Linear equation with unknowns.                 21. A, B, and C can do a piece of work in 10 days. A and B can do it in 12 days, A and C in 20 days. How many days would it take each to do the work alone?

This is a typical problem involving work rates. We will use the concept of rates of work to solve for the individual work rates of A, B, and C.

### Step 1: Define the unknowns
Let:
- \( A \) be the amount of work A can do in one day (A's work rate).
- \( B \) be the amount of work B can do in one day (B's work rate).
- \( C \) be the amount of work C can do in one day (C's work rate).

The total amount of work done is assumed to be 1 unit (the entire piece of work).

### Step 2: Express the work rates in terms of the given conditions

- A, B, and C together can complete the work in 10 days. Hence, their combined rate is:
  \[
  A + B + C = \frac{1}{10} \text{ (since they complete the work in 10 days)}
  \]

- A and B together can complete the work in 12 days. So, their combined rate is:
  \[
  A + B = \frac{1}{12}
  \]

- A and C together can complete the work in 20 days. So, their combined rate is:
  \[
  A + C = \frac{1}{20}
  \]

### Step 3: Set up the system of equations
We now have the following three equations:

1. \( A + B + C = \frac{1}{10} \)
2. \( A + B = \frac{1}{12} \)
3. \( A + C = \frac{1}{20} \)

### Step 4: Solve the system of equations

#### Subtract Equation 2 from Equation 1:
\[
(A + B + C) - (A + B) = \frac{1}{10} - \frac{1}{12}
\]
\[
C = \frac{1}{10} - \frac{1}{12} = \frac{6 - 5}{60} = \frac{1}{60}
\]
Thus, \( C = \frac{1}{60} \), meaning C can do the work in 60 days.

#### Subtract Equation 3 from Equation 1:
\[
(A + B + C) - (A + C) = \frac{1}{10} - \frac{1}{20}
\]
\[
B = \frac{1}{10} - \frac{1}{20} = \frac{2 - 1}{20} = \frac{1}{20}
\]
Thus, \( B = \frac{1}{20} \), meaning B can do the work in 20 days.

#### Substitute \( B \) into Equation 2:
\[
A + B = \frac{1}{12}
\]
\[
A + \frac{1}{20} = \frac{1}{12}
\]
Solving for \( A \):
\[
A = \frac{1}{12} - \frac{1}{20} = \frac{5 - 3}{60} = \frac{2}{60} = \frac{1}{30}
\]
Thus, \( A = \frac{1}{30} \), meaning A can do the work in 30 days.

### Final Answer:
- A can complete the work in 30 days.
- B can complete the work in 20 days.
- C can complete the work in 60 days.

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Would you like further details or explanations on any part? Here are some related questions you might find interesting:

1. How do you handle work problems when more than three workers are involved?
2. How would the solution change if the rates were not constant?
3. Can we apply similar methods to other types of rate problems, like speed or flow rates?
4. How do we handle work rate problems with partial work done by some of the workers?
5. What are the most common mistakes when solving work rate problems?

**Tip:** Always set up your unknowns clearly when dealing with work rate problems—this keeps your equations organized and easy to solve!

A, B, and C can do a piece of work in 10 days. A and B can do it in 12 days, A and C in 20 days. How many days would it take each to do the work alone?

Learn how to solve a work rate problem involving three workers, A, B, and C, working together and individually. Using the system of equations, find how long it takes each to complete the work alone. Suitable for students in Grades 9-12.

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