Last year a state senate consisting of only Republican and Democrat members had 20 more Republican members than Democrat members....

1. Translate the problem requirements: We need to clarify the senate composition changes between two years and find the total number of members this year when Republicans represent \(\frac{2}{3}\) of the senate.
2. Define variables for senate composition: Set up variables to represent the number of Democrats and Republicans in a way that keeps calculations simple.
3. Apply the year-over-year changes: Use the given information about how Republican membership changed between last year and this year.
4. Use the current ratio to solve: Apply the fact that Republicans now make up \(\frac{2}{3}\) of the senate to find the total membership.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what's happening in plain English. We have a state senate with only Republicans and Democrats. The problem gives us information about two different years - last year and this year.

Last year: Republicans outnumbered Democrats by exactly 20 members.
This year: The total number of senators stayed the same, but there are 2 fewer Republicans than last year (which means there are 2 more Democrats than last year). Also, Republicans now make up exactly \(\frac{2}{3}\) of the entire senate.

We need to find the total number of senators this year.

Process Skill: TRANSLATE - Converting the word problem into clear mathematical relationships

2. Define variables for senate composition

Let's use simple variables to keep track of the numbers:

For last year:
- Let D = number of Democrats last year
- Let R = number of Republicans last year
- We know that \(\mathrm{R} = \mathrm{D} + 20\) (Republicans had 20 more than Democrats)
- Total senators last year = \(\mathrm{D} + \mathrm{R} = \mathrm{D} + (\mathrm{D} + 20) = 2\mathrm{D} + 20\)

For this year:
- Democrats this year = \(\mathrm{D} + 2\) (2 more than last year)
- Republicans this year = \(\mathrm{R} - 2 = (\mathrm{D} + 20) - 2 = \mathrm{D} + 18\) (2 fewer than last year)
- Total senators this year = \((\mathrm{D} + 2) + (\mathrm{D} + 18) = 2\mathrm{D} + 20\)

Notice that the total is indeed the same both years: \(2\mathrm{D} + 20\).

3. Apply the year-over-year changes

Now we use the key constraint: this year, Republicans make up \(\frac{2}{3}\) of the senate.

In everyday terms, if Republicans are \(\frac{2}{3}\) of the senate, then out of every 3 senators, 2 are Republican and 1 is Democrat.

So we can write:
Republicans this year = \(\frac{2}{3} \times \text{Total senators this year}\)

Substituting what we know:
\(\mathrm{D} + 18 = \frac{2}{3} \times (2\mathrm{D} + 20)\)

4. Use the current ratio to solve

Let's solve this equation step by step:

\(\mathrm{D} + 18 = \frac{2}{3} \times (2\mathrm{D} + 20)\)

Multiply both sides by 3 to eliminate the fraction:
\(3(\mathrm{D} + 18) = 2(2\mathrm{D} + 20)\)
\(3\mathrm{D} + 54 = 4\mathrm{D} + 40\)

Subtract \(3\mathrm{D}\) from both sides:
\(54 = \mathrm{D} + 40\)

Subtract 40 from both sides:
\(\mathrm{D} = 14\)

So there were 14 Democrats last year.

Now we can find the total number of senators:
Total = \(2\mathrm{D} + 20 = 2(14) + 20 = 28 + 20 = 48\)

Let's verify: This year we have:
- Democrats: \(14 + 2 = 16\)
- Republicans: \(14 + 18 = 32\)
- Total: \(16 + 32 = 48\)
- Republican fraction: \(\frac{32}{48} = \frac{2}{3}\) ✓

4. Final Answer

The senate has 48 members this year.

This matches answer choice (E) 48.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the timing of changes

Students often confuse which year's conditions apply to which constraints. They might mistakenly think that the "20 more Republicans than Democrats" applies to this year instead of last year, or confuse when the "\(\frac{2}{3}\) Republicans" ratio applies. This leads to setting up equations with the wrong year's data.

2. Incorrectly handling the membership changes

Students may misunderstand the direction of changes between years. When the problem states "2 fewer Republican members than last year," some students incorrectly interpret this as Republicans increasing rather than decreasing, or they forget that when Republicans decrease by 2, Democrats must increase by 2 to keep the total constant.

3. Missing the constraint that total membership stays constant

Some students overlook that "the senate has the same number of members as last year" and instead assume the total can change. This leads them to set up separate variables for each year's total membership, creating unnecessary complexity and potential errors.

Errors while executing the approach

1. Fraction manipulation errors

When solving the equation \(\mathrm{D} + 18 = \frac{2}{3} \times (2\mathrm{D} + 20)\), students commonly make algebraic mistakes. They might incorrectly distribute the \(\frac{2}{3}\), forget to multiply both sides by 3 to clear the fraction, or make errors when expanding \(2(2\mathrm{D} + 20) = 4\mathrm{D} + 40\).

2. Sign errors in equation setup

Students often make sign errors when translating "2 fewer Republicans" into the algebraic expression. They might write \(\mathrm{R} + 2\) instead of \(\mathrm{R} - 2\), or incorrectly handle the subtraction when substituting \(\mathrm{R} = \mathrm{D} + 20\) to get \((\mathrm{D} + 20) - 2 = \mathrm{D} + 18\).

Errors while selecting the answer

1. Reporting the wrong quantity

After solving correctly and finding \(\mathrm{D} = 14\), some students report 14 as their final answer instead of calculating the total senate membership. They might confuse what the question is asking for and select the number of Democrats rather than the total number of senators.

2. Failing to verify the solution

Students may arrive at the correct total of 48 but fail to check whether their answer satisfies all the given conditions. Without verification, they might not catch computational errors that led to an incorrect total, or they might second-guess a correct answer and change it to a wrong one.

Alternate Solutions

Smart Numbers Approach

Strategy: Since Republicans make up exactly \(\frac{2}{3}\) of the senate this year, we can work with concrete numbers that satisfy this fraction.

Step 1: Choose a smart number for total senate members this year
Since Republicans = \(\frac{2}{3}\) of total members, the total must be divisible by 3.
Let's try total members = 48 (from answer choice E)
Then Republicans this year = \(\frac{2}{3} \times 48 = 32\)
Democrats this year = \(48 - 32 = 16\)

Step 2: Work backwards to last year
This year has 2 fewer Republicans than last year
So Republicans last year = \(32 + 2 = 34\)
Since total members stayed the same: Democrats last year = \(48 - 34 = 14\)

Step 3: Verify the constraint
Last year: Republicans had 20 more members than Democrats
Check: \(34 - 14 = 20\) ✓

Step 4: Confirm our answer
All conditions are satisfied with 48 total members:
• Last year: 34R, 14D (difference of 20) ✓
• This year: 32R, 16D (2 fewer Republicans) ✓
• This year: \(\frac{32}{48} = \frac{2}{3}\) Republicans ✓

Answer: 48 members