Dick and Jane each saved $3,000 in 1989. In 1990 Dick saved 8% more than in 1989, and together he...

1. Translate the problem requirements: We need to find what percent less Jane saved in 1990 compared to 1989. Both saved \(\$3,000\) in 1989. In 1990, Dick saved 8% more than his 1989 amount, and their combined 1990 savings totaled \(\$5,000\).
2. Calculate Dick's 1990 savings: Apply the 8% increase to his 1989 baseline to find his 1990 amount.
3. Determine Jane's 1990 savings: Use the fact that their combined 1990 savings equal \(\$5,000\) to find Jane's portion.
4. Calculate the percentage decrease for Jane: Compare Jane's 1990 savings to her 1989 baseline to find the percent decrease.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we know in plain English:

• In 1989: Both Dick and Jane each saved \(\$3,000\)
• In 1990: Dick saved 8% more than his 1989 amount
• In 1990: Dick and Jane together saved a total of \(\$5,000\)
• We need to find: What percent less did Jane save in 1990 compared to 1989?

The key insight is that we can find Dick's 1990 savings, then use the total to find Jane's 1990 savings, and finally compare Jane's amounts.

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

2. Calculate Dick's 1990 savings

Dick saved 8% more in 1990 than in 1989. Let's think about this step by step:

• Dick's 1989 savings: \(\$3,000\)
• 8% increase means he saved an additional 8% of \(\$3,000\)
• 8% of \(\$3,000 = 0.08 \times \$3,000 = \$240\)
• So Dick's 1990 savings = \(\$3,000 + \$240 = \$3,240\)

Alternatively, we can think of this as Dick saving 108% of his 1989 amount:

• Dick's 1990 savings = \(1.08 \times \$3,000 = \$3,240\)

3. Determine Jane's 1990 savings

Now we use the fact that together they saved \(\$5,000\) in 1990:

• Total 1990 savings = Dick's 1990 savings + Jane's 1990 savings
• \(\$5,000 = \$3,240 + \text{Jane's 1990 savings}\)
• Jane's 1990 savings = \(\$5,000 - \$3,240 = \$1,760\)

So Jane saved \(\$1,760\) in 1990, compared to \(\$3,000\) in 1989.

4. Calculate the percentage decrease for Jane

Now let's find what percent less Jane saved in 1990 compared to 1989:

• Jane's 1989 savings: \(\$3,000\)
• Jane's 1990 savings: \(\$1,760\)
• Decrease in Jane's savings = \(\$3,000 - \$1,760 = \$1,240\)

To find the percentage decrease, we compare the decrease to her original 1989 amount:

• Percentage decrease = (Decrease ÷ Original amount) × 100%
• Percentage decrease = \((\$1,240 ÷ \$3,000) \times 100\%\)
• Percentage decrease = \(0.4133... \times 100\% \approx 41.3\%\)

Rounding to the nearest whole percent, Jane saved approximately 41% less in 1990 than in 1989.

5. Final Answer

Jane saved approximately 41% less in 1990 than in 1989.

Looking at the answer choices, this matches choice C: 41%.

Answer: C

Common Faltering Points

Errors while devising the approach

1. Misinterpreting what "8 percent more" means: Students might think Dick saved 8% of the total (\(\$5,000\)) more, rather than 8% more than his own 1989 savings of \(\$3,000\). This leads to calculating Dick's 1990 savings incorrectly as \(\$3,000 + 0.08(\$5,000) = \$3,400\) instead of \(\$3,240\).

2. Confusing the base year for percentage comparison: Students might try to calculate Jane's percentage change using her 1990 savings (\(\$1,760\)) as the base instead of her 1989 savings (\(\$3,000\)). This would lead to the wrong formula: (decrease ÷ 1990 amount) instead of (decrease ÷ 1989 amount).

3. Setting up the problem with incorrect total: Students might misread and think the \(\$5,000\) total refers to their combined savings over both years (1989 + 1990) rather than just their 1990 savings, leading to a completely different equation setup.

Errors while executing the approach

1. Arithmetic errors in percentage calculations: When calculating 8% of \(\$3,000\), students might compute \(0.8 \times \$3,000 = \$2,400\) instead of \(0.08 \times \$3,000 = \$240\), leading to Dick's 1990 savings being \(\$5,400\) instead of \(\$3,240\).

2. Sign errors in subtraction: Students might calculate Jane's 1990 savings as \(\$3,240 - \$5,000 = -\$1,760\) instead of \(\$5,000 - \$3,240 = \$1,760\), not realizing they've reversed the subtraction.

3. Decimal conversion mistakes: When converting \(\$1,240 ÷ \$3,000\) to a percentage, students might forget to multiply by 100 or make errors like getting 4.13% instead of 41.3%, or incorrectly calculating the division as 0.041 instead of 0.413.

Errors while selecting the answer

1. Selecting the wrong direction of change: Students might calculate the correct percentage (41%) but select an answer that represents "41% more" rather than "41% less," potentially choosing a value like 141% if available, or getting confused about the direction of Jane's change.

2. Using Dick's percentage instead of Jane's: Students might correctly calculate that Dick saved 8% more but then mistakenly select 8% (choice A) thinking the question asked about Dick's change rather than Jane's change.