Question

X can complete a piece of work in 4 days. Y can complete the same work in 5 days. How many days will it take to complete the same work if X and Y work together?

• A. 20/9
• B. 19/9
• C. 22/9
• D. 23/9

Hint:

The formula \( \frac{t_1 \times t_2}{t_1 + t_2} \) is a very quick and efficient way to find the combined time for two people working together. Memorizing this formula can save valuable time in exams.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
This is a standard time and work problem. We need to find the combined rate of work and then the total time taken.

Step 2: Key Formula or Approach
If two people can do a piece of work in \(t_1\) and \(t_2\) days respectively, then the time taken by them to complete the work together is given by: \[ \text{Time together} = \frac{t_1 \times t_2}{t_1 + t_2} \] Alternatively, we can add their individual rates of work (work per day).

Step 3: Detailed Explanation
Method 1: Adding Rates
Work done by X in 1 day = \( \frac{1}{4} \).
Work done by Y in 1 day = \( \frac{1}{5} \).
Work done by X and Y together in 1 day = \( \frac{1}{4} + \frac{1}{5} \).
To add the fractions, find a common denominator (20): \[ \frac{1}{4} + \frac{1}{5} = \frac{5}{20} + \frac{4}{20} = \frac{9}{20} \] So, together they complete \( \frac{9}{20} \) of the work in one day.
The total time taken to complete the work is the reciprocal of their combined rate: \[ \text{Time together} = \frac{1}{9/20} = \frac{20}{9} \text{ days} \] Method 2: Using the Formula
\(t_1 = 4\) days, \(t_2 = 5\) days. \[ \text{Time together} = \frac{4 \times 5}{4 + 5} = \frac{20}{9} \text{ days} \]
Step 4: Final Answer
Together, X and Y will take \( \frac{20}{9} \) days to complete the work. Therefore, option (A) is the correct answer.