Let's solve the problem step by step.

Given:

• A can complete the work in 10 days.
• B can complete the work in 15 days.
• C can complete the work in 30 days.

Let's calculate their daily work rates:

• Work rate of A = $\frac{1}{10}$ work/day
• Work rate of B = $\frac{1}{15}$ work/day
• Work rate of C = $\frac{1}{30}$ work/day

(a) In 1 day, how much work can be done by A and B?

The combined work rate of A and B: $\text{Work rate of A and B} = \frac{1}{10} + \frac{1}{15}$

Finding the common denominator (which is 30): $\frac{1}{10} = \frac{3}{30}, \quad \frac{1}{15} = \frac{2}{30}$ $\frac{1}{10} + \frac{1}{15} = \frac{3}{30} + \frac{2}{30} = \frac{5}{30} = \frac{1}{6}$

Therefore, A and B together can do $\frac{1}{6}$ of the work in 1 day.

Answer: $\frac{1}{6}$ part

(b) In how many days can they together finish the work?

The combined work rate of A, B, and C: $\frac{1}{10} + \frac{1}{15} + \frac{1}{30}$ Finding the common denominator (which is 30): $\frac{1}{10} = \frac{3}{30}, \quad \frac{1}{15} = \frac{2}{30}, \quad \frac{1}{30} = \frac{1}{30}$ $\frac{1}{10} + \frac{1}{15} + \frac{1}{30} = \frac{3}{30} + \frac{2}{30} + \frac{1}{30} = \frac{6}{30} = \frac{1}{5}$

So, A, B, and C together can complete $\frac{1}{5}$ of the work per day.

The total time to finish the work together is: $\frac{1}{\frac{1}{5}} = 5 \text{ days}$

Answer: 5 days

(c) If after 3 days A left the work unfinished, in how many days will the whole work be finished?

First, let's find out how much work is done in the first 3 days by A, B, and C together.

The work rate of A, B, and C together is $\frac{1}{5}$ per day. Therefore, in 3 days, they will complete: $3 \times \frac{1}{5} = \frac{3}{5}$

The remaining work is: $1 - \frac{3}{5} = \frac{2}{5}$

Now, A leaves, so only B and C are working together. Their combined work rate: $\frac{1}{15} + \frac{1}{30}$ Finding the common denominator (which is 30): $\frac{1}{15} = \frac{2}{30}, \quad \frac{1}{30} = \frac{1}{30}$ $\frac{1}{15} + \frac{1}{30} = \frac{2}{30} + \frac{1}{30} = \frac{3}{30} = \frac{1}{10}$

B and C together can complete $\frac{1}{10}$ of the work per day.

The time to complete the remaining $\frac{2}{5}$ of the work is: $\frac{\frac{2}{5}}{\frac{1}{10}} = 2 \times 2 = 4 \text{ days}$

Thus, the total time to finish the entire work is: $3 + 4 = 7 \text{ days}$

Answer: 7 days

Summary:

• (a) $\frac{1}{6}$ part
• (b) 5 days
• (c) 7 days

Would you like any further details or clarifications?

5 Related Questions:

1. How much work can A and C complete together in 1 day?
2. If only B and C work together, in how many days will they complete the work?
3. What fraction of the work will be completed if A and B work together for 4 days?
4. How many days will A alone take to complete half the work?
5. If C works for 5 days alone, how much work is left to complete?

Tip

Always break down combined work problems by calculating the individual work rates first and then summing them up!