Directions: 20Men completes a work in Todays working 5 hours a day, 30 women completes a work in 20 days working 8 hours a day and 30 boys completes a work in 15 days working 4 hours a day..calculate the efficiency 

To calculate the efficiency of men, women, and boys, we need to find out the total work done by each group and then compare it. We can use the concept that "Work done (W) = Number of Workers (N) × Number of Days (D) × Number of Hours per Day (H) × Efficiency (E)". Let's denote the efficiency of one man, one woman, and one boy as \( E_m \), \( E_w \), and \( E_b \), respectively.

### Step 1: Calculate Total Work Done by Each Group
Assume the total work is the same for each group.

**For 20 men:**
\[
\text{Work done by men} = 20 \times 10 \times 5 \times E_m = 1000 \times E_m
\]

**For 30 women:**
\[
\text{Work done by women} = 30 \times 20 \times 8 \times E_w = 4800 \times E_w
\]

**For 30 boys:**
\[
\text{Work done by boys} = 30 \times 15 \times 4 \times E_b = 1800 \times E_b
\]

### Step 2: Equate the Work Done
Since the total work is the same for all:

\[
1000 \times E_m = 4800 \times E_w = 1800 \times E_b
\]

Now we can express the efficiencies relative to each other:

\[
E_m : E_w : E_b = \frac{1}{1000} : \frac{1}{4800} : \frac{1}{1800}
\]

### Step 3: Simplify the Ratios
To find the ratio, we take the inverse:

\[
E_m : E_w : E_b = 4800 : 1000 : 4800
\]

Simplify by dividing all terms by 400:

\[
E_m : E_w : E_b = 12 : 2.5 : 6
\]

Or, multiplying by 2 to eliminate the decimal:

\[
E_m : E_w : E_b = 24 : 5 : 12
\]

### Conclusion
The efficiency ratio of one man, one woman, and one boy is \( 24 : 5 : 12 \).

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Would you like a deeper explanation of any part of this problem, or do you have any further questions?

Here are five related questions to expand your understanding:

1. How would the efficiency ratio change if the number of hours worked per day were different for each group?
2. What happens to the work completion time if the number of men, women, or boys is doubled?
3. Can you calculate the total work in terms of man-hours if each group had a different total work assigned?
4. How would you calculate the efficiency if another group (e.g., girls) were added to the problem?
5. What if one group takes a break for a day? How does that affect the total work done?

**Tip:** Understanding how to manipulate and solve ratios can simplify a wide range of problems, especially when comparing efficiencies or work done.

Learn how to calculate the efficiency of 20 men, 30 women, and 30 boys working on a task with different work durations and hours per day. Understand the concept of efficiency and solve for ratios between different groups. Suitable for high school students studying work and time problems.

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