2014 AMC 10A Problem 10

Question

Five positive consecutive integers starting with

a

have average

b.

What is the average of

5

consecutive integers that start with

b?

a+3

a+4

a+5

a+6

a+7

The average of five consecutive integers starting at

x

is

x+2

Apply that rule first to

a

, then to

b

Note that the average of

5

consecutive numbers starting with

x

is

\dfrac{5x + 1 + 2 + 3 + 4}{5} = x + 2.

This means that the average of

5

consecutive integers starting with

a

is

a + 2,

which we know is

b.

Furthermore, the average of

5

consecutive numbers starting with

b

is

b + 2 = a + 4.

Thus, B is the correct answer.

Problem 10 in other years