Question

In a nationwide poll, P people were asked 2 questions. If \(\frac{2}{3}\) answered "yes" to question 1, and of those \(\frac{1}{5}\) also answered "yes" to question 2, which of the following represents the number of people polled who did not answer "yes" to both questions?

• A. \(\frac{2}{15} P\)
• B. \(\frac{3}{5} P\)
• C. \(\frac{3}{4} P\)
• D. \(\frac{5}{6} P\)
• E. \(\frac{13}{15} P\)

Hint:

To find how many did not answer "yes" to both questions, subtract the number of people who answered "yes" to both from the total number of people polled.

Answer 1

Answer: (E)

Step 1: Number of people who answered "yes" to question 1.
Since \(\frac{2}{3}\) answered "yes" to question 1, the number of people who answered "yes" to question 1 is: \[ \frac{2}{3} P \] Step 2: Number of people who answered "yes" to both questions.
Of those who answered "yes" to question 1, \(\frac{1}{5}\) also answered "yes" to question 2. So, the number of people who answered "yes" to both questions is: \[ \frac{1}{5} \times \frac{2}{3} P = \frac{2}{15} P \] Step 3: Number of people who did not answer "yes" to both questions.
The number of people who did not answer "yes" to both questions is: \[ P - \frac{2}{15} P = \frac{13}{15} P \] Final Answer: \[ \boxed{\frac{13}{15} P} \]