Question

A can complete a piece of work in 15 days and B can complete the same work in 10 days. In how many days will they complete the work together?

• A. 4 days
• B. 5 days
• C. 6 days
• D. 7 days

Hint:

The LCM method is highly effective for complex time-and-work problems.
Always try to use LCM to convert fraction-based work rates into simple whole numbers representing efficiencies.
This minimizes computational errors and speeds up the calculation process.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
This problem belongs to the topic of Time and Work.
We are given the individual times taken by two people, A and B, to complete a specific task.
We need to determine the total time required for both of them to complete the same work when working together.

Step 2: Key Formula or Approach:
We can approach this using two standard methods:
Method 1 (Formula-based):
If A can do a work in $x$ days and B can do it in $y$ days, the time taken by both working together is:
\[ \text{Time taken} = \frac{x \times y}{x + y} \text{ days} \]
Method 2 (LCM-based/Efficiency method):
Define the total work as the Least Common Multiple (LCM) of the individual days, calculate their daily efficiencies, and divide the total work by their combined efficiency.

Step 3: Detailed Explanation:
$\bullet$ Let us write down the given individual times.
$\bullet$ Time taken by A to complete the work alone ($x$) = 15 days.
$\bullet$ Time taken by B to complete the work alone ($y$) = 10 days.
$\bullet$ Let us apply the LCM method to find the total work and efficiency.
$\bullet$ The LCM of 15 and 10 is 30. Let us assume the Total Work is 30 units.
$\bullet$ Efficiency of A (work done by A per day) is:
\[ \text{Efficiency of A} = \frac{\text{Total Work}}{\text{Days taken by A}} = \frac{30}{15} = 2 \text{ units/day} \]
$\bullet$ Efficiency of B (work done by B per day) is:
\[ \text{Efficiency of B} = \frac{\text{Total Work}}{\text{Days taken by B}} = \frac{30}{10} = 3 \text{ units/day} \]
$\bullet$ Combined efficiency of A and B when working together:
\[ \text{Combined Efficiency} = 2 + 3 = 5 \text{ units/day} \]
$\bullet$ The total time taken by A and B to complete the work together is:
\[ \text{Time taken} = \frac{\text{Total Work}}{\text{Combined Efficiency}} = \frac{30}{5} = 6 \text{ days} \]
$\bullet$ Alternatively, using the formula:
\[ \text{Time taken} = \frac{15 \times 10}{15 + 10} = \frac{150}{25} = 6 \text{ days} \]

Step 4: Final Answer:
Working together, A and B will complete the work in 6 days.
Therefore, the correct option is (C).