2022 AMC 8

Complete problem set with solutions and individual problem pages

Problem 16 Hard

Four numbers are written in a row. The average of the first two is 21, the average of the middle two is 26, and the average of the last two is 30. What is the average of the first and last of the numbers?

  • A.

    24

  • B.

    25

  • C.

    26

  • D.

    27

  • E.

    28

Answer:B

Solution 1 (Arithmetic)

Note that the sum of the first two numbers is 21\cdot2=42, the sum of the middle two numbers is 26\cdot2=52, and the sum of the last two numbers is 30\cdot2=60.

It follows that the sum of the four numbers is 42+60=102, so the sum of the first and last numbers is 102-52=50. Therefore, the average of the first and last numbers is 50\div2=\boxed{\textbf{(B) } 25}.

 

Solution 2 (Algebra)

Let a,b,c, and d be the four numbers in that order. We are given that

\begin{align*} \frac{a+b}{2} &= 21, &(1) \\ \frac{b+c}{2} &= 26, &(2) \\ \frac{c+d}{2} &= 30, &(3) \end{align*}

and we wish to find \frac{a+d}{2}.

We add (1) and (3), then subtract (2) from the result:

\frac{a+d}{2}=21+30-26=\boxed{\textbf{(B) } 25}.

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