AMC 10 · 2018 · #6

Grade 6 arithmetic
Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Percent / Fraction Recovery
percentagelinear-equations-one-var convert-to-algebra ↑ Prerequisites: percentage
📏 Short solution 💡 1 insight
Problem
A video's score goes up 1 for each like and down 1 for each dislike, starting from 0. When the score is 90, exactly 65% of all votes are likes. Find the total number of votes cast.

Pick an answer.

(A)
200
(B)
300
(C)
400
(D)
500
(E)
600

AMC 10 2018 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Convert to Algebra

This is a word problem with one unknown — the total number of votes — so name it with a variable and translate the words into an equation. The score is likes minus dislikes, and both counts are percents of that same total. Seeing dislikes as the complement of likes (35% = 100% − 65%) turns the score into a single percent of the total, which gives a one-step equation to solve.

1STEP 1

Name the total and split the votes

Let N be the total votes. Likes are 65%, so 0.65N; dislikes are the rest, 35%, giving 0.35N.

likes = 0.65N, dislikes = 0.35N
2STEP 2

Write the score as one percent

Score = likes − dislikes = 0.65N − 0.35N = 0.30N, i.e. 30% of the total; matched pairs cancel, leaving only the 30% gap.

score = 0.65N - 0.35N = 0.30N
3STEP 3

Solve for the total

The score is 90, so 0.30N = 90; divide by 0.30 to get N = 300, answer (B).

0.30N = 90 → N = 90/0.30 = 300
Answer
300
Check it forward: 65% of 300 is 195 likes and 35% of 300 is 105 dislikes, and 195 − 105 = 90, the given score. The answer 300 sits in the middle of the choices, which fits — a smaller total like 200 would give a score of only 60, and a larger total like 600 would give 180.
💡Key takeaway

Likes and dislikes cancel out, so only the leftover 30% builds the score — and 30% of the total is 90.

  • Name the total and split the votes
  • Write the score as one percent
  • Solve for the total

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