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# 2.7: Distances or Dimensions Given Scale Measurements

Difficulty Level: At Grade Created by: CK-12
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Mr. Jones lives next door to Alex. He designed a plot with the following scale.

1" = 2.5 feet

Mr. Jones drew a plan for his garden showing a square plot with a side length of 4 inches. What is the actual side length of Mr. Jones' garden? What is the area of the plot? What is the perimeter?

In this Concept you will learn about scale and actual measurements. By the end of the Concept, you will know how to figure out the answers to these questions.

### Guidance

Maps represent real places. Every part of the place has been reduced to fit on a single piece of paper. A map is an accurate representation because it uses a scale.

The scale is a ratio that relates the small size of a representation of a place to the real size of a place.

Maps aren’t the only places that we use a scale. Architects use a scale when designing a house. A blueprint shows a small size of what the house will look like compared to the real house. Any time a model is built, it probably uses a scale. The actual building or mountain or landmark can be built small using a scale.

We use units of measurement to create a ratio that is our scale. The ratio compares two things. It compares the small size of the object or place to the actual size of the object or place.

A scale of 1 inch to 1 foot means that 1 inch on paper represents 1 foot in real space. If we were to write a ratio to show this we would write:

1” : 1 ft-this would be our scale.

If the distance between two points on a map is 2 inches, the scale tells us that the actual distance in real space is 2 feet.

We can make scales of any size. One inch can represent 1,000 miles if we want our map to show a very large area, such as a continent. One centimeter might represent 1 meter if the map shows a small space, such as a room.

How can we figure out actual distances or dimensions using a scale?

Let’s start by thinking about distances on a map. On a map, we have a scale that is usually found in the corner. For example, if we have a map of the state of Massachusetts, this could be a possible scale.

Here $\frac{3}{4}$ is equal to 20 miles.

What is the distance from Boston to Framingham?

To work on this problem, we need to use our scale to measure the distance from Boston to Framingham. We can do this by using a ruler. We know that every $\frac{3}{4}$ on the ruler is equal to 20 miles. From Boston to Framingham measures $\frac{3}{4}$ , therefore the distance is 20 miles. If the scale and map were different, we could use the same calculation method. Let’s use another example that just gives us a scale.

If the scale is 1”:500 miles, how far is a city that measures $5\frac{1}{2}$ on a map? We know that every inch is 500 miles. We have $5\frac{1}{2}$ . Let’s start with the 5.

5 $\times$ 500 $=$ 2500 + \frac{1}{2} \times 500 $=$ 2750 miles

By using arithmetic, we were able to figure out the mileage.

Another way to do this is to write two ratios. We can compare the scale with the scale and the distance with the distance.

Here are a few problems for you to try on your own.

#### Example A

If the scale is 1” : 3 miles, how many miles does 5 inches represent?

Solution: 15 miles

#### Example B

If the scale is 2” : 500 meters, how many meters does 4 inches represent?

Solution: 1000 meters

#### Example C

If the scale is 5 ft : 1000 feet, how many feet is 50 feet?

Solution:10,000 feet

Now back to Mr. Jones' garden. Here is the original problem once again.

Mr. Jones lives next door to Alex. He designed a plot with the following scale.

1" = 2.5 feet

Mr. Jones drew a plan for his garden showing a square plot with a side length of 4 inches. What is the actual side length of Mr. Jones' garden? What is the area of the plot? What is the perimeter?

First, we can use the scale to figure out the actual side length of the plot. The side length is the drawing is 4 inches. That is four times the scale.

$2.5 \times 4 = 10$

The actual side length of the plot is 10 feet. The perimeter of a square is four times the side length, so the perimeter of this plot is 40 feet. The area of the square is found by multiplying the side length by the side length, so the area of this plot is 100 square feet.

### Vocabulary

Here are the vocabulary words in this Concept.

Scale
a ratio that compares a small size to a larger actual size. One measurement represents another measurement in a scale.
Ratio
the comparison of two things
Proportion
a pair of equal ratios, we cross multiply to solve a proportion

### Guided Practice

Here is one for you to solve on your own.

If the scale is 2” : 1 ft, what is the actual measurement if a drawing shows the object as 6” long?

We can start by writing a ratio that compares the scale.

$\frac{1 \ ft}{2}=\frac{x \ ft}{6}$ Here we wrote a proportion. We don’t know how big the object really is, so we used a variable to represent the unknown quantity. Notice that we compared the size to the scale in the first ratio and the size to the scale in the second ratio. We can solve this logically using mental math, or we can cross multiply to solve it.

$1 \times 6 &= 6\\2(x) &= 2x\\2x &= 6 \qquad \text{What times two will give us 6?}\\x &= 3 \ ft$

The object is actually 3 feet long.

### Video Review

Here are a few videos for review.

### Practice

Directions: Use the given scale to determine the actual distance.

Given: Scale 1” = 100 miles

1. How many miles is 2” on the map?

2. How many miles is $2\frac{1}{2} inch$ on the map?

3. How many miles is $\frac{1}{4} inch$ on the map?

4. How many miles is 8 inches on the map?

5. How many miles is 16 inches on the map?

6. How many miles is 12 inches on the map?

7. How many miles is $\frac{1}{2} inch$ on the map?

8. How many miles is $5 \frac{1}{4} inches$ on the map?

Given: 1 cm = 20 mi

9. How many miles is 2 cm on the map?

10. How many miles is 4 cm on the map?

11. How many miles is 8 cm on the map?

12. How many miles is 18 cm on the map?

13. How many miles is 11 cm on the map?

14. How many miles is $\frac{1}{2}$ cm on the map?

15. How many miles is $1 \frac{1}{2}$ cm on the map?

16. How many miles is $4 \frac{1}{4}$ cm on the map?

Oct 29, 2012

Dec 17, 2014