- Translate the problem requirements: We need to find the additional amount Sally will pay this week for the same quantity of gasoline she bought last week, given that prices increased from \(\$1.65\) to \(\$1.82\) per gallon and she paid \(\$26.40\) last week.
- Calculate the quantity of gasoline purchased: Use last week's total payment and price per gallon to determine how many gallons Sally bought.
- Calculate this week's total cost: Apply this week's higher price to the same quantity of gasoline.
- Find the price difference: Subtract last week's cost from this week's cost to determine the additional amount Sally will pay.
Execution of Strategic Approach
Translate the problem requirements
Let's break down what we know and what we need to find in plain English:
- Last week: gasoline cost \(\$1.65\) per gallon
- This week: gasoline costs \(\$1.82\) per gallon
- Sally paid \(\$26.40\) total last week
- We need to find: How much MORE will Sally pay this week for the same amount of gasoline
The key insight is that Sally is buying the exact same number of gallons both weeks - only the price per gallon has changed.
Process Skill: TRANSLATE - Converting the word problem into clear mathematical relationships
Calculate the quantity of gasoline purchased
To find out how many gallons Sally bought, we use the basic relationship: \(\mathrm{Total\ Cost} = \mathrm{Price\ per\ Gallon} \times \mathrm{Number\ of\ Gallons}\)
Since we know Sally's total cost last week (\(\$26.40\)) and the price per gallon last week (\(\$1.65\)), we can find the number of gallons:
\(\mathrm{Number\ of\ gallons} = \mathrm{Total\ Cost} \div \mathrm{Price\ per\ Gallon}\)
\(\mathrm{Number\ of\ gallons} = \$26.40 \div \$1.65\)
\(\mathrm{Number\ of\ gallons} = 16\) gallons
Let's verify: \(16 \times \$1.65 = \$26.40\) ✓
Calculate this week's total cost
Now we know Sally bought 16 gallons last week. This week, she wants to buy the same 16 gallons, but at the new price of \(\$1.82\) per gallon.
\(\mathrm{This\ week's\ total\ cost} = \mathrm{Number\ of\ gallons} \times \mathrm{New\ price\ per\ gallon}\)
\(\mathrm{This\ week's\ total\ cost} = 16 \times \$1.82\)
\(\mathrm{This\ week's\ total\ cost} = \$29.12\)
Find the price difference
The question asks how much MORE Sally will pay this week compared to last week:
\(\mathrm{Additional\ amount} = \mathrm{This\ week's\ cost} - \mathrm{Last\ week's\ cost}\)
\(\mathrm{Additional\ amount} = \$29.12 - \$26.40\)
\(\mathrm{Additional\ amount} = \$2.72\)
Final Answer
Sally will pay \(\$2.72\) more this week for the same amount of gasoline.
Looking at our answer choices, this matches choice D: \(\$2.72\).
We can double-check our work: The price increased by \(\$1.82 - \$1.65 = \$0.17\) per gallon, and Sally bought 16 gallons, so the total increase should be \(16 \times \$0.17 = \$2.72\) ✓
Common Faltering Points
Errors while devising the approach
- Misunderstanding what the question is asking: Students might think they need to find Sally's total cost this week (\(\$29.12\)) rather than how much MORE she will pay. The word 'more' is crucial - it signals we need the difference between costs, not the absolute cost.
- Attempting to use percentage increase incorrectly: Some students might try to calculate the percentage increase in price and apply it directly to the \(\$26.40\), which would lead to computational errors and miss the straightforward approach of finding gallons first.
Errors while executing the approach
- Division error when calculating gallons: When dividing \(\$26.40 \div \$1.65\), students might make arithmetic mistakes, especially if doing this by hand. Getting an incorrect number of gallons (like 15 or 17 instead of 16) will throw off all subsequent calculations.
- Multiplication errors: Even with the correct number of gallons (16), students might incorrectly calculate \(16 \times \$1.82\), potentially getting \(\$29.02\) or \(\$29.22\) instead of \(\$29.12\), leading to wrong final differences.
- Subtraction mistakes in final step: When calculating \(\$29.12 - \$26.40\), students might rush and get \(\$2.62\) or \(\$2.82\) instead of \(\$2.72\), especially under time pressure.
Errors while selecting the answer
- Selecting this week's total cost instead of the difference: Students who calculated \(\$29.12\) correctly but misunderstood the question might look for an answer choice close to \(\$29\), potentially selecting a wrong option or getting confused when they don't see it listed.
- Rounding errors leading to wrong choice: If students made small computational errors that led them to values like \(\$2.64\) (choice C) or \(\$2.55\) (choice B), they might select these instead of recognizing their calculation error.
