In today's competitive world, understanding how team composition impacts work efficiency is crucial for maximizing productivity. This article explores a mathematical problem involving a specific work scenario, providing insights into the optimal allocation of resources to achieve project completion as quickly as possible. By tackling the problem step-by-step, we will illustrate the importance of team dynamics and work rate calculations in real-world applications.

Problem Statement and Objective

A team consisting of 1 man, 2 women, and 3 boys can complete a piece of work in 30 days. Another team of 3 men, 2 women, and 1 boy can accomplish the same work in 20 days. The objective is to determine how many days it would take for a team of 12 men, 12 women, and 12 boys to complete the work together.

Step-by-Step Solution

Step 1: Establish Work Rates

Let's denote the work rates of a man, a woman, and a boy as M, W, and B respectively. The problem provides us with two scenarios:

1 man, 2 women, and 3 boys can complete the work in 30 days. 3 men, 2 women, and 1 boy can complete the work in 20 days.

Mathematically, these scenarios can be represented as:

1M 2W 3B 1/30

3M 2W 1B 1/20

Step 2: Set Up the Equations

From the given information, we can set up the following equations:

Equation 1: 1M 2W 3B 1/30 Equation 2: 3M 2W 1B 1/20

Step 3: Solve the Equations

To solve the equations simultaneously, we can start by isolating one of the variables. First, we isolate M from Equation 1:

M (1/30) - 2W - 3B

Next, we substitute this expression for M into Equation 2:

3[(1/30) - 2W - 3B] 2W B 1/20

Simplify and solve for W and B. After further simplification, we find:

W 2B 1/80

Step 4: Substitute Back to Find Individual Rates

Now, we substitute W 1/80 - 2B back into Equation 1 to find the relationship between M, W, and B:

M (1/30) - 2[(1/80) - 2B] - 3B

Further simplification gives:

M B 1/120

Step 5: Find the Combined Rate of 12 Men, 12 Women, and 12 Boys

The combined rate is:

12M 12W 12B 12(M W B)

Substituting the values of M, W, and B, we find:

12[(B 1/120) (1/80 - 2B) B] 12[1/60]

This simplifies to:

Note: 12[1/48] 1/4

So, the combined rate for 12 men, 12 women, and 12 boys is 1/48, meaning they can complete the work in 4 days.

Final Answer

The team of 12 men, 12 women, and 12 boys will complete the work together in 4 days.