A alone completes 75 percent of a certain work in 50 days. He then calls in B, and together they finish the remaining work in 10 more days. How long would B alone take to complete the entire work?

Introduction / Context:
This problem involves fractional completion of work by one person and then joint completion by two people. We are given the time taken by A to finish three quarters of the job and the time taken by A and B together to finish the remaining one quarter. Based on this, we are asked to find how long B alone would need to complete the entire job.

Given Data / Assumptions:

• A completes 75 percent (that is 3/4) of the work in 50 days.
• The remaining 25 percent (that is 1/4) of the work is finished by A and B together in 10 days.
• Work rates of A and B are constant.
• We want the time B alone would require to complete 100 percent of the work.

Concept / Approach:
First, we determine A's work rate from the information that A finishes 3/4 of the work in 50 days. This gives us A's rate in terms of work per day. Next, we use the joint completion of the remaining 1/4 in 10 days to determine the combined rate of A and B. Subtracting A's rate from this combined rate yields B's rate. The reciprocal of B's rate then gives the time B would take alone to complete the entire job.

Step-by-Step Solution:
Step 1: Let the total work be 1 unit.Step 2: A completes 3/4 of the work in 50 days. So A's daily work rate = (3 / 4) / 50 = 3 / 200 of the work per day.Step 3: Remaining work = 1 - 3 / 4 = 1 / 4 of the job.Step 4: A and B together complete this remaining 1 / 4 in 10 days.Step 5: Combined daily rate of A and B = (1 / 4) / 10 = 1 / 40 of the work per day.Step 6: Rate of B alone = combined rate of A and B minus rate of A alone = 1 / 40 - 3 / 200.Step 7: Use denominator 200: 1 / 40 = 5 / 200, so rate of B = 5 / 200 - 3 / 200 = 2 / 200 = 1 / 100 of the work per day.Step 8: Time taken by B alone to finish the whole work = 1 divided by B's rate = 1 / (1 / 100) = 100 days.

Verification / Alternative check:
We can check by recomputing the phases. In 50 days, A does 50 * (3 / 200) = 150 / 200 = 3 / 4 of the work. In the next 10 days, A adds 10 * (3 / 200) = 30 / 200 = 3 / 20, and B adds 10 * (1 / 100) = 10 / 100 = 1 / 10. Together in the last 10 days they complete 3 / 20 + 1 / 10 = 3 / 20 + 2 / 20 = 5 / 20 = 1 / 4. So total work = 3 / 4 + 1 / 4 = 1, confirming that the rates are consistent and that B alone requires 100 days for the full job.

Why Other Options Are Wrong:

• 200 days: This would imply a rate of 1 / 200 per day, half the correct rate, and would not match the given joint completion of the last quarter in 10 days.
• 50 days: This is the time A spends on the first 3/4 of the work and cannot be B's time for the full job.
• 125 days: This is slower than the correct value and would give too small a contribution from B in the final 10 days.

Common Pitfalls:
Many candidates incorrectly assume that the fraction completed by A in 50 days applies to the whole work done together with B, which is not true. Another error is to forget to convert percentages into fractions correctly. Always translate percentages (here 75 percent and 25 percent) into fractions, compute rates carefully, and keep track of each phase of the work.

Final Answer:
B alone would take 100 days to complete the entire work.