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In working out a family budget, a couple estimates the amounts \(\mathrm{F}\) and \(\mathrm{T}\) for the family's annual food expenditure and tax liability, respectively. They assume that the maximum errors in these estimates are 10 percent and 15 percent, respectively. Based on these assumptions, which of the following is the maximum percent error in the estimate for the sum of these two budget amounts?
Let's start by understanding what this problem is really asking in everyday terms.
Imagine you're planning your family budget and you estimate that you'll spend \$5,000 on food (F) and \$3,000 on taxes (T) this year. But you know your estimates might be wrong. You think your food estimate could be off by up to \(10\%\), and your tax estimate could be off by up to \(15\%\).
The question is asking: "What's the maximum percentage by which your combined estimate of \$8,000 (food + taxes) could be wrong?"
In mathematical terms, we need to find the maximum percent error for the sum \((\mathrm{F} + \mathrm{T})\) when:
Percent error tells us how much our estimate could be off compared to the true value, expressed as a percentage.
Process Skill: TRANSLATE - Converting the budget scenario into a mathematical percent error problem
To find the maximum error, we need to think about when things would go most wrong.
Using our example: if your food estimate is wrong, it could be wrong in two ways - you could overestimate or underestimate. Same with taxes. But when would the total error be biggest?
The total error is biggest when both estimates are wrong in the same direction by their maximum amounts. For instance:
The same logic applies if both are underestimated by their maximum amounts.
In general terms: for maximum error in the sum \((\mathrm{F} + \mathrm{T})\), both individual errors should point in the same direction and be at their maximum values.
Process Skill: CONSIDER ALL CASES - Identifying that maximum error occurs when individual errors compound rather than cancel
Now let's calculate exactly how big this maximum error can be.
If F can be wrong by up to \(10\%\), then the maximum error in F is \(0.1\mathrm{F}\)
If T can be wrong by up to \(15\%\), then the maximum error in T is \(0.15\mathrm{T}\)
When both errors work in the same direction (both too high or both too low), they add up:
Maximum absolute error in the sum = \(0.1\mathrm{F} + 0.15\mathrm{T}\)
Using our numbers: Maximum error = \(0.1(\$5,000) + 0.15(\$3,000) = \$500 + \$450 = \$950\)
This matches what we calculated before!
We now know the maximum absolute error is \((0.1\mathrm{F} + 0.15\mathrm{T})\). But the question asks for the maximum percent error.
Percent error is calculated as:
Percent Error = \(\frac{\text{Absolute Error}}{\text{True Value}} \times 100\%\)
The true value of our sum is \((\mathrm{F} + \mathrm{T})\), so:
Maximum Percent Error = \(\frac{(0.1\mathrm{F} + 0.15\mathrm{T})}{(\mathrm{F} + \mathrm{T})} \times 100\%\)
Using our example:
Maximum Percent Error = \(\frac{\$950}{\$8,000} \times 100\% = 11.875\%\)
The mathematical expression for any values of F and T is:
\(\frac{(0.1\mathrm{F} + 0.15\mathrm{T})}{(\mathrm{F} + \mathrm{T})} \times 100\%\)
Looking at our answer choices, option D matches exactly: \(\frac{(0.1\mathrm{F} + 0.15\mathrm{T})}{(\mathrm{F} + \mathrm{T})} \times 100\%\)
This makes sense because:
The answer is D.
Faltering Point 1: Misunderstanding what "maximum error" means
Students often confuse maximum error with average error or think it means the errors will always be at their maximum values. They might try to find some kind of weighted average of \(10\%\) and \(15\%\) (like \(12.5\%\)) instead of recognizing that maximum error occurs when both individual errors are at their peak values AND working in the same direction. This leads them to incorrectly select answer choice A.
Faltering Point 2: Not recognizing that errors can compound
Many students assume that when you have two estimates with different error rates, the overall error should somehow be between those rates or be their simple average. They fail to understand that to find the maximum error in a sum, you need to consider the worst-case scenario where both estimates are wrong by their maximum amounts in the same direction, causing the errors to add up rather than cancel out.
Faltering Point 3: Confusing absolute error with percent error
Students might correctly identify that the maximum absolute error is \((0.1\mathrm{F} + 0.15\mathrm{T})\) but then think this IS the answer, leading them to select option B. They don't take the crucial next step of converting this absolute error into a percentage of the total estimated amount \((\mathrm{F} + \mathrm{T})\).
Faltering Point 1: Incorrectly calculating the error amounts
Students might write down \(1.1\mathrm{F}\) and \(1.15\mathrm{T}\) instead of \(0.1\mathrm{F}\) and \(0.15\mathrm{T}\) when calculating the error amounts. They confuse the total estimated amount (which would be \(1.1\mathrm{F}\) if overestimated by \(10\%\)) with the actual error amount (which is \(0.1\mathrm{F}\)). This leads them toward answer choices C or E.
Faltering Point 2: Using wrong base for percentage calculation
Even if students correctly find the absolute error as \((0.1\mathrm{F} + 0.15\mathrm{T})\), they might use the wrong denominator when converting to a percentage. Some might use just F, or just T, or some other combination instead of the correct base \((\mathrm{F} + \mathrm{T})\), which represents the true total value.
Faltering Point 1: Selecting the absolute error instead of percent error
After correctly calculating that the maximum absolute error is \((0.1\mathrm{F} + 0.15\mathrm{T})\), students might immediately select answer choice B without realizing that the question specifically asks for the "maximum percent error," which requires dividing by the total \((\mathrm{F} + \mathrm{T})\) and expressing as a percentage.
Faltering Point 2: Getting confused by the mathematical notation in answer choices
Students who have done the math correctly might get intimidated by the fraction format in options D and E, or misread the expressions. They might select option E because it "looks more complete" with the larger numerator values \((1.1\mathrm{F} + 1.15\mathrm{T})\), not recognizing that this represents the total estimated amounts rather than the error amounts.
Step 1: Choose convenient smart numbers
Let's assign specific values to F and T that make calculations straightforward:
These numbers are chosen because they're round numbers that make percentage calculations easy to follow.
Step 2: Calculate maximum errors for each estimate
Step 3: Find the true sum and maximum error in the sum
Note: We add the errors because for maximum error, both estimates would be off in the same direction.
Step 4: Calculate maximum percent error
Maximum percent error = \(\frac{\text{Maximum absolute error}}{\text{True value}} \times 100\%\)
= \(\frac{\$40}{\$300} \times 100\% = \frac{4}{30} \times 100\% = 13.33\%\)
Step 5: Verify using the formula approach
Using our smart numbers in choice D: \(\frac{0.1\mathrm{F} + 0.15\mathrm{T}}{\mathrm{F} + \mathrm{T}} \times 100\%\)
= \(\frac{0.1(100) + 0.15(200)}{100 + 200} \times 100\%\)
= \(\frac{10 + 30}{300} \times 100\% = \frac{40}{300} \times 100\% = 13.33\%\) ✓
This confirms that choice D gives us the correct formula for maximum percent error.