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A marketing firm determined that, of 200 households surveyed, 80 used neither Brand A nor Brand B soap, 60 used only Brand A soap, and for every household that used both brands of soap, 3 used only Brand B soap. How many of the 200 households surveyed used both brands of soap?
Let's break down what we know in everyday language first. We have 200 households total that were surveyed about their soap usage.
From the problem, we can identify these groups:
The key relationship given is: "for every household that used both brands of soap, 3 used only Brand B soap." This means if we have some number of households using both brands, then we have exactly 3 times that number using only Brand B.
Process Skill: TRANSLATE - Converting the word problem into clear mathematical relationships is crucial here
Now let's define our unknown quantities using simple variables.
Let's say \(\mathrm{x}\) = number of households that used both Brand A and Brand B soap.
From the given relationship, if \(\mathrm{x}\) households use both brands, then \(3\mathrm{x}\) households use only Brand B soap.
So our four categories are:
Since we surveyed exactly 200 households total, all our categories must add up to 200.
In plain English: Neither + Only A + Only B + Both = Total households
Substituting our values:
\(80 + 60 + 3\mathrm{x} + \mathrm{x} = 200\)
Simplifying the left side:
\(140 + 4\mathrm{x} = 200\)
Solving for \(\mathrm{x}\):
\(4\mathrm{x} = 200 - 140\)
\(4\mathrm{x} = 60\)
\(\mathrm{x} = 15\)
Therefore, 15 households used both brands of soap.
Process Skill: APPLY CONSTRAINTS - Using the total count constraint is essential to solve this problem
Let's check our answer by making sure all categories add up to 200:
Total: \(80 + 60 + 45 + 15 = 200\) ✓
Also, let's verify the ratio relationship: We have 15 households using both brands and 45 households using only Brand B. The ratio is \(45:15 = 3:1\), which matches "for every household that used both brands, 3 used only Brand B." ✓
The answer is A. 15 households used both brands of soap.
Students often misread "for every household that used both brands of soap, 3 used only Brand B soap" as meaning the ratio of both brands to only Brand A is \(1:3\), rather than correctly understanding it as the ratio of both brands to only Brand B is \(1:3\). This leads to setting up the wrong variables and relationships from the start.
Students may confuse "used only Brand A" with "used Brand A" (which would include households using both brands). The word "only" is crucial - it means Brand A but NOT Brand B. Missing this distinction leads to double-counting households and incorrect variable setup.
Some students focus so much on the ratio relationship that they forget the fundamental constraint that all categories must sum to the total of 200 households. Without recognizing that Neither + Only A + Only B + Both = 200, they cannot create the equation needed to solve for the unknown.
When combining like terms in the equation \(80 + 60 + 3\mathrm{x} + \mathrm{x} = 200\), students may incorrectly add \(3\mathrm{x} + \mathrm{x}\) as \(3\mathrm{x}\) instead of \(4\mathrm{x}\), or make arithmetic errors when calculating \(200 - 140 = 60\), leading to the wrong value for \(\mathrm{x}\).
Even if students correctly identify that there should be \(3\mathrm{x}\) households using only Brand B, they might substitute this incorrectly in their equation setup, perhaps writing \(3 + \mathrm{x}\) instead of \(3\mathrm{x}\), or placing the \(3\mathrm{x}\) in the wrong category.
After finding \(\mathrm{x} = 15\), students might mistakenly select 45 (which represents the number of households using only Brand B) instead of 15 (the number using both brands). This happens because they calculate both values during verification and confuse which one answers the original question.
Instead of setting up algebraic equations, we can strategically test convenient numbers that satisfy the given \(3:1\) ratio relationship.
Step 1: Choose a convenient smart number
Since "for every household that used both brands, 3 used only Brand B," let's try 15 households using both brands (chosen because it's a reasonable size and will give us a clean multiple of 3).
Step 2: Calculate corresponding values
If 15 households use both brands, then:
Households using only Brand B = \(3 \times 15 = 45\)
Step 3: Check if our smart number works
Let's verify this adds up to 200 total households:
• Used neither brand: 80
• Used only Brand A: 60
• Used only Brand B: 45
• Used both brands: 15
• Total: \(80 + 60 + 45 + 15 = 200\) ✓
Step 4: Verify the ratio constraint
Ratio check: \(45 \div 15 = 3\), which matches "for every household that used both brands, 3 used only Brand B" ✓
Answer: 15 households used both brands of soap
Note: The smart number 15 was chosen because it's small enough to be reasonable for a subset of 200 households, creates a clean \(3:1\) ratio (\(45:15\)), and when we test it, all constraints are perfectly satisfied.