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2020 AMC 10A Problem 2

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2020 AMC 10A Problem 2AMC 10 Step by Step

Name: Date: September 13, 2026

Ratios, Percents & Averages · difficulty 1 of 5 · about 1 min · position P1-10

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

A)
$0$
B)
$15$
C)
$30$
D)
$45$
E)
$60$

Answer

Key insight

An average of 15 over five numbers means the total is 75; remove 3 + 5 + 7 = 15 and halve what remains.

Solution

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 $\boxed{\textbf{(C)}\ 30}$ .

Why this works

Averages are awkward to combine directly, but totals add. Convert the given mean into a sum, subtract the known part, and convert back to a mean at the end. Note that $a$ and $b$ individually are not determined, yet their average is.

The trap

Reporting a + b = 60 (choice E) instead of the average of a and b.

Common mistakes

Reporting a + b = 60 (choice E) instead of the average of a and b.
Averaging the three known numbers ( $5$ ) and trying to balance it against $15$ without weighting by how many numbers are in each group.

Techniques

Set up the equation/formula and compute; no special trick needed