2016 AMC 8 Problems/Problem 8

Problem

Find the value of the expression $100-98+96-94+92-90+\cdots+8-6+4-2.$ $\textbf{(A) }20\qquad\textbf{(B) }40\qquad\textbf{(C) }50\qquad\textbf{(D) }80\qquad \textbf{(E) }100$

Solutions

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 $\boxed{\textbf{(C) }50}$

Solution 3

Note: This solution is very similar to Solution 1, but simplified.

The value of the expression increases by 2 after every pair as shown:

Pair 1: [100-98= $2$ ]

Pair 1 + 2: [2+96-94= $4$ ]

Pair 1 + 2 + 3: [4+92-90= $6$ ]

Since there is 25 pairs in the expression, the value will increase by two 25 times. Thus, the sum of the expression is $2 \cdot 25=\boxed{\textbf{(C) }50}$

Video Solution

https://youtu.be/iuUwextm334?si=LrUYCG3Cvo0zKqym

A solution so simple a 12-year-old made it!

~Elijahman~

Video Solution by savannahsolver

https://youtu.be/TwIwA0XkzoI

~savannahsolver

Video Solution by OmegaLearn

https://youtu.be/51K3uCzntWs?t=645

~ pi_is_3.14

2016 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 7	Followed by Problem 9
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AJHSME/AMC 8 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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