Question

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?

• A. 15
• B. 20
• C. 30
• D. 40
• E. 45

Hint:

For survey problems, drawing a Venn diagram can be very helpful to visualize the different groups. Alternatively, setting up an algebraic equation based on the sum of the distinct parts is a quick and reliable method.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This problem involves analyzing survey data using set theory. The total number of surveyed households is composed of four distinct, non-overlapping groups: those who use only Brand A, those who use only Brand B, those who use both, and those who use neither.
Step 2: Key Formula or Approach:
Let's define the groups with variables.
Let \(x\) be the number of households that used both brands of soap.
From the problem, for every household that used both, 3 used only Brand B. So, the number of households that used only Brand B is \(3x\).
The total number of households is the sum of the four groups: \[ \text{Total} = (\text{Only A}) + (\text{Only B}) + (\text{Both}) + (\text{Neither}) \] Step 3: Detailed Explanation:
We are given the following information:
Total households = 200
Used only Brand A = 60
Used neither Brand A nor Brand B = 80
Used both brands = \(x\)
Used only Brand B = \(3x\)
Now, we can set up an equation using the formula from Step 2: \[ 200 = 60 + 3x + x + 80 \] First, combine the constant terms and the \(x\) terms: \[ 200 = 140 + 4x \] Next, isolate the term with \(x\) by subtracting 140 from both sides: \[ 200 - 140 = 4x \] \[ 60 = 4x \] Finally, solve for \(x\) by dividing both sides by 4: \[ x = \frac{60}{4} = 15 \] Step 4: Final Answer:
The value of \(x\) represents the number of households that used both brands of soap. Therefore, 15 households used both brands.