One man or two women or three children can do a work in 55 min. How long will it take for one man one woman and one child to do the work?

Let’s break down the problem and solve it step by step.

We know that:
- One man can do the work in 55 minutes.
- Two women together can do the same work in 55 minutes, so one woman alone would take \( 2 \times 55 = 110 \) minutes to complete the work.
- Three children together can do the work in 55 minutes, so one child alone would take \( 3 \times 55 = 165 \) minutes.

Now, we need to determine how long it would take for one man, one woman, and one child working together to complete the same task.

### Step 1: Calculate individual work rates.
The rate of work is the reciprocal of the time taken to complete the work:
- One man's rate of work: \( \frac{1}{55} \) of the work per minute.
- One woman's rate of work: \( \frac{1}{110} \) of the work per minute.
- One child's rate of work: \( \frac{1}{165} \) of the work per minute.

### Step 2: Sum of their work rates.
When working together, their combined work rate is the sum of their individual rates:
\[
\text{Combined rate} = \frac{1}{55} + \frac{1}{110} + \frac{1}{165}
\]

### Step 3: Calculate the combined work rate.
To add these fractions, we find a common denominator:
\[
\frac{1}{55} + \frac{1}{110} + \frac{1}{165} = \frac{6}{330} + \frac{3}{330} + \frac{2}{330} = \frac{11}{330}
\]

Thus, the combined work rate is \( \frac{11}{330} \), which simplifies to \( \frac{1}{30} \).

### Step 4: Find the total time.
Since the combined rate is \( \frac{1}{30} \) of the work per minute, it will take \( 30 \) minutes for one man, one woman, and one child to complete the work together.

### Final Answer:
It will take **30 minutes** for one man, one woman, and one child to complete the work together.

Would you like further details or have any other questions? 

Here are some related questions:
1. How do you calculate combined work rates for different groups?
2. What is the relationship between time and work rate in such problems?
3. How can you generalize this problem for other groups of people?
4. How would you solve this problem if the work rates weren't given directly?
5. What happens if multiple people work at different paces on different tasks?

**Tip:** In work-related problems, always sum the rates of individuals when they are working together, then find the reciprocal of the total rate to get the total time.

Discover how to calculate the time it takes for one man, one woman, and one child to complete a task together. This problem explores concepts in work and time, including individual and combined work rates. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation