The average of 11 players of a cricket team is decreased by 2 months when two of them aged 17 years and 20 years are replaced by two new players. The average age of the new players is

### Concept & Strategy When members are replaced and the average decreases, the incoming members must weigh less (or be younger) than the outgoing members. The total difference in age between the old and new players equals the total decrease in the group's average. ### Step-by-Step Solution * **Calculate the total age decrease:** The average of all 11 players decreases by 2 months. $$Total\ Decrease = 11 \times 2\text{ months} = 22\text{ months}$$ * **Calculate the total age of the departing players:** Player 1 is 17 years old, Player 2 is 20 years old. $$Sum_{old} = 17 + 20 = 37\text{ years}$$ * **Calculate the total age of the new players:** The new players must be 22 months younger collectively than the players they replaced. $$Sum_{new} = 37\text{ years} - 22\text{ months}$$ To subtract properly, convert 1 year from the 37 years into months: $37\text{ years} = 35\text{ years} + 24\text{ months}$ $$Sum_{new} = (35\text{ years} + 24\text{ months}) - 22\text{ months}$$ $$Sum_{new} = 35\text{ years and } 2\text{ months}$$ * **Calculate the average age of the two new players:** Divide their total age by 2. $$Average_{new} = \frac{35\text{ years } 2\text{ months}}{2}$$ Convert the odd year to months to make division by 2 perfectly clean: $$35\text{ years } 2\text{ months} = 34\text{ years } + 14\text{ months}$$ $$Average_{new} = 17\text{ years and } 7\text{ months}$$ ### Exam Strategy & Shortcut Instead of converting to large totals of months, handle the years and the deviation separately. The average of the departing players is $\frac{37}{2} = 18.5\text{ years}$, which is 18 years 6 months. The total deficit they bring is 22 months, meaning *each* new player is individually responsible for an 11-month deficit from that outgoing average. $18\text{ years } 6\text{ months} - 11\text{ months} = 17\text{ years } 7\text{ months}$. ### Common Pitfall Struggling with the base-12 time conversion. Students often treat months as base-10 decimals, incorrectly assuming 35 years 2 months divided by 2 is 17.5 years 1 month. Always borrow 12 months when carrying over years. ### Final Answer **Therefore, the correct answer is 17 years 7 months.**