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2018 AMC 10A Problem 6

2018 AMC 10A Problem 6AMC 10 Step by Step

Name: Date: October 28, 2026

Ratios, Percents & Averages · difficulty 1 of 5 · about 2 min · position P1-10

Sangho uploaded a video to a website where viewers can vote that they like or dislike a video. Each video begins with a score of $0$ , and the score increases by $1$ for each like vote and decreases by $1$ for each dislike vote. At one point Sangho saw that his video had a score of $90$ , and that $65\%$ of the votes cast on his video were like votes. How many votes had been cast on Sangho's video at that point?

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

Answer

Key insight

Out of every 100 votes, 65 are likes and 35 are dislikes, so each 100 votes adds 30 to the score. A score of 90 needs 300 votes.

Solution

Start with a small concrete batch of votes to see how the score grows.

Look at $100$ votes. $65\%$ of them are likes, so $65$ likes and $35$ dislikes.
Score from those $100$ votes. Each like adds $1$ and each dislike takes away $1$ , so the score is $65 - 35 = 30$ .
Scale up. Every $100$ votes (in the same $65$ to $35$ split) adds $30$ to the score. The score is $90 = 3 \cdot 30$ , so there must be $3 \cdot 100 = 300$ votes.
Check. $65\%$ of $300$ is $195$ likes, leaving $105$ dislikes. Score: $195 - 105 = 90$ . Correct.

The answer is $\boxed{\textbf{(B)}\ 300}$ .

Why this works

Likes and dislikes are both fixed shares of the same total. Their difference, $65\% - 35\% = 30\%$ , is also a fixed share of the total, and that difference is exactly the score. So the score is $30\%$ of all votes, and $30\%$ of $300$ is $90$ .

Alternative approach

Try the answer choices. With $200$ votes: $130$ likes and $70$ dislikes, score $60$ . Too small. With $300$ votes: $195 - 105 = 90$ . That matches, and more votes would only raise the score, so $300$ is the answer.

The trap

Treating the score as the number of likes, so that 65% of the votes equals 90. Each dislike takes a point away, so only the gap between likes and dislikes counts.

Concise solution

Let $T$ be the total number of votes. Likes make up $65\%$ of them and dislikes the remaining $35\%$ , so

$\text{likes} = 0.65T, \qquad \text{dislikes} = 0.35T.$

The score is likes minus dislikes:
$0.65T - 0.35T = 0.30T = 90.$

Hence $T = \dfrac{90}{0.30} = 300$ .

Check: $195$ likes and $105$ dislikes give a score of $90$ , and $195/300 = 65\%$ .

The answer is $\boxed{\textbf{(B)}\ 300}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 6: Ratios and Percents
Introduction to Algebra, Chapter 3: One-Variable Linear Equations (§3.3, §3.4)
Introduction to Algebra, Chapter 7: Proportion (§7.1, §7.2, §7.3)
Prealgebra, Chapter 13: Data and Statistics (§13.1)

Techniques

Set up the equation/formula and compute; no special trick needed