2014 AMC 10A Problem 10

P1-10~1 minPrint

Ratios, Percents & Averages Number Properties Set up the equation/formula and compute; no special trick needed Test small/specific values or special cases to find or verify the answer

Five positive consecutive integers starting with $a$ have average $b$ . What is the average of $5$ consecutive integers that start with $b$ ?

Show solution without answering

Key insight

The average of five consecutive whole numbers is the middle one. So b is a + 2, and the second list averages b + 2, which is a + 4.

Solution

Try a small case first. If $a = 1$ , the five numbers are $1, 2, 3, 4, 5$ . Their sum is $15$ and their average is $3$ , which is the middle number. That always happens: the numbers are spread evenly around the middle one, so the ones below it and above it balance out.

The first list is $a, a+1, a+2, a+3, a+4$ . Its middle number is $a + 2$ , so $b = a + 2$ .
The second list starts at $b$ : $b, b+1, b+2, b+3, b+4$ . Its average is its middle number, $b + 2$ .
Put in $b = a + 2$ : the average is $(a + 2) + 2 = a + 4$ .

Check with $a = 1$ : then $b = 3$ , and the list $3, 4, 5, 6, 7$ has average $5$ , which is $1 + 4$ .

The answer is $\boxed{\textbf{(B)}\ a+4}$ .

Why this works

When numbers go up by the same step and there is an odd number of them, the average is the middle one, because each number below the middle is matched by one the same distance above it. Using that fact twice makes the problem a two-line calculation. Checking with a small value like $a = 1$ is a quick way to be sure.

Concise solution

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

The five integers $a, a+1, a+2, a+3, a+4$ are evenly spaced, so their average is the middle one: $b = a + 2$ .

The five consecutive integers starting with $b$ are $b, b+1, b+2, b+3, b+4$ , whose average is likewise the middle term $b + 2$ . Substituting $b = a + 2$ ,
$b + 2 = (a + 2) + 2 = a + 4.$

The answer is $\boxed{\textbf{(B)}\ a+4}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 6: Ratios and Percents
Introduction to Number Theory, Chapter 1: Integers: The Basics
Introduction to Algebra, Chapter 7: Proportion (§7.1, §7.2, §7.3)
Introduction to Algebra, Chapter 20: Special Functions (§20.3)