Question

A can do work in 4 days, B in 6 days and C in 10 days. Find the time taken by A, B and C to do the work togather

• A. 41/60
• B. 51/60
• C. 61/60
• D. 31/60

Hint:

Always check if the options represent the daily rate of work or the total time. If the options are less than 1 (as in this question), they refer to the rate of work done per day.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This problem deals with rate of work.
The total amount of work is typically designated as 1 unit.
If a person can complete the work in $D$ days, their rate of work per day is $\frac{1}{D}$.
Key Formula or Approach:
The combined work rate of A, B, and C working together in a single day is: \[ \text{Combined Rate} = \text{Rate}_A + \text{Rate}_B + \text{Rate}_C = \frac{1}{D_A} + \frac{1}{D_B} + \frac{1}{D_C} \] The total time to complete the work is the reciprocal of this combined daily rate.

Step 2: Detailed Explanation:
Given the individual completion times:
- A completes the work in $4$ days $\implies$ A's rate = $\frac{1}{4}$ of the work per day.
- B completes the work in $6$ days $\implies$ B's rate = $\frac{1}{6}$ of the work per day.
- C completes the work in $10$ days $\implies$ C's rate = $\frac{1}{10}$ of the work per day.
Now, calculate their combined work done in one day: \[ \text{Combined Rate} = \frac{1}{4} + \frac{1}{6} + \frac{1}{10} \] Find the Least Common Multiple (LCM) of denominators 4, 6, and 10.
The prime factorizations are: - $4 = 2^2$
- $6 = 2 \times 3$
- $10 = 2 \times 5$
The LCM is $2^2 \times 3 \times 5 = 40 \times 1.5 = 60$.
Convert each fraction to have a denominator of 60: \[ \text{Combined Rate} = \frac{15}{60} + \frac{10}{60} + \frac{6}{60} \] \[ \text{Combined Rate} = \frac{15 + 10 + 6}{60} = \frac{31}{60} \] Thus, working together, they complete $\frac{31}{60}$ of the total work in one day.
The total time taken to complete the work is: \[ \text{Total Time} = \frac{60}{31} \text{ days} \] The options provided in the question represent the combined daily work rate of the three individuals rather than the total days. Under this convention, the corresponding correct choice is $\frac{31}{60}$.

Step 3: Final Answer:
The fraction of work done by A, B, and C together in one day is 31/60.