A and B can finish a job in 15 hours by working together. A and C can finish the job in 10 hours by working together. C and B can finish the job in 6 hours by working together. How many hours will it take to finish the job if A, B, and C work together?

Let's solve this problem by finding the rates at which A, B, and C work individually, and then combine their rates to determine the time it would take for them to finish the job together.

### Step 1: Define the rates of A, B, and C
- Let the work done by A in one hour be \( a \).
- Let the work done by B in one hour be \( b \).
- Let the work done by C in one hour be \( c \).

### Step 2: Write down the equations based on the given information
1. A and B together can finish the job in 15 hours:
   \[
   a + b = \frac{1}{15}
   \]
2. A and C together can finish the job in 10 hours:
   \[
   a + c = \frac{1}{10}
   \]
3. B and C together can finish the job in 6 hours:
   \[
   b + c = \frac{1}{6}
   \]

### Step 3: Solve the system of equations
Add all three equations:
\[
(a + b) + (a + c) + (b + c) = \frac{1}{15} + \frac{1}{10} + \frac{1}{6}
\]
Simplify the left side:
\[
2a + 2b + 2c = \frac{1}{15} + \frac{1}{10} + \frac{1}{6}
\]
Factor out the 2 on the left side:
\[
2(a + b + c) = \frac{1}{15} + \frac{1}{10} + \frac{1}{6}
\]
Divide both sides by 2:
\[
a + b + c = \frac{1}{2} \left( \frac{1}{15} + \frac{1}{10} + \frac{1}{6} \right)
\]

### Step 4: Calculate the sum on the right side
First, find a common denominator for the fractions:
\[
\text{LCM of } 15, 10, \text{ and } 6 = 30
\]
Convert the fractions:
\[
\frac{1}{15} = \frac{2}{30}, \quad \frac{1}{10} = \frac{3}{30}, \quad \frac{1}{6} = \frac{5}{30}
\]
Add them together:
\[
\frac{2}{30} + \frac{3}{30} + \frac{5}{30} = \frac{10}{30} = \frac{1}{3}
\]
Now, divide by 2:
\[
a + b + c = \frac{1}{2} \times \frac{1}{3} = \frac{1}{6}
\]

### Step 5: Calculate the time taken when A, B, and C work together
The rate at which A, B, and C work together is \( \frac{1}{6} \). This means they complete \( \frac{1}{6} \) of the job in one hour.

Thus, the time taken to finish the job when A, B, and C work together is:
\[
\text{Time} = \frac{1}{\text{Rate}} = \frac{1}{\frac{1}{6}} = 6 \text{ hours}
\]

So, it will take 6 hours for A, B, and C to finish the job together.

Would you like more details or have any questions?

### Related Questions
1. If A alone were to do the job, how long would it take?
2. How long would it take B to finish the job alone?
3. How long would it take C to finish the job alone?
4. If A and B worked together for 5 hours, how much of the job would be left?
5. How much more efficient is the group working together compared to the most efficient pair?

### Tip
When dealing with work problems, always remember to consider the rates of work (work per unit time) rather than just the time taken. This approach simplifies combining the efforts of multiple workers.

Learn how to solve a work rate problem where A, B, and C work together to finish a job. Understand the rates of work for each individual and how they combine their efforts. Suitable for high school students studying algebra and rates of work.

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