On a map, 1 inch represents 150 miles. What is the actual distance between two cities if they are `3 1/2` inches apart on the map ?

Options

A. 225
B. 300
C. 450
D. 525

Answer

To solve the problem, we need to determine the actual distance between two cities based on the distance represented on a map. Let's break it down step by step:

### Step 1: Understand the given information
We know that on the map, 1 inch represents 150 miles. We also know that the distance between the two cities on the map is \(3 \frac{1}{2}\) inches.

### Step 2: Convert the mixed fraction to an improper fraction
To convert the mixed fraction \(3 \frac{1}{2}\) inches into an improper fraction, we can use the formula:

\[ \text{Improper fraction} = a \times c + b \div c \]

Where \(a\) is the whole number, \(b\) is the numerator, and \(c\) is the denominator. Here, \(a = 3\), \(b = 1\), and \(c = 2\).

Calculating it:

\[ 3 \frac{1}{2} = \frac{3 \times 2 + 1}{2} = \frac{6 + 1}{2} = \frac{7}{2} \text{ inches} \]

### Step 3: Use the unitary method to find the actual distance
Now that we have the distance on the map in improper fraction form, we can use the unitary method to find the actual distance.

Since 1 inch represents 150 miles, we can find the distance for \( \frac{7}{2} \) inches:

\[ \text{Actual distance} = \left(\frac{7}{2} \text{ inches}\right) \times 150 \text{ miles/inch} \]

Calculating this:

\[ \text{Actual distance} = \frac{7 \times 150}{2} = \frac{1050}{2} = 525 \text{ miles} \]

### Final Answer
The actual distance between the two cities is **525 miles**.

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