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# Proportions to Find Scale Ratios

## Understand and use scale factor.

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Proportions to Find Scale Ratios

Have you ever thought about proportions and buildings? Take a look at this dilemma.

At the supermarket, Sarah was looking at the cereal boxes as she was putting them away. On the back of one of the boxes, there was a picture of a skyscraper. Underneath the picture was a ratio.

It said: 1inch30feet\begin{align*} \frac{1 inch}{30 feet}\end{align*}

Sarah wasn't sure what this meant. Do you know?

This Concept is all about scale ratios and factors. At the end of the Concept, you will understand what Sarah saw.

### Guidance

Now that you have figured out how to use proportions to figure out actual and scale dimensions, we can look at figuring out scale factors.

What is a scale factor?

A scale factor is another name for a scale ratio. When looking for a scale factor, you can look at the relationship between the scale measurement and the actual measurement to determine what scale was used. This scale is called the scale factor.

A fence is actually 16 feet long. If the fence is drawn as four inches, what is the scale factor? To figure this out, we need to write a ratio to compare the drawing of the fence to the actual measurement.

4"16 ft\begin{align*}\frac{4"}{16\ ft}\end{align*}

Now we want to figure out the scale factor. To do this, we simplify the ratio using the greatest common factor. The greatest common factor of 4 and 16 is 4.

4÷416÷4=1=4\begin{align*}4 \div 4 & = 1 \\ 16 \div 4 & = 4\end{align*}

The scale factor is 14 ft\begin{align*}\frac{1^{\prime\prime}}{4\ ft}\end{align*}.

Use this information to simplify and find the following scale factors.

#### Example A

5′′25\begin{align*}\frac{5''}{25'}\end{align*}

Solution: 15\begin{align*}\frac{1}{5}\end{align*}

#### Example B

2′′50\begin{align*}\frac{2''}{50'}\end{align*}

Solution: 125\begin{align*} \frac{1}{25}\end{align*}

#### Example C

12.5′′25\begin{align*}\frac{12.5''}{25'}\end{align*}

Solution: 12\begin{align*} \frac{1}{2}\end{align*}

Now back to Sarah. Here is the original problem once again.

At the supermarket, Sarah was looking at the cereal boxes as she was putting them away. On the back of one of the boxes, there was a picture of a skyscraper. Underneath the picture was a ratio.

It said: 1inch30feet\begin{align*} \frac{1 inch}{30 feet}\end{align*}

Sarah wasn't sure what this meant. Do you know?

Sarah has seen a scale ratio. Inches are being compared to feet. In other words, the scale is saying that for every inch of the building on the cereal box that it represents 30 feet of actual height.

We can figure out the scale factor if we can simplify this ratio. However, this ratio is already in simplest form so this scale ratio is also the scale factor.

### Vocabulary

Proportion
two equal ratios
Scale Drawing
a drawing used when a real life object is too big to draw with its actual dimensions.
Scale
the relationship of the size of a drawing to the size of the real object
Scale Factor
the relationship between the measurement of the drawing and the measurement of the real object.

### Guided Practice

Here is one for you to try on your own.

What is the scale factor?

312\begin{align*} \frac{3}{12}\end{align*}

To figure this out, we divide the denominator by the numerator. Once the fraction is simplified, it will show the scale factor.

14\begin{align*} \frac{1}{4}\end{align*}

### Practice

Directions: Simplify each ratio to find the scale factor.

1. 4′′6 ft\begin{align*}\frac{4''}{6\ ft}\end{align*}

2. 12′′24 ft\begin{align*}\frac{12''}{24\ ft}\end{align*}

3. 6′′18 ft\begin{align*}\frac{6''}{18\ ft}\end{align*}

4. 9′′27 ft\begin{align*}\frac{9''}{27\ ft}\end{align*}

5. 4′′16 ft\begin{align*}\frac{4''}{16\ ft}\end{align*}

6. 5′′30 ft\begin{align*}\frac{5''}{30\ ft}\end{align*}

7. 3′′30 ft\begin{align*}\frac{3''}{30\ ft}\end{align*}

8. 3′′60 miles\begin{align*}\frac{3''}{60\ miles}\end{align*}

9. 4′′100 miles\begin{align*}\frac{4''}{100\ miles}\end{align*}

10. 5′′1000 km\begin{align*}\frac{5''}{1000\ km}\end{align*}

11. 6′′1200 km\begin{align*}\frac{6''}{1200\ km}\end{align*}

12. 8′′24000 m\begin{align*}\frac{8''}{24000\ m}\end{align*}

13. 11′′11,000 km\begin{align*}\frac{11''}{11,000\ km}\end{align*}

14. 15′′3000 km\begin{align*}\frac{15''}{3000\ km}\end{align*}

15. 45′′15,000 m\begin{align*}\frac{45''}{15,000\ m}\end{align*}

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### Vocabulary Language: English

TermDefinition
Proportion A proportion is an equation that shows two equivalent ratios.
Scale Scale is the relationship between the size of a drawing and the size of the real object.
Scale Drawing A scale drawing is a drawing that is done with a scale so that specific small units of measure represent larger units of measure.
Scale Factor A scale factor is a ratio of the scale to the original or actual dimension written in simplest form.