Printable

2018 AMC 10A Problem 6

2018 AMC 10A Problem 6AMC 10 Step by Step

Name: Date: September 13, 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

If 65% of the votes are likes then 35% are dislikes, so the score is 30% of the total; 30% of T equals 90.

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 $\boxed{\textbf{(B)}\ 300}$ .

Why this works

Both vote counts are percentages of the same total, so their difference is also a percentage of that total: $65\% - 35\% = 30\%$ . Once the score is expressed as a single percent of $T$ , the problem is a one-step equation. Whenever two quantities are described as parts of one whole, work with the whole as the unknown.

Alternative approach

Test the choices: with $300$ votes, $65\%$ is $195$ likes, leaving $105$ dislikes, and $195 - 105 = 90$ . With $200$ votes the score would be $130 - 70 = 60$ , too small, and the score grows with $T$ , so $300$ is the only fit.

The trap

Setting the score equal to 65% of the votes, forgetting that dislikes subtract and only the 65% - 35% difference survives.

Common mistakes

Setting the score equal to 65% of the votes, forgetting that dislikes subtract and only the 65% - 35% difference survives; this gives $T \approx 138$ , not a choice, and leads to guessing.
Computing likes as $90$ and then dividing by $0.65$ , which confuses the score with the number of like votes.

Techniques

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