Question

A market survey showed that 76 percent of the visitors at a certain resort came from Pacific or southwestern states. Of these, \(\frac{2}{3}\) were from California, and 87 percent of the Californians were from southern California. Approximately what percent of the visitors at the resort were from southern California?

• A. 40%
• B. 45%
• C. 50%
• D. 55%
• E. 65%

Hint:

When multiplying decimals and fractions, it's often easier to convert everything to fractions to look for simplifications (like 87/3) or convert everything to decimals and estimate. For example, \(2/3 \approx 0.67\), so the calculation is roughly \(0.76 \times 0.67 \times 0.87\), which you can estimate as \(0.75 \times 0.67 \times 0.87 \approx (3/4) \times (2/3) \times 0.87 = (1/2) \times 0.87 = 0.435\), or 43.5%. This points directly to 45% as the best answer.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This is a nested percentage problem. We need to find a percentage of a fraction of another percentage of a total. We can find the final percentage by multiplying the successive proportions.
Step 2: Detailed Explanation:
Let's find the fraction of total visitors who were from southern California by multiplying the given proportions.
\begin{itemize} \item Proportion from Pacific/SW states = 76% = 0.76 \item Proportion of those who were from California = \(\frac{2}{3}\) \item Proportion of those Californians from southern California = 87% = 0.87 \end{itemize} To find the proportion of total visitors from southern California, we multiply these values together:
\[ \text{Proportion} = 0.76 \times \frac{2}{3} \times 0.87 \]
Let's perform the multiplication:
\[ = \frac{76}{100} \times \frac{2}{3} \times \frac{87}{100} \]
We can simplify before multiplying. Notice that 87 is divisible by 3 (\(8+7=15\)).
\[ \frac{87}{3} = 29 \]
So the expression becomes:
\[ = \frac{76}{100} \times 2 \times \frac{29}{100} \]
\[ = \frac{76 \times 2 \times 29}{10000} \]
\[ = \frac{152 \times 29}{10000} \]
Now, calculate \(152 \times 29\):
\[ 152 \times 29 = 152 \times (30 - 1) = (152 \times 30) - 152 = 4560 - 152 = 4408 \]
So the proportion is:
\[ \frac{4408}{10000} = 0.4408 \]
To express this as a percentage, we multiply by 100:
\[ 0.4408 \times 100% = 44.08% \]
The question asks for the approximate percent. The value 44.08% is closest to 45%.
Step 3: Final Answer:
Approximately 45% of the visitors at the resort were from southern California.