Question

A and B can complete a work in 12 days and 15 days respectively. They work together for 5 days and then A leaves. In how many more days will B complete the remaining work?

• A. \(3.75 \text{ days} \)
• B. \(6 \text{ days} \)
• C. \(7.5 \text{ days} \)
• D. \(8 \text{ days} \)

Hint:

Using unit-based mapping avoids messy fractions: - Total units = \(60\) - A gives \(5\), B gives \(4\). - In 5 days, they complete: \(5 \times 9 = 45\). - Remaining units = \(15\). - Always verify your arithmetic layout quickly to spot options that rely on unsimplified fractions or common examiner approximations!

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Time and work problems are best managed by determining individual operational output rates per day. By finding a common multiple for the raw timelines, we can calculate the total work units required and evaluate how many units are finished when both individuals cooperate versus when one person operates solo.

Step 2: Key Formula or Approach:
1. Assume Total Work = \(\text{LCM of individual days}\)
2. \(\text{Efficiency (Units/Day)} = \frac{\text{Total Work Units}}{\text{Total Days Required}}\)
3. \(\text{Remaining Work} = \text{Total Work} - \text{Completed Work}\)
4. \(\text{Days taken by B} = \frac{\text{Remaining Work Units}}{\text{B's Efficiency}}\)

Step 3: Detailed Explanation:
Let's find the Least Common Multiple (LCM) of A's time (12 days) and B's time (15 days): \[ \text{Total Work} = \text{LCM}(12, 15) = 60 \text{ units} \] Now determine their individual daily work efficiencies: \[ \text{Efficiency of A} = \frac{60}{12} = 5 \text{ units/day} \] \[ \text{Efficiency of B} = \frac{60}{15} = 4 \text{ units/day} \] Calculate their joint daily productivity when working together: \[ \text{Combined Efficiency (A + B)} = 5 + 4 = 9 \text{ units/day} \] They cooperate for exactly 5 days before A leaves. Calculate the volume of work finished during this initial phase: \[ \text{Work Completed in 5 Days} = 9 \text{ units/day} \times 5 \text{ days} = 45 \text{ units} \] Determine the volume of remaining work that B must finish alone: \[ \text{Remaining Work} = 60 - 45 = 15 \text{ units} \] Calculate the additional time required for B to finish these remaining 15 units at his solo rate of 4 units per day: \[ \text{Additional Days for B} = \frac{\text{Remaining Work}}{\text{Efficiency of B}} \] \[ \text{Additional Days for B} = \frac{15}{4} = 3.75 \text{ days} \]

Step 4: Final Answer:
The remaining work will be completed by B in 3.75 days.