Ten boys can complete a piece of work in 7 days and ten girls can complete the same work in 14 days. If 5 boys and 10 girls work together, in how many days will they finish the entire work?

Introduction / Context:
This time and work question compares the efficiencies of boys and girls working on the same job. We are given the time taken by groups of boys and girls to complete the work separately and must find the time when a mixed group works together. The problem tests understanding of individual work rates and how to combine them correctly.

Given Data / Assumptions:
- Ten boys can complete the full work in 7 days. - Ten girls can complete the same work in 14 days. - We want the time taken when 5 boys and 10 girls work together. - Work rate is constant for each boy and each girl. - Total work is treated as one complete unit.

Concept / Approach:
If a group finishes a job in T days, its daily work rate is 1 / T of the job. From the given group rates, we can find the rate of a single boy and a single girl. Then we calculate the combined rate of 5 boys and 10 girls. Finally, we find the total time by taking the reciprocal of that combined rate. This is a direct application of work rate and proportionality concepts commonly used in aptitude exams.

Step-by-Step Solution:
Step 1: Daily rate of 10 boys = 1 / 7 job per day. Step 2: Daily rate of 1 boy = (1 / 7) / 10 = 1 / 70 job per day. Step 3: Daily rate of 10 girls = 1 / 14 job per day. Step 4: Daily rate of 1 girl = (1 / 14) / 10 = 1 / 140 job per day. Step 5: Rate of 5 boys = 5 * (1 / 70) = 5 / 70 = 1 / 14 job per day. Step 6: Rate of 10 girls = 10 * (1 / 140) = 10 / 140 = 1 / 14 job per day. Step 7: Combined rate of 5 boys and 10 girls = 1 / 14 + 1 / 14 = 2 / 14 = 1 / 7 job per day. Step 8: Time to finish one full job = 1 / (1 / 7) = 7 days.

Verification / Alternative check:
Notice that 5 boys do the same work per day as 10 girls, since both groups have rate 1 / 14 job per day. So, 5 boys and 10 girls together are equivalent to two groups of 1 / 14 job per day, giving 1 / 7 job per day, exactly matching what 10 boys alone would do. Since 10 boys finish the job in 7 days, the mixed group must also take 7 days, confirming our result.

Why Other Options Are Wrong:
13 days and 10 days: These are larger than 7 days and would imply a lower combined work rate than ten boys alone, which is illogical because we are using both boys and girls together.
5 days and 8 days: Five days is too short for this rate, and eight days does not match the calculated 1 / 7 job per day. Only 7 days is consistent with the combined rate.

Common Pitfalls:
A common mistake is to simply average the days (7 and 14) without considering rates, which gives a wrong value. Another error is to assume that 5 boys and 10 girls will finish in half the time of 10 boys, which is not correct unless the total rate doubles. Always convert times into rates first, add or compare rates, and then invert the final rate to get time.

Final Answer:
5 boys and 10 girls together will complete the work in 7 days.