- Translate the problem requirements: Kim worked exactly 30 hours total, earning \(\$8/\mathrm{hour}\) for \(\mathrm{x}\) hours and \(\$12/\mathrm{hour}\) for \(\mathrm{y}\) hours, where both \(\mathrm{x}\) and \(\mathrm{y}\) must be whole numbers (integers). We need to find which total pay amount is possible.
- Set up the constraint equations: Establish the relationship between hours worked (\(\mathrm{x} + \mathrm{y} = 30\)) and total pay (\(8\mathrm{x} + 12\mathrm{y} = \mathrm{total\ pay}\)), recognizing that \(\mathrm{x}\) and \(\mathrm{y}\) must be non-negative integers.
- Determine the range of possible outcomes: Find the minimum possible pay (all hours at \(\$8/\mathrm{hour}\)) and maximum possible pay (all hours at \(\$12/\mathrm{hour}\)) to establish boundaries for valid answers.
- Test answer choices against integer constraints: For each answer choice, solve for the required values of \(\mathrm{x}\) and \(\mathrm{y}\) and verify that both are non-negative integers that sum to 30.
Execution of Strategic Approach
1. Translate the problem requirements
Let's start by understanding what Kim's work situation looks like. Kim worked a total of 30 hours last week. She earned money at two different rates: \(\$8\) per hour for some hours (let's call this \(\mathrm{x}\) hours) and \(\$12\) per hour for the remaining hours (let's call this \(\mathrm{y}\) hours).
The key constraints we need to remember are:
- Kim worked exactly 30 hours total, so \(\mathrm{x} + \mathrm{y} = 30\)
- Both \(\mathrm{x}\) and \(\mathrm{y}\) must be whole numbers (integers) - you can't work a fraction of an hour
- Both \(\mathrm{x}\) and \(\mathrm{y}\) must be non-negative (Kim can't work negative hours)
- Kim's total pay = (hours at \(\$8/\mathrm{hour}\)) × \(\$8\) + (hours at \(\$12/\mathrm{hour}\)) × \(\$12\) = \(8\mathrm{x} + 12\mathrm{y}\)
Process Skill: TRANSLATE - Converting the word problem into mathematical relationships
2. Set up the constraint equations
Now let's write down our mathematical relationships clearly:
- Hours constraint: \(\mathrm{x} + \mathrm{y} = 30\)
- Pay equation: Total pay = \(8\mathrm{x} + 12\mathrm{y}\)
- Integer constraints: \(\mathrm{x}\) and \(\mathrm{y}\) must both be whole numbers ≥ 0
Since \(\mathrm{x} + \mathrm{y} = 30\), we can substitute \(\mathrm{y} = 30 - \mathrm{x}\) into our pay equation:
Total pay = \(8\mathrm{x} + 12(30 - \mathrm{x}) = 8\mathrm{x} + 360 - 12\mathrm{x} = 360 - 4\mathrm{x}\)
This is a much simpler formula: Kim's total pay equals \(360 - 4\mathrm{x}\), where \(\mathrm{x}\) is the number of hours she worked at the lower \(\$8/\mathrm{hour}\) rate.
3. Determine the range of possible outcomes
Let's think about the extreme cases to understand what total pay amounts are even possible:
Minimum pay scenario: If Kim worked all 30 hours at the lower rate (\(\$8/\mathrm{hour}\))
- This means \(\mathrm{x} = 30\), \(\mathrm{y} = 0\)
- Total pay = \(360 - 4(30) = 360 - 120 = \$240\)
Maximum pay scenario: If Kim worked all 30 hours at the higher rate (\(\$12/\mathrm{hour}\))
- This means \(\mathrm{x} = 0\), \(\mathrm{y} = 30\)
- Total pay = \(360 - 4(0) = \$360\)
So Kim's total pay must be between \(\$240\) and \(\$360\). Looking at our answer choices, this immediately eliminates choice E (\(\$400\)) since it's above the maximum possible.
Also, since Total pay = \(360 - 4\mathrm{x}\) and \(\mathrm{x}\) must be a whole number, Kim's total pay must be of the form 360 minus a multiple of 4.
