Scale measurements

Anytime you are given a scale drawing (or the scale for something) in a word problem, you can be fairly sure that you are going to have to use a proportion.

What is a scale drawing?

A scale drawing (or a scale figure) is a representation of something that is drawn proportionally. A scale drawing isn't a little map that we draw on a napkin or a post-it note. Think of it more like something that an architect or designer would use. Scales are also used in maps. The scale tells you what fraction of the size of the real item the map, or drawing, or figure is. So, on a map, one inch might represent 10 miles.  On a blueprint, one centimeter might represent 20 feet. In a scale picture of a cell in your biology book, one inch might represent $\dfrac{1}{100}$ of a millimeter$ (in this case, the microscopic cell is scaled up so that you can see it).

A scale is basically a conversion ratio. So, like when you do measurement conversions, you'll use proportions to solve scale problems.  

Like in conversions, you start with the ratio that you know is true (the scale), use labels, fill in the other piece of information that you know, and solve for the final piece of information.

Let's try one. 

Example: 

Katya uses graph paper to draw a scale model of her apartment. In her model, one square on the graph paper represents 2 feet in the apartment.  If her room is 6 squares long on the graph paper, how long is her room?

The first thing you want to do is set up a proportion.  On one side of the proportion, you put the ratio that you know, which is the scale:

$\dfrac{1\text{ square (model)}}{2 \text{ feet (real)}} = $

On the other side of the proportion, you match up the information you know, and the information you want to find out (write it in as a variable).

Notice how we label everything. Your labels should match horizontally. When your labels match horizontally, you know that you have it set up properly.

$\dfrac{1\text{ square (model)}}{2 \text{ feet (real)}} = \dfrac{6\text{ squares (model)}}{x \text{ feet (real)}}$

Remember, in a true proportion, the top left times the bottom right equals the top right times the bottom left. Thus, you can cross multiply to solve for $x$.

$$\eqalign{\dfrac{1\text{ square (model)}}{2 \text{ feet (real)}} &= \dfrac{6\text{ squares (model)}}{x \text{ feet (real)}}\\1 \times x &= 2 \times 6\\ x &= 12}$$

Katya's room is 12 feet long.

Very often in scale problems, you are given a picture.  You will get at least some of the information that you need to complete the problem from the picture.

Example: 

[[{"type":"media","view_mode":"media_large","fid":"873","field_deltas":{"1":{}},"fields":{},"attributes":{"height":"208","width":"364","style":"margin: 5px auto; vertical-align: middle; display: block;","class":"media-image media-element file-media-large","data-delta":"1"}}]]

The picture above shows a new homeowner's scale drawing of her yard.  She plans to plan fruit trees around the border, but wants to plant grass in the interior portion of the yard.  The depth of the area she wants to plant grass in is 28 feet.  What is the length of the portion of the yard she wants to plant grass in?

In this problem, we are not given the actual scale of the picture.  But we are given one actual length and one drawing length, which we can use to find the scale.

Set up the ratio of measurements you know:

$\dfrac{\text{drawing}}{\text{real}}=\dfrac{7\text{ inches (drawing)}}{28 \text{ feet (real)}} = $

We can reduce this fraction to find the actual scale ration, but we don't need to.  In order to set up an proportion, you just need one ratio that is true.  The ratio of 7 inches in the drawing to 28 feet in real life, is true.

Now, just put in the one other dimension that you know and the one that you don't.  You know that the length of the grass part of the yard is 10 inches in the drawing, but you don't know what it is in real life, so you will use a variable.

$\dfrac{7\text{ inches (drawing)}}{28 \text{ feet (real)}}= \dfrac{10\text{ inches (drawing)}}{x \text{ feet (real)}}$

Cross multiply to solve for $x$.

$$\eqalign{\dfrac{7\text{ inches (drawing)}}{28 \text{ feet (real)}}&= \dfrac{10\text{ inches (drawing)}}{x \text{ feet (real)}}\\7 \times x &= 10 \times 28\\ 7x &= 280\\ \div 7& \; \div 7\\x&=40}$$

The grass part of the yard is 40 feet long.

So, whenever you see a question that asks about scale, use the scale (either the info in the problem, the info from the drawing, or a combination of the two) to create a $\dfrac{\text{model}}{\text{real}}$ ratio, then create a proportion to solve for your unknown dimension.