Question

X and Y can complete a piece of work in 20 days and X alone in 40 days. In how many days Y alone can complete the work?

• A. 20
• B. 40
• C. 10
• D. 15

Hint:

Another popular method is the LCM method. Assume the total work is the LCM of the given days (LCM of 20 and 40 is 40 units).

Efficiency of (X+Y) = \(\frac{40 \text{ units}}{20 \text{ days}}\) = 2 units/day.
Efficiency of X = \(\frac{40 \text{ units}}{40 \text{ days}}\) = 1 unit/day.
Efficiency of Y = Efficiency of (X+Y) - Efficiency of X = 2 - 1 = 1 unit/day.
Time for Y = \(\frac{\text{Total Work}}{\text{Efficiency of Y}}\) = \(\frac{40 \text{ units}}{1 \text{ unit/day}}\) = 40 days.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
This is a work and time problem. We can solve it by calculating the rate of work (work done per day) for each person.

Step 2: Key Formula or Approach
If a person can complete a work in 'n' days, their one day's work is \( \frac{1}{n} \).
(X and Y's one day's work) = (X's one day's work) + (Y's one day's work)

Step 3: Detailed Explanation
1. Express work rates as fractions:
Work done by X and Y together in 1 day = \( \frac{1}{20} \).
Work done by X alone in 1 day = \( \frac{1}{40} \).
2. Calculate Y's one day's work:
Y's one day's work = (X and Y's one day's work) - (X's one day's work)
\[ \text{Y's work per day} = \frac{1}{20} - \frac{1}{40} \] To subtract the fractions, find a common denominator, which is 40.
\[ \text{Y's work per day} = \frac{2}{40} - \frac{1}{40} = \frac{2-1}{40} = \frac{1}{40} \] So, Y completes \( \frac{1}{40} \) of the work in one day.
3. Find the total time taken by Y:
If Y does \( \frac{1}{40} \) of the work per day, the total number of days Y will take to complete the whole work is the reciprocal of this rate.
Time taken by Y = \( \frac{1}{1/40} = 40 \) days.

Step 4: Final Answer
Y alone can complete the work in 40 days. Therefore, option (B) is the correct answer.