A, B,C started working on a project , but A left the work 9 days before the completion,and B left the work 7 days before the completion , if had A worked for 1 more day the total work would have been completed 2 days before actual time, if B would have worked for 2 more days , the total work would have been completed 3 days before the actual time, In how many days A,B,C can together finish a work if C can alone complete the work in 18 days?

Let’s define the variables for the rate of work for each person:
- \( r_A \) = Rate at which A works (in work per day)
- \( r_B \) = Rate at which B works (in work per day)
- \( r_C \) = Rate at which C works (in work per day)

We know that C can complete the entire work in 18 days, so:
\[
r_C = \frac{1}{18}
\]

Let \( T \) be the total number of days it took to complete the work. From the problem, we know:
- A worked for \( T - 9 \) days.
- B worked for \( T - 7 \) days.
- C worked for the entire \( T \) days.

The total work done by A, B, and C is equal to 1 (since the whole project is completed). Thus, the total work equation can be written as:
\[
(T-9)r_A + (T-7)r_B + T r_C = 1
\]

We are also given two conditions about finishing the project earlier:
1. If A had worked for 1 more day (total \( T - 8 \) days), the project would have finished 2 days earlier (i.e., in \( T - 2 \) days):
\[
(T-8)r_A + (T-7)r_B + (T-2) r_C = 1
\]
2. If B had worked for 2 more days (total \( T - 5 \) days), the project would have finished 3 days earlier (i.e., in \( T - 3 \) days):
\[
(T-9)r_A + (T-5)r_B + (T-3)r_C = 1
\]

We now have three equations:
1. \((T-9)r_A + (T-7)r_B + T r_C = 1\)
2. \((T-8)r_A + (T-7)r_B + (T-2)r_C = 1\)
3. \((T-9)r_A + (T-5)r_B + (T-3)r_C = 1\)

We’ll solve these equations to find the rates of A and B, and then determine how long A, B, and C together would take to complete the project.

Let me calculate this for you.The solution to the system of equations is:

- \( T = 10.33 \) days (approximately 10 days and 8 hours to complete the project),
- \( r_A = \frac{1}{9} \), meaning A can complete the work in 9 days if working alone,
- \( r_B = \frac{1}{12} \), meaning B can complete the work in 12 days if working alone.

Now, to find how long A, B, and C together can complete the work, we sum their rates:
\[
\text{Total rate} = r_A + r_B + r_C = \frac{1}{9} + \frac{1}{12} + \frac{1}{18}
\]

Let me calculate that.A, B, and C can together complete the entire work in 4 days.

Would you like further clarification on any part of this? Here are some related questions for deeper exploration:

1. How can we calculate individual contributions to

A, B, and C started working on a project, but A left the work 9 days before the completion, and B left the work 7 days before the completion. If A had worked for 1 more day, the total work would have been completed 2 days before actual time, and if B had worked for 2 more days, the total work would have been completed 3 days before actual time. In how many days can A, B, and C together finish the work if C can alone complete the work in 18 days?

Explore how to solve a work-rate problem where A, B, and C work together to finish a project. Using system of equations and algebraic methods, we determine how long A, B, and C can complete the work if C alone finishes in 18 days. Suitable for students in Grades 10-12.

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