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. $45$

Answer 1

The correct option is A $15$

Since it is given that $80$ households use neither Brand A nor Brand B, then $200-80=120$ must use Brand A, Brand B, or both. It is also given that $60$ households use only Brand A and that three times as many households use Brand B exclusively as use both brands. If $x$ is the number of households that use both Brand A and Brand B, then $3x$ use Brand B alone. A Venn diagram can be helpful for visualizing the logic of the given information for this item:

All the sections in the circles can be added up and set equal to $120$, and then the equation can be solved for $x$:

$60+x+3x=120$

$60+4x=120$ combine like terms

$4x=60$ subtract $60$ from both sides

$x=15$$dividebothsidesby$4$

The correct answer is A.