Alice's take-home pay last year was the same each month, and she saved the same fraction of her take-home pay...

1. Translate the problem requirements: Alice saves the same fraction each month, and her total annual savings equals 3 times her monthly spending (the portion she doesn't save). We need to find what fraction of monthly pay she saves.
2. Set up variables for monthly finances: Define monthly take-home pay and use a variable for the savings fraction to express both monthly savings and monthly spending.
3. Express the key relationship algebraically: Convert the statement "annual savings = 3 × monthly spending" into an equation using our variables.
4. Solve for the savings fraction: Simplify the equation to find the fraction of monthly pay that Alice saves, then verify against answer choices.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what Alice's financial situation looks like in everyday terms. Alice gets the same paycheck every month, and she consistently saves the same portion of it. The key insight is understanding what "3 times the amount she did NOT save" means.

If Alice saves some portion of her monthly pay, then the rest (what she did NOT save) is what she spent that month. The problem tells us that her total savings for the entire year equals 3 times this monthly spending amount.

So we need to find: what fraction of her monthly take-home pay did Alice save?

Process Skill: TRANSLATE - Converting the complex problem language into clear financial relationships

2. Set up variables for monthly finances

Let's use concrete terms to make this clear. Say Alice's monthly take-home pay is P dollars.

If Alice saves a fraction f of her pay each month, then:

• Monthly savings = \(\mathrm{f} \times \mathrm{P}\)
• Monthly spending (what she did NOT save) = \((1 - \mathrm{f}) \times \mathrm{P}\)

For example, if she saves \(\frac{1}{4}\) of her pay, then she spends \(\frac{3}{4}\) of her pay each month.

Since she saves the same amount each month for 12 months:

• Annual savings = 12 × (monthly savings) = \(12 \times \mathrm{f} \times \mathrm{P}\)

3. Express the key relationship algebraically

Now we can translate the main condition: "The total amount she saved at the end of the year was 3 times the amount of that portion of her monthly take-home pay that she did NOT save."

In plain English: Annual savings = 3 × Monthly spending

Using our variables:
\(12 \times \mathrm{f} \times \mathrm{P} = 3 \times (1 - \mathrm{f}) \times \mathrm{P}\)

Notice that the monthly pay P appears on both sides, so we can divide both sides by P:
\(12 \times \mathrm{f} = 3 \times (1 - \mathrm{f})\)

Process Skill: SIMPLIFY - Eliminating the common factor P to make the equation more manageable

4. Solve for the savings fraction

Now we solve the equation \(12\mathrm{f} = 3(1 - \mathrm{f})\):

Expanding the right side:
\(12\mathrm{f} = 3 - 3\mathrm{f}\)

Adding 3f to both sides:
\(12\mathrm{f} + 3\mathrm{f} = 3\)
\(15\mathrm{f} = 3\)

Dividing both sides by 15:
\(\mathrm{f} = \frac{3}{15} = \frac{1}{5}\)

Let's verify this makes sense: If Alice saves \(\frac{1}{5}\) of her monthly pay, she spends \(\frac{4}{5}\) of her monthly pay. Her annual savings would be \(12 \times \frac{1}{5} \times \mathrm{P} = \frac{12\mathrm{P}}{5}\), and 3 times her monthly spending would be \(3 \times \frac{4}{5} \times \mathrm{P} = \frac{12\mathrm{P}}{5}\). ✓

Final Answer

Alice saved \(\frac{1}{5}\) of her take-home pay each month.

The answer is D. \(\frac{1}{5}\)

Common Faltering Points

Errors while devising the approach

1. Misinterpreting "the amount she did NOT save"

Students often confuse this phrase and think it refers to her total annual spending (12 months of spending) rather than her monthly spending from just one month. This leads them to set up the equation as: \(12 \times \mathrm{f} \times \mathrm{P} = 3 \times 12 \times (1-\mathrm{f}) \times \mathrm{P}\), which gives the wrong answer.

2. Setting up the relationship backwards

Some students flip the relationship and think that her monthly spending equals 3 times her annual savings, leading to the equation: \((1-\mathrm{f}) \times \mathrm{P} = 3 \times 12 \times \mathrm{f} \times \mathrm{P}\). This fundamental misreading of the problem statement leads to an incorrect setup.

3. Confusing savings rate vs spending rate

Students may correctly understand the relationship but then solve for the spending fraction instead of the savings fraction, thinking that \((1-\mathrm{f})\) is what the question is asking for rather than f.

Errors while executing the approach

1. Arithmetic errors in algebraic manipulation

When expanding \(12\mathrm{f} = 3(1-\mathrm{f})\), students may incorrectly get \(12\mathrm{f} = 3 + 3\mathrm{f}\) instead of \(12\mathrm{f} = 3 - 3\mathrm{f}\), or make sign errors when moving terms to one side of the equation.

2. Incorrect fraction simplification

After getting \(\mathrm{f} = \frac{3}{15}\), some students fail to reduce this to \(\frac{1}{5}\), or make calculation errors that lead them to think \(\frac{3}{15}\) equals \(\frac{1}{6}\) or some other incorrect fraction.

Errors while selecting the answer

1. Selecting the spending fraction instead of savings fraction

Students who correctly solve \(\mathrm{f} = \frac{1}{5}\) but then remember that Alice spends \(\frac{4}{5}\) of her income might mistakenly think the question asks for the spending rate and look for \(\frac{4}{5}\) among the choices. When they don't find it, they may guess or select a close-looking option like \(\frac{1}{2}\).

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a convenient monthly take-home pay
Let's set Alice's monthly take-home pay = \$100
(We choose \$100 because it makes percentage and fraction calculations straightforward)

Step 2: Set up the relationship using our concrete number
• If Alice saves fraction f of her monthly pay:
• Monthly savings = \(\mathrm{f} \times \\$100\)
• Monthly spending (amount NOT saved) = \((1 - \mathrm{f}) \times \\$100\)
• Annual savings = \(12 \times \mathrm{f} \times \\$100\)

Step 3: Apply the given constraint
We're told: Annual savings = 3 × Monthly spending
\(12 \times \mathrm{f} \times \\$100 = 3 \times (1 - \mathrm{f}) \times \\$100\)

Step 4: Solve for the savings fraction
Divide both sides by \$100:
\(12\mathrm{f} = 3(1 - \mathrm{f})\)
\(12\mathrm{f} = 3 - 3\mathrm{f}\)
\(12\mathrm{f} + 3\mathrm{f} = 3\)
\(15\mathrm{f} = 3\)
\(\mathrm{f} = \frac{3}{15} = \frac{1}{5}\)

Step 5: Verify with concrete numbers
If Alice saves \(\frac{1}{5}\) of \$100 = \$20 monthly:
• Monthly spending = \$100 - \$20 = \$80
• Annual savings = 12 × \$20 = \$240
• Check: 3 × monthly spending = 3 × \$80 = \$240 ✓

Answer: Alice saves \(\frac{1}{5}\) of her monthly take-home pay.