A marketing firm determined that, of 200 households surveyed, 80 used neither Brand A nor Brand B soap, 60 used...

1. Translate the problem requirements: Convert the word problem into clear mathematical relationships. We have 200 total households with specific usage patterns for Brand A and Brand B soap, and need to find how many use both brands.
2. Set up variables for the unknown quantities: Define variables for households using both brands and those using only Brand B, using the given 3:1 ratio relationship.
3. Apply the total constraint: Use the fact that all household categories must sum to 200 to create an equation and solve for the number using both brands.
4. Verify the solution: Check that our answer makes sense by ensuring all categories add up to 200 households.

Execution of Strategic Approach

1. Translate the problem requirements

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:

• 80 households used neither Brand A nor Brand B soap
• 60 households used only Brand A soap (not Brand B)
• Some households used both brands (this is what we need to find)
• Some households used only Brand B soap (not Brand A)

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

2. Set up variables for the unknown quantities

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:

• Neither brand: 80 households
• Only Brand A: 60 households
• Only Brand B: \(3\mathrm{x}\) households
• Both brands: \(\mathrm{x}\) households

3. Apply the total constraint

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

4. Verify the solution

Let's check our answer by making sure all categories add up to 200:

• Neither brand: 80 households
• Only Brand A: 60 households
• Only Brand B: \(3(15) = 45\) households
• Both brands: 15 households

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." ✓

Final Answer

The answer is A. 15 households used both brands of soap.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the ratio relationship

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.

2. Incorrectly categorizing overlap groups

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.

3. Failing to recognize the constraint equation

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.

Errors while executing the approach

1. Algebraic manipulation errors

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 \(\mathrm{x}\).

2. Substitution errors with the ratio

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.

Errors while selecting the answer

1. Selecting the wrong calculated value

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.

Alternate Solutions

Smart Numbers Approach

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.