Three taps are emptying a tank. Tap A can empty the full tank in 8 hours, tab B in 7 hours and tap C in 5 hours. If all taps are open together, how long would it take to empty the full tank?

To solve this problem, we calculate the rate at which each tap empties the tank and combine their rates to find the total time required to empty the tank when all taps are open.

### Step 1: Rate of emptying by each tap
- **Tap A** empties the tank in 8 hours. Its rate is:  
  \[
  \text{Rate of A} = \frac{1}{8} \, \text{tanks per hour}.
  \]

- **Tap B** empties the tank in 7 hours. Its rate is:  
  \[
  \text{Rate of B} = \frac{1}{7} \, \text{tanks per hour}.
  \]

- **Tap C** empties the tank in 5 hours. Its rate is:  
  \[
  \text{Rate of C} = \frac{1}{5} \, \text{tanks per hour}.
  \]

### Step 2: Total rate when all taps are open
The combined rate when all three taps are open is the sum of their individual rates:  
\[
\text{Total rate} = \frac{1}{8} + \frac{1}{7} + \frac{1}{5}.
\]

To calculate this, we find the least common denominator (LCD) of \(8, 7, 5\), which is \(280\). Expressing each fraction with the LCD:  
\[
\frac{1}{8} = \frac{35}{280}, \quad \frac{1}{7} = \frac{40}{280}, \quad \frac{1}{5} = \frac{56}{280}.
\]

Adding these fractions:  
\[
\text{Total rate} = \frac{35}{280} + \frac{40}{280} + \frac{56}{280} = \frac{131}{280}.
\]

### Step 3: Time to empty the tank
The time to empty the tank is the reciprocal of the total rate:  
\[
\text{Time} = \frac{1}{\text{Total rate}} = \frac{1}{\frac{131}{280}} = \frac{280}{131} \, \text{hours}.
\]

### Final Answer:
\[
\text{Time to empty the tank} = \frac{280}{131} \, \text{hours} \approx 2.14 \, \text{hours (rounded to two decimal places)}.
\]

Would you like further clarification or more details about this calculation?

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### Related Questions:
1. How is the least common denominator calculated for fractions?
2. What is the role of reciprocals in rate and time problems?
3. Can this problem be extended to include a tank filling tap (opposing rate)?
4. How do such calculations apply to real-world engineering problems?
5. What methods can be used to check the accuracy of the final result?

### Tip:
When solving combined rate problems, always work with fractions rather than decimals until the final step to ensure precision.

Three taps are emptying a tank. Tap A can empty the full tank in 8 hours, tap B in 7 hours, and tap C in 5 hours. If all taps are open together, how long would it take to empty the full tank?

Discover how to calculate the time needed to empty a tank using three taps working together. Tap A empties the tank in 8 hours, Tap B in 7 hours, and Tap C in 5 hours. Includes step-by-step solutions and concepts like rates of work, least common denominators, and reciprocals. Suitable for Grades 6-8.

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