## How to Evaluate Decimal Distances on Maps

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Budding cartographers and number crunchers! Today's expedition takes us to the intriguing terrain of maps and decimals. We'll specifically be exploring how to calculate decimal distances on maps. This is a crucial skill in both math and geography, enabling us to decode real-world distances from map scales. So, let's embark on this fascinating journey!

### Interpreting Decimal Distances on Maps

When navigating maps, scale is our compass. It’s a ratio that provides a comparison between a distance on the map to the real distance on the ground. As an example, a map scale could be \(1:50,000\), signifying that 1 unit (like an inch or centimeter) on the map equates to \(50,000\) of the same units on earth.

Often, map distances are represented as decimals. For instance, \(0.3 \, \text{cm}\) could stand for \(3 \, \text{km}\) on the ground, dictated by the map’s scale.

### Guiding Steps to Evaluate Decimal Distances on Maps

**Step 1: Decode the Scale**

Before any distance calculations, the scale of the map needs to be understood. It's typically denoted somewhere on the map.

**Step 2: Spot the Points**

Identify the two points whose distance you want to calculate.

**Step 3: Measure the Distance**

Use a ruler to measure the distance between the two points on the map. Record this distance with precision.

**Step 4: Leverage the Scale**

Multiply the map distance you measured by the map's scale to ascertain the real-world distance.

**Step 5: Verify Your Answer**

Ensure your answer makes sense in the real world. If the calculated distance seems unusually short or long, revisit your measurements and calculations.

#### Example:

Let's suppose that you have a map with a scale of \(1:100,000\). This means that 1 cm on the map is equivalent to 100,000 cm (or 1 km) in real life.

Now, you want to know the real distance between two cities on the map that are 3.5 cm apart. Here's how you would calculate it:

**Step 1: Decode the Scale**

The scale of the map is \(1:100,000\), which means 1 cm on the map equals 1 km in real life.

**Step 2: Spot the Points**

You have identified the two cities on the map.

**Step 3: Measure the Distance**

Using a ruler, you find that the cities are \(3.5 \, \text{cm}\) apart on the map.

**Step 4: Leverage the Scale**

Multiply the map distance by the scale to get the real-world distance. So, \(3.5 \, \text{cm} \times 1 \, \text{km/cm} = 3.5 \, \text{km}\).

**Step 5: Verify Your Answer**

The calculated distance of \(3.5 \, \text{km}\) seems reasonable, so you can be confident in your calculation!

So, using the map's scale and measuring the distance on the map, you've found that the two cities are 3.5 km apart in real life.

### Exercises for Evaluate Decimal Distances on the Map

1) If a map has a scale of 1:50000 and two cities are 1.2 cm apart on the map, what is the real-world distance between the cities?

2) On a map with a scale of 1:25000, two landmarks are 4.8 cm apart. What is the actual distance between the landmarks?

3) A river is shown as 3.5 cm long on a map with a scale of 1:100000. What is the real length of the river?

4) On a map with a scale of 1:20000, a trail is shown as 6.3 cm long. How long is the trail in real life?

5) If a map has a scale of 1:75000 and two towns are 2.4 cm apart on the map, what is the actual distance between the towns?

6) On a map with a scale of 1:50000, a road is shown as 5.6 cm long. What is the actual length of the road?

7) If a map has a scale of 1:40000 and two villages are 1.8 cm apart on the map, what is the real-world distance between the villages?

8) On a map with a scale of 1:60000, a lake is shown as 2.7 cm wide. What is the actual width of the lake?

9) If a map has a scale of 1:50000 and a park is shown as 3.9 cm wide on the map, what is the actual width of the park?

1)

\[ \begin{align*} \text{Map Scale} &= 1:50000 \\ \text{Map Distance} &= 1.2 \, \text{cm} \\ \text{Real Distance} &= 1.2 \, \text{cm} \times 50000 \\ &= 60000 \, \text{cm} \\ &= 600 \, \text{m} \\ &= 0.6 \, \text{km} \end{align*} \]2)

\[ \begin{align*} \text{Map Scale} &= 1:25000 \\ \text{Map Distance} &= 4.8 \, \text{cm} \\ \text{Real Distance} &= 4.8 \, \text{cm} \times 25000 \\ &= 120000 \, \text{cm} \\ &= 1200 \, \text{m} \\ &= 1.2 \, \text{km} \end{align*} \]3)

\[ \begin{align*} \text{Map Scale} &= 1:100000 \\ \text{Map Distance} &= 3.5 \, \text{cm} \\ \text{Real Distance} &= 3.5 \, \text{cm} \times 100000 \\ &= 350000 \, \text{cm} \\ &= 3500 \, \text{m} \\ &= 3.5 \, \text{km} \end{align*} \]4)

\[ \begin{align*} \text{Map Scale} &= 1:20000 \\ \text{Map Distance} &= 6.3 \, \text{cm} \\ \text{Real Distance} &= 6.3 \, \text{cm} \times 20000 \\ &= 126000 \, \text{cm} \\ &= 1260 \, \text{m} \\ &= 1.26 \, \text{km} \end{align*} \]5)

\[ \begin{align*} \text{Map Scale} &= 1:75000 \\ \text{Map Distance} &= 2.4 \, \text{cm} \\ \text{Real Distance} &= 2.4 \, \text{cm} \times 75000 \\ &= 180000 \, \text{cm} \\ &= 1800 \, \text{m} \\ &= 1.8 \, \text{km} \end{align*} \]6)

\[ \begin{align*} \text{Map Scale} &= 1:50000 \\ \text{Map Distance} &= 5.6 \, \text{cm} \\ \text{Real Distance} &= 5.6 \, \text{cm} \times 50000 \\ &= 280000 \, \text{cm} \\ &= 2800 \, \text{m} \\ &= 2.8 \, \text{km} \end{align*} \]7)

\[ \begin{align*} \text{Map Scale} &= 1:40000 \\ \text{Map Distance} &= 1.8 \, \text{cm} \\ \text{Real Distance} &= 1.8 \, \text{cm} \times 40000 \\ &= 72000 \, \text{cm} \\ &= 720 \, \text{m} \\ &= 0.72 \, \text{km} \end{align*} \]8)

\[ \begin{align*} \text{Map Scale} &= 1:60000 \\ \text{Map Distance} &= 2.7 \, \text{cm} \\ \text{Real Distance} &= 2.7 \, \text{cm} \times 60000 \\ &= 162000 \, \text{cm} \\ &= 1620 \, \text{m} \\ &= 1.62 \, \text{km} \end{align*} \]9)

\[ \begin{align*} \text{Map Scale} &= 1:50000 \\ \text{Map Distance} &= 3.9 \, \text{cm} \\ \text{Real Distance} &= 3.9 \, \text{cm} \times 50000 \\ &= 195000 \, \text{cm} \\ &= 1950 \, \text{m} \\ &= 1.95 \, \text{km} \end{align*} \]