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

Introduction / Context:
This is a comparative time and work question involving three workers A, B, and C. Instead of giving direct times for the same job, the problem compares how much work each can do in different numbers of days. From these relationships, we must determine the relative efficiencies and finally compute how long C would take to complete a piece of work that A can finish in 7 days.

Given Data / Assumptions:
- A does as much work in 4 days as B does in 5 days.
- B does as much work in 6 days as C does in 7 days.
- A can complete a certain piece of work in 7 days.
- We need the time taken by C to do the same piece of work.

Concept / Approach:
Let the daily work rates of A, B, and C be a, b, and c units per day. The statements about equal work lead to proportionality equations: 4a = 5b and 6b = 7c. From these, we express b and c in terms of a. Since A completes the whole job in 7 days, total work is 7a units. Using C’s rate in terms of a, we find the time required for C to do the same total work by dividing 7a by c.

Step-by-Step Solution:
Step 1: Let the daily rate of A = a units per day. Step 2: From 4a = 5b, we get b = (4a) / 5. Step 3: From 6b = 7c, substitute b to obtain 6 * (4a / 5) = 7c. Step 4: Simplify: 24a / 5 = 7c, so c = (24a) / (35). Step 5: A completes the required piece of work in 7 days, so total work W = 7a units. Step 6: Time taken by C to complete the same work W is W / c. Step 7: Time for C = 7a / (24a / 35) = 7a * 35 / 24a. Step 8: Cancel a: Time for C = 245 / 24 days. Step 9: Convert 245 / 24 into mixed form: 24 * 10 = 240, remainder 5, so time = 10 5/24 days.

Verification / Alternative check:
We can test relative efficiencies. From 4a = 5b, A is more efficient than B. From 6b = 7c, B is more efficient than C. Hence C should take more time than A for the same work. A takes 7 days, and C takes about 10.21 days (10 5/24), which is indeed larger, aligning with the efficiency ordering. The algebraic steps satisfy both proportionality equations and the final answer fits the logical trend.

Why Other Options Are Wrong:
- 4 4/5 and 6 8/15 days: These are less than 7 days and would imply that C is faster than A, contradicting the efficiency chain A faster than B and B faster than C.
- 12 6/19 days: This overestimates C’s time relative to the proportional relationships derived from the given conditions.
- 8 days: Slightly greater than A’s time but does not satisfy the precise ratio calculations from 4a = 5b and 6b = 7c.

Common Pitfalls:
Learners often misinterpret the statements about equal work and incorrectly set up the equations, or they may try to directly guess times without using algebra. It is important to clearly define symbols for daily rates and use the equal-work conditions to relate these rates. Mistakes in manipulating fractions and converting improper fractions to mixed numbers are also common, so extra care is needed in the final steps.

Final Answer:
C will take 10 5/24 days to complete the same piece of work that A can do in 7 days.