**OFF TOPIC**: Announcing the release of my new book:

*Weight Loss Made a Bit Easier: Realistic and Practical Advice for Healthy Eating and Exercise*

Available on Amazon.com in paperback/Kindle formats for $6.25/$2.99. Please click here for details.

Home | My Math and Education Books | Math Lessons | Ask a Math Question

Site Info | Contact Info | Tutoring Info | LarryZafran.com

# Lesson 88: Ratios, Rates, Proportions, Unit Conversion, Metric Unit Prefixes

*Ratios and rates are just a comparison between two numbers, and a
proportion is a way of using that comparison as applied to some other
number. This lesson explains. *

Let's say we have a bag of 3 red marbles and 5 blue marbles.
We can say that the **ratio** of red to blue marbles is
3 to 5. We can also represent this in fraction format as 3/5,
or we can represent it as 3:5. A ratio is just a comparison
between two numbers with the same units. In this cases, we're
comparing marbles to marbles.

A rate is just like a ratio, but we are comparing numbers with different units. An example would be "2 apples for $1", or "45 miles per hour".

Here is how we can use ratios and rates to solve problems. Let's say
apples are priced at 4 for $7. How much would 8 apples cost?
You can probably do this in your head. They will cost $14.
We're buying twice as many. What you actually did was solve
the problem using a **proportion**. A proportion
is just an equation involving a known ratio or rate.

What would we do if we weren't given numbers that were so easy to work with? Let's say we wanted to buy 9 apples instead of 8. Here is what we would do. Set up an equation with the known rate or ratio on one side. We'll write 4/$7. On the other side, we'll write 9/x. Our equation is 4/$7 = 9/x. What this really means is, if 4 apples cost $7, what is the cost of 9 apples? We're using x to represent the unknown price.

We
solve this by doing what we call **cross multiplying**.
We'll multiply the numerator of the first fraction by the
denominator of the second fraction, and then multiply the numerator
of the second fraction by the denominator of the first fraction.
At right you can see that we're actually multiplying diagonally.

Each of these products must be set equal to each other.
The **cross products** in a proportion are always
equal. When we cross multiply one way, we get 4x, and we we cross multiply
the other way, we get 9 times $7, or $63. Our equation is then
4x = $63. Remember, we need to get x by itself, because that
is the price of the 9 apples. To do this, we need to divide
each side of the equation by 4. We're allowed to do the same
thing to each side the equation, and by doing this, the 4's on the
left will cancel out, leaving us with just x. We get x = $63/4
= $15.75. This means that if 4 apples cost $7, then 9 apples cost $15.75.
We used a rate and a proportion to figure out our answer.

We often use ratios and proportions to help us convert from one
unit of measurement to another. In order to do so, it is
important to be familiar with some basic units of measurements.
Be sure to **memorize** the equivalent measurements
below.

12 in = 1 ft | 5280 ft = 1 mile | 2 cups = 1 pt | 16 oz = 1 lb |

3 ft = 1 yd | 2000 lbs = 1 ton | 2 pts = 1 qt | 4 qts = 1 gal |

Make sure that you are also familiar with some common metric
prefixes. *Kilo* means 1000 times as much. For
example, a kilometer (km) equals 1000 meters, and a kilogram (kg)
equals 1000 grams. *Centi* means 100 times as small.
For example, 100 centimeters equals one meter, or, stated another
way, one centimeter equals 1/100 of a meter. *Milli*
means 1000 times as small. For example, 1000 millimeters
equals one meter, or, stated another way, one millimeter equals
1/1000 of a meter.

To convert from one unit of measurement to another, multiply by a
**unit conversion factor** so that the unwanted unit
will cancel out, and you'll be left with the desired unit. For
example, let's convert 4 feet to inches. We can use the unit
conversion factor

12 inches / 1 foot. That fraction is really equivalent to 1,
so we're allowed to multiply by it. We can multiply (4 feet /
1) times (12 inches / 1 foot). The units of feet cancel, since
they appear in the numerator and denominator of the product.
The units of inches remain, and multiplying across gives us an
answer of 48 inches.

You'll have more practice doing this in later lessons. Make
sure that you fully understand the concept, since this type of
exercise comes up quite a bit, not only on math exams, but in
everyday life as well.

Remember that you can
ask a math question if you have additional
questions about a topic, or you can contact me if you have any comments
or suggestions for this site.