How Many Days Will 12 People Take to Complete a Piece of Work if 16 People Can Finish It in 9 Days?

Often, in problem-solving and productivity tasks, determining the number of days required for a different number of people to complete a piece of work can be complex. This article leads you through a detailed explanation of how to solve such problems, focusing on the relationship between the number of people, working hours, and the total time required to complete a task.

Understanding the Work Rate Concept

The work rate concept is a fundamental principle that helps us understand how the number of people, their working hours, and the total time required to complete a task interrelate. By utilizing this concept, we can calculate the exact number of days required for a different number of people to accomplish the same work. Let’s dive into a detailed example to clarify this concept.

Example: 16 People Complete Work in 9 Days

Let's consider the initial scenario where 16 people complete their work in 9 days. We need to determine how many days 12 people would take to complete the same work. This type of problem often requires the use of the formula:

M1 × D1 M2 × D2

Where:

M1 Number of people in the first scenario D1 Number of days in the first scenario M2 Number of people in the second scenario D2 Number of days in the second scenario, which we need to find

In our example, M1 16, D1 9, and M2 12. We need to solve for D2.

Step-by-Step Solution

16 × 9 144 man hours per day

12 × 8 96 man hours per day

144/96 × D2 10 days

Therefore, 12 men can do the same work in 10 days.

Explanation:

The total work done is first calculated by multiplying the number of people (16) by their working hours (9), giving 144 man hours. Next, we need to find out the number of man hours a group of 12 people can complete in one day. This is given by 12 × 8, which equals 96 man hours per day. Equating these two expressions, we get 144/96 × D2 10. Solving for D2, we get: 144/96 1.5 1.5 × D2 10 D2 10 / 1.5 D2 10/1.5 100/15 20/3 ≈ 6.67

Alternative Calculation Method

Another method to solve the same problem is to directly calculate the total man hours required and then find out how many days it would take for 12 people to complete the work. Here’s how:

Person 16

Working hour 9/day

No of days 10

So total man hours 16×9 ×10 1440

Now person 12

Working hours 8

Let no of days X

So total man hours 12 ×8 ×X

12× 8 ×X 1440.

X 1440/9615

Days reqd. 15

Explanation:

Total man hours required to complete the work is 16 × 9 × 10 1440 man hours. Now, for 12 people working 8 hours a day, the total man hours required to complete the work is 12 × 8 × X, where X is the number of days required. Equating the two expressions, we get: 12 × 8 × X 1440 192X 1440 X 1440 / 192 X 15 days

Indirect Relationship and Work Rate

The relationship between the number of people and the time required to complete a piece of work is inversely proportional. This means that as the number of people increases, the time required to complete the work decreases, and vice versa. This inverse relationship can be visually represented using the formula: 16 × 9 12 × X

Solving for X:

144 12X X 144 / 12 X 12

Therefore, 12 people can complete the work in 12 days, which aligns with the calculations provided previously.

Practical Application

This principle of work rate is widely applicable in various fields, including project management, construction, and manufacturing. Understanding how to calculate the number of days required or the number of people needed to complete a task based on the available resources is essential for efficient planning and organization.

Conclusion

In conclusion, when 16 people can complete a piece of work in 9 days, 12 people can do the same work in approximately 15 days. This is based on the principle that the total work required remains constant, and the relationship between the number of people and the time to complete the work is inversely proportional.