Process Skill: APPLY CONSTRAINTS - Using the integer requirement to limit possible answers
4. Test answer choices against integer constraints
Let's check each remaining answer choice to see if it gives us integer values for \(\mathrm{x}\) and \(\mathrm{y}\):
For each choice, we'll use Total pay = \(360 - 4\mathrm{x}\) to find \(\mathrm{x}\), then check if \(\mathrm{x}\) is a non-negative integer ≤ 30.
Choice A: \(\$200\)
\(200 = 360 - 4\mathrm{x} \rightarrow 4\mathrm{x} = 160 \rightarrow \mathrm{x} = 40\)
But \(\mathrm{x} = 40\) means Kim worked 40 hours at \(\$8/\mathrm{hour}\), which is impossible since she only worked 30 hours total. ❌
Choice B: \(\$280\)
\(280 = 360 - 4\mathrm{x} \rightarrow 4\mathrm{x} = 80 \rightarrow \mathrm{x} = 20\)
If \(\mathrm{x} = 20\), then \(\mathrm{y} = 30 - 20 = 10\)
Check: 20 hours at \(\$8/\mathrm{hour}\) + 10 hours at \(\$12/\mathrm{hour}\) = \(\$160 + \$120 = \$280\) ✓
Both \(\mathrm{x} = 20\) and \(\mathrm{y} = 10\) are valid integers. ✓
Choice C: \(\$325\)
\(325 = 360 - 4\mathrm{x} \rightarrow 4\mathrm{x} = 35 \rightarrow \mathrm{x} = 8.75\)
Since \(\mathrm{x} = 8.75\) is not an integer, this is impossible. ❌
Choice D: \(\$350\)
\(350 = 360 - 4\mathrm{x} \rightarrow 4\mathrm{x} = 10 \rightarrow \mathrm{x} = 2.5\)
Since \(\mathrm{x} = 2.5\) is not an integer, this is impossible. ❌
Process Skill: CONSIDER ALL CASES - Systematically checking each answer choice
4. Final Answer
Only choice B (\(\$280\)) satisfies all our constraints. This corresponds to Kim working 20 hours at \(\$8/\mathrm{hour}\) and 10 hours at \(\$12/\mathrm{hour}\), giving her a total pay of \(\$280\).
The answer is B.
Common Faltering Points
Errors while devising the approach
- Misunderstanding the constraint relationships: Students may fail to recognize that \(\mathrm{x} + \mathrm{y} = 30\) is a crucial constraint, or they might set up the problem thinking that \(\mathrm{x}\) and \(\mathrm{y}\) are independent variables rather than interdependent ones where knowing one determines the other.
- Overlooking the integer requirement: Students often miss that "\(\mathrm{x}\) and \(\mathrm{y}\) are integers" is a critical constraint that eliminates many potential answers. They may approach this as a continuous optimization problem rather than recognizing it as a discrete constraint problem.
- Not recognizing the substitution opportunity: Students might try to solve this by testing random combinations of \(\mathrm{x}\) and \(\mathrm{y}\) values instead of substituting \(\mathrm{y} = 30 - \mathrm{x}\) into the pay equation to get the simplified form Total pay = \(360 - 4\mathrm{x}\).
Errors while executing the approach
- Arithmetic errors in substitution: When simplifying \(8\mathrm{x} + 12(30 - \mathrm{x})\), students commonly make errors like getting \(8\mathrm{x} + 360 + 12\mathrm{x} = 360 + 20\mathrm{x}\) instead of the correct \(8\mathrm{x} + 360 - 12\mathrm{x} = 360 - 4\mathrm{x}\).
- Incorrect boundary calculations: Students may miscalculate the minimum and maximum possible pay values. For example, they might think the minimum is when \(\mathrm{x} = 0\) instead of \(\mathrm{x} = 30\), confusing which rate gives the lower total pay.
- Sign errors when solving for x: When solving equations like \(280 = 360 - 4\mathrm{x}\), students often make sign errors, getting \(4\mathrm{x} = 360 + 280\) instead of \(4\mathrm{x} = 360 - 280\), leading to incorrect values for \(\mathrm{x}\).
Errors while selecting the answer
- Choosing the first valid-looking answer: Students might find that choice B works and immediately select it without checking if other choices could also be valid, missing the systematic verification process.
- Misinterpreting fractional results: When testing answer choices that yield non-integer values for \(\mathrm{x}\) (like \(\mathrm{x} = 8.75\) for choice C), students might round these to the nearest integer and incorrectly conclude that the answer choice is valid.
