A man can do a work in 10 days. The man along with a boy can do the same work…

Concept

Work rate is the fraction of a job completed per unit time, so for a person who finishes a job in T days, the rate is 1/T job per day.

When two people work together, their individual rates add. Therefore, an unknown rate can be found by subtracting a known individual rate from the combined rate.

Application

1. The man completes the work in 10 days, so his rate is 1/10 of the work per day.

2. Together, the man and the boy complete the work in 8 days, so their combined rate is 1/8 of the work per day.

3. The boy’s rate is the combined rate minus the man’s rate: 1/8 − 1/10 = (5 − 4)/40 = 1/40 of the work per day.

4. Hence, the ratio of the man’s efficiency to the boy’s efficiency is (1/10):(1/40). Multiplying both terms by 40 gives 4:1.

Cross-check

Take the total work as 40 units. The man does 4 units per day and the boy does 1 unit per day; together they do 5 units per day and finish 40 units in 8 days. The man alone finishes the same work in 10 days.

Result

Therefore, the efficiency ratio of the man to the boy is 4:1.