ram can do a piece of work in 8hrs. shyam and naresh can do it in 9hrs. ram and naresh can do it in 12hrs. how long will shyam alone take to do it?

Let's denote the total work as 1 unit.

- Ram can do the work in 8 hours, so Ram's work rate is \( \frac{1}{8} \) units per hour.
- Shyam and Naresh together can do the work in 9 hours, so their combined work rate is \( \frac{1}{9} \) units per hour.
- Ram and Naresh together can do the work in 12 hours, so their combined work rate is \( \frac{1}{12} \) units per hour.

We need to determine how long Shyam alone will take to complete the work.

### Step 1: Express Naresh's rate in terms of Shyam's rate.

Let the work rates of Shyam and Naresh be \( S \) and \( N \) units per hour, respectively. From the given information:

1. Shyam and Naresh together can work at a rate of:
   \[
   S + N = \frac{1}{9}
   \]

2. Ram and Naresh together can work at a rate of:
   \[
   \frac{1}{8} + N = \frac{1}{12}
   \]

### Step 2: Solve for Naresh's rate \( N \).

From the equation \( \frac{1}{8} + N = \frac{1}{12} \), we solve for \( N \):

\[
N = \frac{1}{12} - \frac{1}{8}
\]
\[
N = \frac{2}{24} - \frac{3}{24} = -\frac{1}{24}
\]

Thus, Naresh's work rate is \( N = -\frac{1}{24} \).

### Step 3: Solve for Shyam's rate \( S \).

Now, substitute \( N = -\frac{1}{24} \) into the equation \( S + N = \frac{1}{9} \):

\[
S - \frac{1}{24} = \frac{1}{9}
\]
\[
S = \frac{1}{9} + \frac{1}{24}
\]
\[
S = \frac{8}{72} + \frac{3}{72} = \frac{11}{72}
\]

So, Shyam's work rate is \( S = \frac{11}{72} \) units per hour.

### Step 4: Calculate the time Shyam alone will take.

The time Shyam alone will take to complete the work is the reciprocal of his work rate:

\[
\text{Time} = \frac{1}{S} = \frac{1}{\frac{11}{72}} = \frac{72}{11} \approx 6.55 \, \text{hours}.
\]

Thus, Shyam alone will take approximately **6.55 hours** to complete the work.

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Would you like any further details on the steps or the methodology?

### Related Questions:
1. How long will it take Ram, Shyam, and Naresh working together to complete the work?
2. If Ram worked for 4 hours, how much of the work would be left for Shyam and Naresh?
3. If Shyam and Naresh started the work together, but Shyam left after 3 hours, how long will Naresh take to finish the remaining work?
4. What would happen to the total time if Ram, Shyam, and Naresh all work together for 2 hours and then Ram stops?
5. What is the ratio of work done by Shyam and Ram in 4 hours?

### Tip:
Always double-check the units and consistency of rates when solving work problems, especially when combining work rates of different individuals.

Ram can do a piece of work in 8 hours. Shyam and Naresh can do it in 9 hours. Ram and Naresh can do it in 12 hours. How long will Shyam alone take to do it?

Find out how long Shyam will take to complete a piece of work alone given the time taken by Ram, Shyam, and Naresh working in different pairs. This work-rate problem involves algebraic calculations and is suitable for students in Grades 8-10.

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