In an apartment complex, the number of people aged 51 years and above is 30 and there are at most 39 people whose ages are below 51 years. The average age of all the people in the apartment complex is 38 years. What is the largest possible average age, in years, of the people whose ages are below 51

Question

In an apartment complex, the number of people aged 51 years and above is 30 and there are at most 39 people whose ages are below 51 years. The average age of all the people in the apartment complex is 38 years. What is the largest possible average age, in years, of the people whose ages are below 51 years?

Accepted Answer

28 — Easy There are $30$ people aged $\ge 51$ (their average $x \ge 51$) and at most $39$ people below $51$ (average $y$, what we maximise). The overall average is $38$. To make $y$ as large as possible, take the maximum count of the below-51 group (39) and the smallest possible $x$, namely $x = 51$. 1 Total-age equation. With $39$ people below 51 and $30$ at or above 51, the whole group of $69$ averages $38$. $$\begin{aligned} &30x + 39y = 69	imes 38 = 2622 \end{aligned}$$ 2 Maximise $y$. $y$ is la

CAT 2018 Slot 1 — QA Question 6

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