems.

Revision Table: Work and Time Concepts

Additional Information: Applying Inverse Proportion

Inverse proportion isn't just limited to work problems like bridge repair. It appears in various situations:

• Speed and Time: If you travel a fixed distance, increasing your speed decreases the time taken. \( \text{Speed} \times \text{Time} = \text{Distance} \) (constant).
• Pressure and Volume (Boyle's Law): For a fixed mass of gas at constant temperature, pressure is inversely proportional to volume. \( P_1 V_1 = P_2 V_2 \).
• Number of Items and Cost Per Item: If you spend a fixed amount of money, the number of items you can buy is inversely proportional to the price per item. \( \text{Number of Items} \times \text{Cost per Item} = \text{Total Money} \) (constant).

Understanding inverse proportion is key to solving many mathematical and real-world problems efficiently.