2018 AMC 10A Problem 6

P1-10~2 minPrint

Ratios, Percents & Averages Linear Equations & Word Problems Set up the equation/formula and compute; no special trick needed

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?

Show solution without answering

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.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

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)