A alone takes as much time as B and C together take to complete a piece of work. If A and B together take 10 days and B alone takes 50 days to complete it, in what time can A and C together do the work?

Options

A. `7(1)/(2)` days
B. `7(1)/(7)` days
C. `8(1)/(7)` days
D. `15(1)/(7)` days

Answer

To solve the problem step by step, we will use the information given in the question to find out how long A and C can complete the work together.

### Step 1: Understand the work done by A and B together
We know that A and B together can complete the work in 10 days. Therefore, their combined work rate per day is:
\[ \text{Work rate of A + B} = \frac{1}{10} \text{ (work per day)} \]

**Hint:** The work rate is the reciprocal of the time taken to complete the work.

### Step 2: Determine the work done by B alone
B alone takes 50 days to complete the work. Thus, B's work rate per day is:
\[ \text{Work rate of B} = \frac{1}{50} \text{ (work per day)} \]

**Hint:** Again, the work rate is the reciprocal of the time taken.

### Step 3: Calculate the work done by A alone
To find A's work rate, we can subtract B's work rate from the combined work rate of A and B:
\[ \text{Work rate of A} = \text{Work rate of A + B} - \text{Work rate of B} \]
\[ \text{Work rate of A} = \frac{1}{10} - \frac{1}{50} \]

To perform this subtraction, we need a common denominator. The least common multiple of 10 and 50 is 50:
\[ \frac{1}{10} = \frac{5}{50} \]
Thus,
\[ \text{Work rate of A} = \frac{5}{50} - \frac{1}{50} = \frac{4}{50} = \frac{2}{25} \]

**Hint:** When subtracting fractions, find a common denominator.

### Step 4: Relate A's work to B and C
According to the problem, A alone takes as much time as B and C together. This means:
\[ \text{Work rate of A} = \text{Work rate of B + C} \]
Thus,
\[ \text{Work rate of B + C} = \frac{2}{25} \]

**Hint:** This establishes a relationship between A's work rate and the combined work rate of B and C.

### Step 5: Calculate C's work rate
Now, we can find C's work rate by subtracting B's work rate from the combined work rate of B and C:
\[ \text{Work rate of C} = \text{Work rate of B + C} - \text{Work rate of B} \]
\[ \text{Work rate of C} = \frac{2}{25} - \frac{1}{50} \]

Again, we need a common denominator. The least common multiple of 25 and 50 is 50:
\[ \frac{2}{25} = \frac{4}{50} \]
Thus,
\[ \text{Work rate of C} = \frac{4}{50} - \frac{1}{50} = \frac{3}{50} \]

**Hint:** Use the common denominator to simplify calculations.

### Step 6: Calculate the combined work rate of A and C
Now that we have the work rates of A and C, we can find their combined work rate:
\[ \text{Work rate of A + C} = \text{Work rate of A} + \text{Work rate of C} \]
\[ \text{Work rate of A + C} = \frac{2}{25} + \frac{3}{50} \]

Finding a common denominator (which is 50):
\[ \frac{2}{25} = \frac{4}{50} \]
Thus,
\[ \text{Work rate of A + C} = \frac{4}{50} + \frac{3}{50} = \frac{7}{50} \]

**Hint:** When adding fractions, ensure they have the same denominator.

### Step 7: Calculate the time taken by A and C together
The time taken by A and C together to complete the work is the reciprocal of their combined work rate:
\[ \text{Time taken by A + C} = \frac{1}{\text{Work rate of A + C}} = \frac{1}{\frac{7}{50}} = \frac{50}{7} \text{ days} \]

**Hint:** The time taken is the reciprocal of the work rate.

### Final Answer
A and C together can complete the work in \(\frac{50}{7}\) days.

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