Conversions and conversion factors

Links and useful resources

• gr7's Prealgebra Course Outline START HERE<<<
• AoPS Online Textbook
• IXL Grade 7 index
• IXL Grade 8 index
• AoPS Alcumus
• IXL prealgebra practice index
• OpenSTAX Prealgebra-1 textbook

Lesson-specific resource links

• AoPS Prealgebra conversions chapter

Concept summary and lesson

• using proportions to convert amounts

Two ways to do conversions

1. Scale the ratios to match the interior terms
2. Make conversion factors

Scaling ratios

Remember that you can multiply a ratio by any number you want that isn't zero, and it'll work. What if we want to "join up" a few ratios? We can do that by making the common terms match:

$12\text{ inches}:1\text{ feet}$, $3\text{ feet}:1\text{ yard}$

We notice that both ratios have a feet term (even though one of them only has one feet. Grammar is optional in math). We can scale the inches to feet ratio up so that it has three feet:

$36\text{ inches}:3\text{ feet}$

Now, we can just directly make a three-way ratio!

$36\text{ inches}:3\text{ feet}:1\text{ yard}$

So how many inches per yard? 36, it says right in the ratio.

Making conversion factors

You can also make conversion factors by noticing that the ratio represents an equation:

$12\text{ inches}:1\text{ feet}$ means the same thing as $12\text{ inches}=1\text{ feet}$
So, we can make a fraction by dividing both sides by one side or the other:

$\frac{12\text{ inches}}{1\text{ foot}}=1$ or $\frac{1\text{ foot}}{12\text{ inches}}=1$

That gave us two different conversion factors, and both of them are equal to 1! Remember that you can always multiply by one without changing the value of the expression, so if you have something in feet and you multiply it by the $\frac{12\text{ inches}}{1\text{ foot}}$ conversion factor, you'll end up with the same distance, expressed in inches (it's the same because you multiplied by one, even though the unit has now changed)!

How do you know which conversion factor to use? Pick the one that has the unit you want in the right place. Sometimes you're trying to convert the numerator of a fraction, sometimes the denominator, sometimes both. The trick is to use the conversion factor that has the unit you're looking for in the correct spot. Here's an example: convert $70\frac{\text{ miles}}{\text{ hour}}$ into $\frac{\text{feet}}{\text{second}}$:

We have miles in the numerator position on the left, and feet on the right, so we want to convert that part from miles to feet:

$1\text{ mile}=5280\text{ feet}$, so our factor with feet in the numerator would be $\frac{5280\text{ feet}}{\text{mile}}$.

We have hours in the denominator on the left, and seconds on the right, so we want the factor that converts hours to seconds and has seconds in the denominator. This one is two steps long, because it's easier to remember hours to minutes and minutes to seconds:

$1\text{ hour}:60\text{ minutes}$, and $1\text{ minutes}:60\text{ seconds}$. If we scale the minutes:seconds, we get $1\text{ hour}:60\text{ minutes}:3600\text{ seconds}$, so now we have the equation $1\text{ hour}=3600\text{ seconds}$. Since we need to have seconds in our denominator in the result we need the conversion factor $\frac{1\text{ hour}}{3600\text{ seconds}}=1$.

Now we can string our conversion factors together:

$\frac{70\text{ miles}}{1\text{ hours}}\times \frac{1\text{ hours}}{3600\text{ seconds}}\times \frac{5280\text{ feet}}{1\text{ miles}}=102\frac{2}{3}\frac{\text{feet}}{\text{second}}$

Worked examples

Media resources

• Youtube search for "using proportions to convert amounts"

Guided practice

• If there are 7 frobs to 3 jits, and it takes 18 jits to make five wobs, how many wobs would you get from 84 frobs?
• A tablespoon is half of a fluid ounce, a cup is 8 fluid ounces, and a gallon is 16 cups. How many tablespoons are in a gallon?
• An inch is exactly 2.54 centimeters. How many square centimeters are in a square inch?

Homework

• #hw (math) (gr7) 7.4 Conversions (p. 296) 2025-02-05