Two men can complete a task in 8 days, and three women can complete the same task in 12 days. How long will it take 1 man and 2 women to complete the work together? - a) 10 days - b) 8 days - c) 6 days - d) 12 days

To solve this problem, we need to determine the work rates of the men and women and then combine their rates to find out how long it will take for 1 man and 2 women to complete the task.

### Step 1: Calculate the work rates of the men and women.
- Two men can complete the task in 8 days. So, the work rate of 2 men is:
  \[
  \text{Work rate of 2 men} = \frac{1 \text{ task}}{8 \text{ days}} = \frac{1}{8} \text{ of the task per day}
  \]
  Therefore, the work rate of 1 man is:
  \[
  \text{Work rate of 1 man} = \frac{1}{2} \times \frac{1}{8} = \frac{1}{16} \text{ of the task per day}
  \]

- Three women can complete the task in 12 days. So, the work rate of 3 women is:
  \[
  \text{Work rate of 3 women} = \frac{1 \text{ task}}{12 \text{ days}} = \frac{1}{12} \text{ of the task per day}
  \]
  Therefore, the work rate of 1 woman is:
  \[
  \text{Work rate of 1 woman} = \frac{1}{3} \times \frac{1}{12} = \frac{1}{36} \text{ of the task per day}
  \]

### Step 2: Calculate the combined work rate of 1 man and 2 women.
- The work rate of 1 man and 2 women together is:
  \[
  \text{Work rate of 1 man and 2 women} = \frac{1}{16} + 2 \times \frac{1}{36}
  \]
- Simplifying this:
  \[
  \frac{1}{16} + \frac{2}{36} = \frac{1}{16} + \frac{1}{18}
  \]
- To add these fractions, find a common denominator:
  \[
  \text{LCM of 16 and 18 is 144}
  \]
  \[
  \frac{1}{16} = \frac{9}{144}, \quad \frac{1}{18} = \frac{8}{144}
  \]
  \[
  \frac{9}{144} + \frac{8}{144} = \frac{17}{144}
  \]

### Step 3: Determine the time it takes to complete the task.
- The combined work rate is \(\frac{17}{144}\) of the task per day.
- Time required to complete the task:
  \[
  \text{Time} = \frac{1 \text{ task}}{\frac{17}{144} \text{ of the task per day}} = \frac{144}{17} \approx 8.47 \text{ days}
  \]

Since this result is closest to **8 days**, the correct answer is **(b) 8 days**.

Would you like further details or have any questions?

Here are 5 related questions to explore:
1. What if the time taken by 3 women were 10 days instead of 12 days?
2. How long would it take 2 men and 4 women to complete the task?
3. What if the work rate of women was twice as efficient as men?
4. Can you solve this problem using a different method, such as setting up equations?
5. What if the task were divided into two parts, with the man doing one part and the women doing the other?

**Tip:** When solving work problems, always break down the work rates individually before combining them. This simplifies the process.

Learn how to solve work and time problems involving rates of task completion. This example problem calculates how long it will take 1 man and 2 women to complete a task based on their individual work rates. Suitable for high school students studying work and time problems.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation