Two typists of different speeds can together complete a typing job in 6 minutes. If the first typist types alone for 4 minutes and then the second typist types alone for 6 minutes, 1/5 of the job is still left unfinished. How many minutes would it take the slower typist to complete the entire job when working alone?

Introduction / Context:
This problem describes two typists with different typing speeds. Together they can finish a job in a short time, and we also know how much of the job is completed when each works alone for a specific number of minutes. The scenario tests your understanding of work rates, partial work contribution by different persons, and identifying the slower worker based on calculated rates.

Given Data / Assumptions:

• The two typists together can finish the job in 6 minutes.
• If the first typist works alone for 4 minutes and then the second typist works alone for 6 minutes, 1/5 of the job remains incomplete.
• Therefore, 4 minutes of the first typist plus 6 minutes of the second typist completes 4/5 of the job.
• We need the time taken by the slower typist to do the whole job alone.
• Total work is taken as 1 unit.

Concept / Approach:
Assign work rates to the two typists and translate the verbal description into equations. The combined work time of 6 minutes gives a relation between their rates. The second condition that 4 minutes of the first plus 6 minutes of the second typist complete 4/5 of the job gives another equation. Solving these simultaneously provides individual rates. The typist with the smaller rate is the slower one, and the reciprocal of that rate gives the time to complete the job alone.

Step-by-Step Solution:
Step 1: Let the rate of the first typist be f jobs per minute and the rate of the second typist be s jobs per minute. Step 2: Together they complete the job in 6 minutes, so f + s = 1/6. Step 3: When the first works for 4 minutes and the second for 6 minutes, they complete 4/5 of the job. So, 4f + 6s = 4/5. Step 4: From f + s = 1/6, we can write f = 1/6 - s. Step 5: Substitute f into the second equation: 4(1/6 - s) + 6s = 4/5. Step 6: Simplify: 4/6 - 4s + 6s = 4/5, so 2/3 + 2s = 4/5. Step 7: Subtract 2/3 from both sides. 4/5 - 2/3 = (12/15 - 10/15) = 2/15, so 2s = 2/15. Step 8: Therefore s = 1/15 jobs per minute. This is the rate of the second typist. Step 9: Since f + s = 1/6, we get f = 1/6 - 1/15. Find the LCM of 6 and 15 which is 30, so 1/6 = 5/30 and 1/15 = 2/30. Step 10: Thus f = 5/30 - 2/30 = 3/30 = 1/10 jobs per minute. Step 11: The slower typist has the smaller rate, which is s = 1/15 jobs per minute. Hence, time taken by the slower typist alone = 1 / (1/15) = 15 minutes.

Verification / Alternative check:
We can verify the combined rate: f + s = 1/10 + 1/15 = (3/30 + 2/30) = 5/30 = 1/6, so they indeed finish the job in 6 minutes together. For the second condition, 4f + 6s = 4 * (1/10) + 6 * (1/15) = 4/10 + 6/15 = 2/5 + 2/5 = 4/5, which matches the statement that only 1/5 of the job remains. This confirms the correctness of the calculations.

Why Other Options Are Wrong:

• 10 minutes: This corresponds to the faster typist's time, not the slower one.
• 12 minutes: There is no rate consistent with all conditions that would yield 12 minutes as the slower typist's time.
• 20 minutes: This implies an even slower typist and would not satisfy the equations based on the given data.

Common Pitfalls:
Common errors include misinterpreting "1/5 of the work is left" as "1/5 completed," which reverses the fraction in the equation. Another pitfall is incorrectly setting up or solving the system of simultaneous equations, especially when working with fractions. Careful algebraic manipulation and clear identification of which typist is slower based on their rate are crucial for correct solutions.

Final Answer:
The slower typist would take 15 minutes to complete the typing job alone.