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