Seven men can dig 145ft of trench in 6.5 hours. Three men can  backfill 100 ft of trench in 4 hours. How long will it take 6 men  to dig and fill 200ft of trench, in hours? SOLVE ALGEBRAICALLY

Let's break down the problem step-by-step and solve it algebraically.

### 1. Digging Rate Calculation
First, calculate the rate at which one man digs the trench.

- Seven men dig 145 feet in 6.5 hours.
- Rate of seven men digging: \(\text{Rate}_{\text{7 men}} = \frac{145}{6.5} = 22.3077 \, \text{ft/hour}\).

- Rate of one man digging: 
\[
\text{Rate}_{\text{1 man}} = \frac{22.3077}{7} = 3.1868 \, \text{ft/hour}
\]

### 2. Time for 6 Men to Dig 200 Feet
- The rate of 6 men digging: 
\[
\text{Rate}_{\text{6 men}} = 6 \times 3.1868 = 19.1208 \, \text{ft/hour}
\]

- Time to dig 200 feet:
\[
\text{Time}_{\text{dig}} = \frac{200}{19.1208} \approx 10.46 \, \text{hours}
\]

### 3. Backfilling Rate Calculation
Now, calculate the rate at which one man backfills the trench.

- Three men backfill 100 feet in 4 hours.
- Rate of three men backfilling: \(\text{Rate}_{\text{3 men}} = \frac{100}{4} = 25 \, \text{ft/hour}\).

- Rate of one man backfilling: 
\[
\text{Rate}_{\text{1 man}} = \frac{25}{3} \approx 8.3333 \, \text{ft/hour}
\]

### 4. Time for 6 Men to Backfill 200 Feet
- The rate of 6 men backfilling: 
\[
\text{Rate}_{\text{6 men}} = 6 \times 8.3333 = 50 \, \text{ft/hour}
\]

- Time to backfill 200 feet:
\[
\text{Time}_{\text{backfill}} = \frac{200}{50} = 4 \, \text{hours}
\]

### 5. Total Time to Dig and Backfill 200 Feet
The total time to dig and backfill 200 feet is the sum of the digging time and the backfilling time.

\[
\text{Total Time} = \text{Time}_{\text{dig}} + \text{Time}_{\text{backfill}} = 10.46 + 4 = 14.46 \, \text{hours}
\]

Hence, it will take approximately 14.46 hours for 6 men to dig and backfill 200 feet of trench.

Do you have any questions or would like more details? 

Here are some related questions:

1. How would the time change if there were only 5 men available?
2. What if the trench needed to be 300 feet long instead of 200 feet?
3. Can we find a formula for the time taken based on the number of men?
4. What happens if the backfill rate decreases by 10%?
5. How long would it take if only 4 men were available for backfilling?

**Tip:** Always double-check your unit conversions and calculations when dealing with rates to ensure accuracy!

Seven men can dig 145ft of trench in 6.5 hours. Three men can backfill 100 ft of trench in 4 hours. How long will it take 6 men to dig and fill 200ft of trench, in hours?

Learn how to solve a work and time problem using rates and algebra. Calculate how long it will take 6 men to dig and backfill 200 feet of trench based on their individual and group digging and backfilling rates. Suitable for students in Grades 8-10.

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