Three men A, B and C workin together can do a job in 2 hours less time than A alone, in 5 hour less time than B alone and in one half the time needed by C when working alone. Then A,B and C together can do the job in how many days?
In short trick

To solve this problem using a shortcut or trick, let's break it down step by step:

### Step 1: Assign variables to the time taken by A, B, and C alone to complete the job.
- Let \( A \) take \( x \) hours to complete the job alone.
- Let \( B \) take \( y \) hours to complete the job alone.
- Let \( C \) take \( z \) hours to complete the job alone.

### Step 2: Set up the given conditions.
1. **A, B, and C together can do the job in 2 hours less time than A alone:**
   \[
   \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{1}{x - 2}
   \]
2. **A, B, and C together can do the job in 5 hours less time than B alone:**
   \[
   \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{1}{y - 5}
   \]
3. **A, B, and C together can do the job in half the time needed by C alone:**
   \[
   \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{2}{z}
   \]

### Step 3: Simplify using a trick.

Notice from the third condition:
\[
\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{2}{z}
\]

Now, let's express the times in terms of a single variable \( T \), which is the time taken by A, B, and C together:
\[
T = \frac{1}{\frac{1}{x} + \frac{1}{y} + \frac{1}{z}}
\]
From the equation \( \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{2}{z} \), we get:
\[
T = \frac{z}{2}
\]

Since \( T = \frac{z}{2} \) and it satisfies all conditions, the time taken by A, B, and C together is:

### Final Answer:
The time taken by A, B, and C together to do the job is **\( \frac{z}{2} \) days**.

If we consider \( z \) in terms of hours, you can convert it into days depending on the context or unit given.

Would you like to see the detailed calculations or have any questions?

Here are five related questions:
1. What if A, B, and C work at different rates?
2. How would the problem change if B's time alone were given differently?
3. Can you derive an exact value for \( z \) given additional information?
4. How can this approach be applied to more than three people?
5. What if one of the workers had a variable work rate?

**Tip:** When solving work problems, expressing rates as reciprocals of time can simplify complex expressions.

Learn how to solve the problem where three men A, B, and C work together can complete a job in less time compared to working alone. This math problem involves understanding rates of work and using a shortcut trick to find the time taken by A, B, and C together. Suitable for students and enthusiasts interested in solving work rate problems.

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