M can complete a piece of work in 10 days. N is 33.33 percent less efficient than M. How many days will M and N working together take to complete the entire work?

Introduction / Context:
This problem is a straightforward example of comparing efficiencies and then combining rates to find the time required to complete a task. It introduces a percentage reduction in efficiency and asks for the time when both workers cooperate. Such questions are very common in aptitude tests and are designed to check whether the candidate correctly interprets percentage efficiency changes in terms of work rates and not in terms of days directly.

Given Data / Assumptions:

• M completes the work alone in 10 days.
• N is 33.33 percent less efficient than M, which is approximately one third less efficient.
• Both work together at their respective constant rates.
• We must find the total number of days required when M and N work together.

Concept / Approach:
If N is 33.33 percent less efficient than M, then N's efficiency is 66.67 percent of M's efficiency, which is equivalent to two thirds of M's rate. First we compute M's work rate in jobs per day. Then we find N's rate as two thirds of M's rate. Adding these rates gives the combined daily rate, and the reciprocal of that combined rate gives the time taken to finish one whole job when they work together.

Step-by-Step Solution:
Step 1: M completes the work in 10 days, so M's rate is 1 / 10 of the job per day. Step 2: N is 33.33 percent less efficient than M, so N's rate is two thirds of 1 / 10, which equals 1 / 15 of the job per day. Step 3: Combined daily rate of M and N is 1/10 plus 1/15. Step 4: 1/10 plus 1/15 equals (3 + 2) / 30 which is 5 / 30, simplifying to 1 / 6 of the job per day. Step 5: Time required for them together to finish the work is 1 divided by 1 / 6 which equals 6 days.

Verification / Alternative check:
Another way is to assume the total work is 30 units. Then M, who finishes in 10 days, does 3 units per day. N, being two thirds as efficient, does 2 units per day. Together they do 5 units per day. To complete 30 units, time needed is 30 divided by 5, which equals 6 days. This unit method confirms the fraction method and reinforces that the final answer is consistent and correct.

Why Other Options Are Wrong:

• 8 days would imply a combined daily rate of 1/8, which is smaller than 1/6 and does not match the sum of individual rates.
• 11 days and 13 days both give even smaller implied rates, which are inconsistent with M and N working together at the calculated speeds.
• Only 6 days corresponds exactly to the combined rate of 1/6 of the work per day.

Common Pitfalls:
A common mistake is to interpret 33.33 percent less efficient as meaning that the time taken increases by the same percentage, which is not correct. Efficiency refers to rate, not time. Another pitfall is subtracting one third from the number of days instead of from the rate. Remember that if efficiency decreases, the time required increases, and these two quantities are inversely related. Careful handling of percentages and inverses avoids these errors.

Final Answer:
M and N working together will complete the work in 6 days.