Question

The scale on a map of Ohio says that on the map, 2 inches equals 35 miles. What is the distance between Cleveland and Cincinnati, which are 14 inches apart on the map? A. 140 miles B. 175 miles C. 210 miles D. 245 miles

Short answer

D. 245 miles.

Step-by-step solution

- Understand the Scale

Identify the given scale on the map. The problem states that 2 inches on the map equals 35 miles in real life.

- Set Up the Proportion

Set up a proportion to relate the inches on the map to the actual miles. The proportion will be \(\frac{2 \text{ inches}}{35 \text{ miles}} = \frac{14 \text{ inches}}{x \text{ miles}}\).

- Solve the Proportion

Cross-multiply to solve for \(x\). \(2 \times x = 35 \times 14\). So, \(2x = 490\).

- Find the Distance

Divide both sides by 2 to find the value of \(x\). \(x = \frac{490}{2} = 245\) miles.

- Verify the Answer

Compare the calculated answer with the given choices. 245 miles (Option D) matches the calculated distance.

Key concepts

Proportions

Proportions are mathematical equations that show two ratios are equivalent. This is key in scaling problems, like our map scale problem. In this exercise, we set up a proportion to relate the map distance to the real distance. A common example is \( \frac{2 \text{ inches}}{35 \text{ miles}} = \frac{14 \text{ inches}}{x \text{ miles}} \). Here, 14 inches on the map relate to a real-life distance we need to find. Proportions help maintain consistency between the map and actual distances, making it easy to calculate unknown values.

Distance Calculation

Calculating distance on a map involves understanding the scale, and using it to find real-world measurements. Here, the scale states 2 inches equals 35 miles. To find the distance between Cleveland and Cincinnati, we set up the proportion mentioned earlier. Using proportions, we can solve for the real-world distance easily. Let's break it down in steps: identify the scale, set up a proportion, cross-multiply, and solve for the unknown distance. This method assures that our calculations are accurate and verifiable.

Cross Multiplication

Cross multiplication is a method to solve proportions. It involves multiplying the numerator of one ratio by the denominator of the other ratio. For our exercise, we have the proportion \( \frac{2 \text{ inches}}{35 \text{ miles}} = \frac{14 \text{ inches}}{x \text{ miles}} \). We cross-multiply to get 2 \( \times \) x = 35 \( \times \) 14. This simplifies to 2x = 490. Finally, we solve for x by dividing both sides by 2, resulting in x = 245 miles. Cross multiplication keeps calculations straightforward and manageable.

Map Reading

Map reading is a crucial skill that involves interpreting symbols and scales to determine distances and directions. Understanding the given scale on a map is the first step, as it converts map measurements into real-world distances. In our example, the scale tells us that 2 inches is equal to 35 miles. This allows us to set up a proportion to find the distance between cities. By correctly applying ratios and proportions, we can accurately interpret map data to navigate and understand geographic locations.