2016 AMC 8

Complete problem set with solutions and individual problem pages

Problem 8 Easy

Find the value of the expression

100-98+96-94+92-90+\cdots+8-6+4-2.

  • A.

    20

  • B.

    40

  • C.

    50

  • D.

    80

  • E.

    100

Answer:C

Solution 1

We can group each subtracting pair together:

(100-98)+(96-94)+(92-90)+ \ldots +(8-6)+(4-2).

After subtracting, we have:

2+2+2+\ldots+2+2=2(1+1+1+\ldots+1+1).

There are 50 even numbers, therefore there are \dfrac{50}{2}=25 even pairs. Therefore the sum is 2 \cdot 25=\boxed{\textbf{(C) }50}

 

Solution 2

Since our list does not end with one, we divide every number by 2 and we end up with

50-49+48-47+ \ldots +4-3+2-1

We can group each subtracting pair together:

(50-49)+(48-47)+(46-45)+ \ldots +(4-3)+(2-1).

There are now 25 pairs of numbers, and the value of each pair is 1. This sum is 25. However, we divided by 2 originally so we will multiply 2*25 to get the final answer of \boxed{\textbf{(C) }50}

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