2020 AMC 10A Problem 2

P1-10~1 minPrint

Ratios, Percents & Averages Statistics & Data Set up the equation/formula and compute; no special trick needed

The numbers $3, 5, 7, a,$ and $b$ have an average (arithmetic mean) of $15$ . What is the average of $a$ and $b$ ?

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Key insight

Five numbers averaging 15 add up to 75. Take away 3 + 5 + 7 = 15 to get a + b = 60, then halve it.

Solution

Averages are hard to split apart, but totals are easy. So turn the average into a total first.

Five numbers with average $15$ add up to $5 \times 15 = 75$ .
The three numbers we know add up to $3 + 5 + 7 = 15$ .
So the other two make up the rest: $a + b = 75 - 15 = 60$ .
The average of $a$ and $b$ is their total divided by $2$ : $\frac{60}{2} = 30$ .

The answer is $\boxed{\textbf{(C)}\ 30}$ .

Why this works

"Average $\times$ how many $=$ total" lets you switch between averages and totals. Totals can be added and subtracted piece by piece, so do the work with totals and switch back to an average only at the end. Notice that we never learn $a$ or $b$ by itself, and we do not need to.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

Five numbers with mean $15$ have total $5 \cdot 15 = 75$ . The three known numbers contribute $3 + 5 + 7 = 15$ , so
$a + b = 75 - 15 = 60.$
The average of $a$ and $b$ is therefore $\frac{60}{2} = 30$ .

The answer is $\boxed{\textbf{(C)}\ 30}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 6: Ratios and Percents
Prealgebra, Chapter 13: Data and Statistics
Introduction to Algebra, Chapter 7: Proportion (§7.1, §7.2, §7.3)