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On a certain road 10% of the motorists exceed the posted speed limit and receive speeding tickets, but 20% of the motorists who exceed the posted speed limit do not receive speeding tickets. What percent of the motorists on the road exceed the posted speed limit?
Let's break down what the problem is telling us in everyday language:
First, we're told that "10% of the motorists exceed the posted speed limit and receive speeding tickets." This means that if we look at ALL motorists on the road, 10% of them both speed AND get caught with a ticket.
Second, we're told that "20% of the motorists who exceed the posted speed limit do not receive speeding tickets." This is talking about a different group - it's looking only at people who speed, and saying that 20% of those speeders manage to avoid getting tickets.
Our goal is to find what percent of ALL motorists exceed the speed limit - both the ones who get caught and the ones who don't.
Process Skill: TRANSLATE - Converting the problem language into clear mathematical understanding is crucial here since the percentages refer to different populations.
Now let's think about this logically. Every person who speeds falls into exactly one of two categories:
• Speeders who GET tickets
• Speeders who DON'T get tickets
There's no overlap and no one is left out - if you speed, you either get a ticket or you don't. This means:
Total Speeders = Speeders with tickets + Speeders without tickets
Let's say S% represents the percentage of ALL motorists who exceed the speed limit. This is what we're trying to find.
From the problem, we know:
• 20% of speeders DON'T get tickets
• This means 80% of speeders DO get tickets (since \(100\% - 20\% = 80\%\))
So we can express both groups:
• Speeders who get tickets = \(80\%\) of \(\mathrm{S}\%\) = \(0.8\mathrm{S}\%\) of all motorists
• Speeders who don't get tickets = \(20\%\) of \(\mathrm{S}\%\) = \(0.2\mathrm{S}\%\) of all motorists
Here's the key connection: We know that 10% of ALL motorists get speeding tickets.
But we just figured out that speeders who get tickets represent \(0.8\mathrm{S}\%\) of all motorists.
These must be the same group! So:
\(0.8\mathrm{S}\% = 10\%\)
Solving for S:
\(0.8\mathrm{S} = 10\)
\(\mathrm{S} = 10 ÷ 0.8\)
\(\mathrm{S} = 12.5\)
Therefore, 12.5% of motorists exceed the speed limit.
Process Skill: INFER - Recognizing that the "speeders who get tickets" must equal the "10% of all motorists who get speeding tickets" is the crucial insight that allows us to solve the problem.
12.5% of the motorists on the road exceed the posted speed limit.
Let's verify: If 12.5% speed, then:
• 80% of speeders get tickets: \(0.8 × 12.5\% = 10\%\) ✓
• 20% of speeders don't get tickets: \(0.2 × 12.5\% = 2.5\%\)
• Total: \(10\% + 2.5\% = 12.5\%\) ✓
The answer is B. 12.5%
1. Misinterpreting what the percentages refer to: Students often confuse which population each percentage is describing. The "10% of motorists" refers to ALL motorists on the road, while "20% of motorists who exceed the speed limit" refers only to the subset of speeders. This confusion leads to incorrect equation setup.
2. Failing to recognize the complementary relationship: Many students miss that if 20% of speeders don't get tickets, then 80% of speeders DO get tickets. They might try to work directly with the 20% figure, making the problem much more complex than needed.
3. Not identifying the key connection: Students may struggle to realize that "speeders who get tickets" and "10% of all motorists who get speeding tickets" represent the same group of people, just described differently. Without this insight, they cannot set up the crucial equation.
1. Incorrect percentage calculations: When converting "80% of S%" into mathematical form, students might write 80S instead of 0.8S, or forget to handle the percentage notation properly, leading to answers that are off by factors of 10 or 100.
2. Setting up the wrong equation: Even if students understand the relationships, they might incorrectly write equations like "0.2S = 10" (using the 20% who don't get tickets) instead of the correct "0.8S = 10" (using the 80% who do get tickets).
No likely faltering points - the calculation leads directly to 12.5%, which matches answer choice B exactly, making selection straightforward once the math is completed correctly.
Step 1: Choose a smart number for total motorists
Let's use 100 motorists as our total population. This makes percentage calculations very straightforward since each person represents 1%.
Step 2: Calculate motorists who get speeding tickets
10% of all motorists get speeding tickets
\(10\%\) of \(100 = 10\) motorists get tickets
Step 3: Analyze the speeder categories
We know that 20% of speeders don't get tickets, which means 80% of speeders DO get tickets.
Step 4: Find total number of speeders
If 80% of speeders get tickets, and we know 10 motorists get tickets:
\(80\%\) of speeders \(= 10\) motorists
\(0.8 × (\text{total speeders}) = 10\)
Total speeders \(= 10 ÷ 0.8 = 12.5\) motorists
Step 5: Convert to percentage
12.5 speeders out of 100 total motorists = 12.5%
Verification:
• Total speeders: 12.5
• Speeders who get tickets: \(80\%\) of \(12.5 = 10\) ✓
• Speeders who don't get tickets: \(20\%\) of \(12.5 = 2.5\)
• \(10 + 2.5 = 12.5\) ✓