A can do as much work in 4 days as B can do in 5, and B can do as much work in 6 days as C in 7. In what time will C do a piece of work which A can do in a week?

Options

A. `10(5)/(24)` days
B. `4(4)/(5)` days
C. `6(8)/(15)` days
D. `12(6)/(19)` days

Answer

To solve the problem step by step, we will first determine the work efficiencies of A, B, and C based on the information provided.

### Step 1: Determine the work done by A and B
We know that A can do as much work in 4 days as B can do in 5 days.

Let’s denote:
- A's efficiency = \( a \) (work done per day)
- B's efficiency = \( b \) (work done per day)

From the information given:
\[ 4a = 5b \]
This implies:
\[ \frac{a}{b} = \frac{5}{4} \]
So, the ratio of A's efficiency to B's efficiency is \( 5:4 \).

### Step 2: Determine the work done by B and C
Next, we know that B can do as much work in 6 days as C can do in 7 days.

Let’s denote:
- C's efficiency = \( c \) (work done per day)

From the information given:
\[ 6b = 7c \]
This implies:
\[ \frac{b}{c} = \frac{7}{6} \]
So, the ratio of B's efficiency to C's efficiency is \( 7:6 \).

### Step 3: Find the combined ratio of A, B, and C
We have the ratios:
1. \( \frac{a}{b} = \frac{5}{4} \)
2. \( \frac{b}{c} = \frac{7}{6} \)

To find the combined ratio of A, B, and C, we need to express them with a common base. We can express B in both ratios.

Let’s make B common:
- From \( \frac{a}{b} = \frac{5}{4} \), we can express B in terms of A:
\[ b = \frac{4}{5}a \]

- From \( \frac{b}{c} = \frac{7}{6} \), we can express C in terms of B:
\[ c = \frac{6}{7}b \]

Substituting the value of B from the first equation into the second:
\[ c = \frac{6}{7} \left(\frac{4}{5}a\right) = \frac{24}{35}a \]

Now we have:
- A = \( 35k \)
- B = \( 28k \)
- C = \( 24k \)

### Step 4: Find the total work done by A in a week
A can do a piece of work in 7 days. The total work done by A in a week is:
\[ \text{Total work} = 7 \times a = 7 \times 35k = 245k \]

### Step 5: Calculate the time taken by C to do the same work
Now, we need to find out how long it will take C to do the same amount of work (245k):
Using C's efficiency:
\[ \text{Time taken by C} = \frac{\text{Total work}}{c} = \frac{245k}{24k} = \frac{245}{24} \]

### Step 6: Simplifying the time taken by C
Now, simplifying \( \frac{245}{24} \):
\[ \frac{245}{24} = 10 \frac{5}{24} \]

Thus, C will take \( 10 \frac{5}{24} \) days to complete the work that A can do in a week.

### Final Answer
C will take \( 10 \frac{5}{24} \) days to complete the work.

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