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In a certain district, the ratio of the number of registered Republicans to the number of registered Democrats was \(\frac{3}{5}\). After 600 additional Republicans and 500 additional Democrats registered, the ratio was \(\frac{4}{5}\). After these registrations, there were how many more voters in the district registered as Democrats than as Republicans?
Let's break down what this problem is asking us to find. We have a district where voters are registered as either Republicans or Democrats. The problem tells us about two different time periods:
Before new registrations: For every 3 Republicans, there were 5 Democrats
After new registrations: 600 more Republicans and 500 more Democrats registered, and now for every 4 Republicans, there are 5 Democrats
Our goal is to find: After all the new registrations, how many more Democrats are there than Republicans?
Process Skill: TRANSLATE - Converting the ratio language into a clear mathematical setup
When we say the ratio of Republicans to Democrats was \(\frac{3}{5}\), this means that if we imagine dividing all the Republicans into groups of 3 and all the Democrats into groups of 5, we'd have the same number of groups.
Let's call this number of groups 'k'. This means:
• Originally, there were \(3\mathrm{k}\) Republicans
• Originally, there were \(5\mathrm{k}\) Democrats
For example, if k = 100, there would be 300 Republicans and 500 Democrats, giving us the 3:5 ratio.
Mathematically: Initial Republicans = \(3\mathrm{k}\), Initial Democrats = \(5\mathrm{k}\)
Now let's see what happens after the new registrations:
• Republicans after new registrations = \(3\mathrm{k} + 600\)
• Democrats after new registrations = \(5\mathrm{k} + 500\)
We're told that after these changes, the new ratio becomes \(\frac{4}{5}\). This means:
\(\frac{3\mathrm{k} + 600}{5\mathrm{k} + 500} = \frac{4}{5}\)
In plain English: The new number of Republicans divided by the new number of Democrats equals \(\frac{4}{5}\).
To solve this equation, we can cross-multiply (multiply both sides to eliminate the fractions):
\(5 \times (3\mathrm{k} + 600) = 4 \times (5\mathrm{k} + 500)\)
\(15\mathrm{k} + 3000 = 20\mathrm{k} + 2000\)
Now let's solve for k by getting all terms with k on one side:
\(3000 - 2000 = 20\mathrm{k} - 15\mathrm{k}\)
\(1000 = 5\mathrm{k}\)
\(\mathrm{k} = 200\)
Process Skill: MANIPULATE - Systematic algebraic manipulation to find the key variable
Now that we know \(\mathrm{k} = 200\), we can find the actual numbers after all registrations:
Final number of Republicans: \(3\mathrm{k} + 600 = 3(200) + 600 = 600 + 600 = 1,200\)
Final number of Democrats: \(5\mathrm{k} + 500 = 5(200) + 500 = 1,000 + 500 = 1,500\)
Let's verify our ratio: \(\frac{1,200}{1,500} = \frac{4}{5}\) ✓
The difference: \(1,500 - 1,200 = 300\)
Therefore, there are 300 more Democrats than Republicans registered in the district.
The answer is (B) 300. After all registrations, there are 300 more voters registered as Democrats than as Republicans.
1. Misinterpreting ratio notation
Students often confuse the ratio \(\frac{3}{5}\) as meaning "3 out of 8 total voters are Republicans and 5 out of 8 are Democrats" instead of understanding it as "for every 3 Republicans, there are 5 Democrats." This leads to setting up fractions like \(\frac{3\mathrm{k}}{3\mathrm{k}+5\mathrm{k}}\) instead of the correct \(3\mathrm{k}\) approach with separate scaling.
2. Incorrectly setting up the final ratio equation
When translating "the ratio was \(\frac{4}{5}\)" after new registrations, students might write the equation as \((3\mathrm{k} + 600) + (5\mathrm{k} + 500) = \text{some expression involving } \frac{4}{5}\), thinking about total voters instead of maintaining the ratio relationship \(\frac{3\mathrm{k} + 600}{5\mathrm{k} + 500} = \frac{4}{5}\).
3. Misunderstanding what the question asks for
Students may focus on finding the initial numbers or the number of new registrants instead of recognizing that the question specifically asks for the difference between Democrats and Republicans AFTER all the new registrations are complete.
1. Cross-multiplication errors
When solving \(\frac{3\mathrm{k} + 600}{5\mathrm{k} + 500} = \frac{4}{5}\), students frequently make mistakes in the cross-multiplication step, writing incorrect equations like \(4(3\mathrm{k} + 600) = 5(5\mathrm{k} + 500)\) instead of the correct \(5(3\mathrm{k} + 600) = 4(5\mathrm{k} + 500)\), or making sign errors during the expansion.
2. Algebraic manipulation mistakes
After expanding \(15\mathrm{k} + 3000 = 20\mathrm{k} + 2000\), students often make errors when collecting like terms, such as incorrectly calculating \(3000 - 2000 = 500\) or \(20\mathrm{k} - 15\mathrm{k} = 10\mathrm{k}\), leading to wrong values of k and consequently incorrect final answers.
3. Arithmetic errors in final calculations
Even with the correct \(\mathrm{k} = 200\), students may make simple arithmetic mistakes when calculating the final numbers: computing \(3(200) + 600 = 1200\) incorrectly, or \(5(200) + 500 = 1500\) incorrectly, or making errors in the final subtraction \(1500 - 1200\).
1. Reporting the wrong quantity
Students may correctly calculate that there are 1,200 Republicans and 1,500 Democrats after registrations, but then mistakenly report 1,200 (number of Republicans) or 1,500 (number of Democrats) as their final answer instead of the difference 300, forgetting that the question asks "how many MORE Democrats than Republicans."
2. Sign confusion in the difference
Students might calculate the difference as \(1,200 - 1,500 = -300\) instead of \(1,500 - 1,200 = 300\), either due to careless calculation order or misunderstanding which group should be subtracted from which, potentially leading them to look for 300 among the choices but select it for the wrong reason.