Simple Conversions (Proportions)

Many word problems ask students to convert units.  Sometimes they explicitly ask how many inches are in 5 feet.  But, the "conversion" part of the question often does not seem that obvious.

However, anytime you have a problem that provides one unit and asks for answers in a different unit, you will need to do a conversion (unless you are adding a dimension, such as going from inches to square inches, in those cases, the conversion is part of the actual calculation, e.g., length in inches times width in inches, gives area in square inches, because length is one dimensional and area is two dimensional).

This lesson will focus on problems in which conversion is the main operation, but these strategies also work when you have to do other calculations (usually, the conversion is one of the last calculations you do before marking an answer).

We find that using proportions is one of the simplest ways to do conversions. And, we firmly believe that students should label their proportions as they write them. Labeling can seem silly, or like a waste of time, but labels help prevent you from making careless errors!

Let's start with a simple problem:

Example: 

You are buying a sofa for your living room. You measure the space and you want a sofa that is 7 feet long. But, when you get to the store, they show the dimensions of the sofas in inches. How many inches do you want your sofa to be? (1 foot = 12 inches).

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 (the conversion rate).  

$\dfrac{1\text{ foot}}{12 \text{ inches}} = $

On the other side of the proportion, you match up the measurement you know (7 feet), and the one 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.

$\dfrac{1\text{ foot}}{12 \text{ inches}} = \dfrac{7\text{ feet}}{x \text{ inches}}$

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 $x$.

$$\eqalign{\dfrac{1\text{ foot}}{12 \text{ inches}} &= \dfrac{7\text{ feet}}{x \text{ inches}}\\1 \times x &= 7 \times 12\\ x &= 84}$$

You are looking for a sofa that is 84 inches long.

All conversion problems actually are this simple. But some seem harder when you read them. Also, while some provide the conversion, others will expect you to know the conversion (e.g., how many inches are in a foot).

Example: 

Jonathon exercised for 2 hours.  How many minutes did he exercise for?

Again, 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 (the conversion rate).  Do you know how many minutes are in an hour?  60.

$\dfrac{1\text{ hour}}{60 \text{ minutes}} = $

Note: it does not matter if you put hours or minutes on top.  Just make sure, whatever label is on top in this ratio, the same label is on top in the next.

On the other side of the proportion, you match up the measurement you know (2 hours), and the one 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.

$\dfrac{1\text{ hour}}{60 \text{ minutes}} = \dfrac{2\text{ hours}}{x \text{ minutes}}$

Now, just cross multiply to solve for $x$.

$$\eqalign{\dfrac{1\text{ hour}}{60 \text{ minutes}} &= \dfrac{2\text{ hours}}{x \text{ minutes}}\\1 \times x &= 2 \times 60\\ x &= 120}$$

Jonathon exercised for 120 minutes.

Some conversion problems contain extra information, that you don't really need, and that can confuse you.

Example: 

April rode her bike at a speed of 15 miles per hour.  She wants to tell her friend in France how many kilometers per hour she rides.  What is April's approximate speed in kilometers? (1 mile $\approx$ 1.6 kilometers)

This question can seem a little tricky, because it's talking about miles per hour and kilometers per hour, and those seem like ratios already.  In fact, they ARE ratios already ($\dfrac{miles}{hour}$).  But you don't need to include "hour" in your calculation.  You're trying to talk about your riding speed, in miles or kilometers, per hour.  The hour is constant in both measurements. So you can ignore it.  All you have to do is convert miles to kilometers. 

So, as usual, the first thing you do is write the conversion rate as a fraction (use your labels!).

$\dfrac{1\text{ mile}}{1.6 \text{ kilometers}} = $

On the other side of the proportion, you match up the measurement you know (15 miles), and the one you want to find out (write it in as a variable).

$\dfrac{1\text{ mile}}{1.6 \text{ kilometers}} = \dfrac{15\text{ miles}}{x \text{ kilometers}}$

Now, just cross multiply to solve for $x$.

$$\eqalign{\dfrac{1\text{ mile}}{1.6 \text{ kilometers}} &= \dfrac{15\text{ miles}}{x \text{ kilometers}}\\1 \times x &= 1.6 \times 15\\ x &= 24}$$

April rides about 24 kilometers per hour.

So, whenever you are faced with a conversion -- either by itself in a word problem or as part of another word problem -- isolate the conversion rate, set up a proportion, use labels, make sure the labels line up horizontally, cross multiply, and you'll have your answer.