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In May Mrs Lee's earnings were \(60\%\) of the Lee family's total income. In June Mrs Lee earned \(20\%\) more than in May. If the rest of the family's income was the same both months, then, in June, Mrs Lee's earnings were approximately what percent of the Lee family's total income?
Let's break down what's happening in plain English first. Mrs. Lee earned 60% of her family's total income in May. This means the rest of the family earned the remaining 40%. In June, Mrs. Lee got a raise - she earned 20% more than she did in May. Everyone else in the family earned exactly the same as they did in May. We need to figure out what percentage of the total family income Mrs. Lee's new earnings represent in June.
The key insight here is that when Mrs. Lee's income goes up but everyone else's stays the same, both her share of the total AND the total itself will change.
Process Skill: TRANSLATE - Converting the percentage relationships into a clear mathematical understanding
To make this easier to work with, let's use concrete numbers instead of trying to work with variables and fractions. Since Mrs. Lee earned 60% of the total in May, let's say the family's total income in May was $100. This makes the math clean and easy to follow.
With a $100 total in May:
• Mrs. Lee earned: \(60\% \text{ of } \$100 = \$60\)
• Rest of family earned: \(40\% \text{ of } \$100 = \$40\)
These numbers are much easier to work with than abstract percentages, and we can always convert back to percentages at the end.
Now let's figure out Mrs. Lee's June earnings. She earned 20% more in June than she did in May.
Mrs. Lee's May earnings: $60
Her 20% increase: \(20\% \text{ of } \$60 = 0.20 \times \$60 = \$12\)
Mrs. Lee's June earnings: \(\$60 + \$12 = \$72\)
Alternatively, we could think of this as: Mrs. Lee earned 120% of her May income in June, which is \(1.20 \times \$60 = \$72\).
Now we need to find the new family total for June and calculate what percentage Mrs. Lee's earnings represent.
June family income breakdown:
• Mrs. Lee: $72 (increased from May)
• Rest of family: $40 (same as May)
• Total family income in June: \(\$72 + \$40 = \$112\)
Mrs. Lee's percentage of family income in June:
\(\$72 \div \$112 = 0.6429... = 64.29\%\)
Since we need an approximate answer, this rounds to 64%.
Process Skill: SIMPLIFY - Using round numbers ($100 total) made all calculations straightforward without complex fractions
Mrs. Lee's earnings in June were approximately 64% of the Lee family's total income.
The answer is (A) 64%.
To verify: We can see this makes sense because even though Mrs. Lee got a 20% raise, her percentage of the family income only went from 60% to 64%. This is because the family's total income also increased when her income went up, so her share didn't increase by the full 20%.
1. Misunderstanding what stays constant vs. what changes: Students often get confused about which income components change and which remain the same. They might incorrectly assume that Mrs. Lee's percentage simply increases by 20% (from 60% to 80%) without recognizing that the rest of the family's income stays constant while the total family income changes.
2. Failing to recognize that the total family income changes: A common conceptual error is assuming the total family income remains $100 in June. Students might think that if Mrs. Lee earns 20% more, this comes from redistributing the existing family income rather than understanding that her raise increases the overall family total.
3. Misinterpreting "20% more than in May": Some students might incorrectly interpret this as Mrs. Lee earning 20% of the total family income in June, rather than understanding it means her individual earnings increased by 20% from her May amount.
1. Arithmetic errors in percentage calculations: When calculating 20% of $60, students might make basic computational mistakes, getting $10 instead of $12, or when finding 120% of $60, they might calculate it as \(\$60 + \$20 = \$80\) instead of the correct $72.
2. Incorrect division for final percentage: In the final step of dividing $72 by $112, students often make calculation errors. They might get confused with the decimal conversion or incorrectly calculate \(72 \div 112\), potentially arriving at values like 0.54 or 0.72 instead of the correct 0.6429.
1. Selecting the wrong approximation: Students who correctly calculate 64.29% might choose answer choice (B) 68% thinking it's closer, or they might round incorrectly to 65% and feel uncertain between choices (A) 64% and (B) 68%, not recognizing that 64% is the appropriate approximation.
2. Choosing the intuitive but incorrect 80%: Students who made conceptual errors in the approach phase might arrive at 80% (thinking 60% + 20% = 80%) and select choice (E), not realizing this completely ignores the changing total family income.
This problem is well-suited for the smart numbers method because we're dealing with percentage relationships and can choose convenient values that make calculations straightforward.
Since Mrs. Lee earned 60% of the family's total income in May, let's choose a total family income that makes this percentage easy to work with. Let's set the May total family income to $100.
This means:
In June, Mrs. Lee earned 20% more than in May:
Mrs. Lee's June earnings = \(\$60 + (20\% \text{ of } \$60) = \$60 + \$12 = \$72\)
The rest of the family's income remained the same at $40.
June total family income = Mrs. Lee's June earnings + Rest of family's income
June total family income = \(\$72 + \$40 = \$112\)
Mrs. Lee's percentage in June = \(\frac{\text{Mrs. Lee's June earnings}}{\text{June total family income}} \times 100\%\)
Mrs. Lee's percentage in June = \(\frac{\$72}{\$112} \times 100\% = 0.643 \times 100\% \approx 64\%\)
Answer: (A) 64%
The smart numbers approach makes this problem much more concrete and easier to follow than working with variables, while still maintaining the same mathematical relationships.