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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.

Cross MultiplicationWe 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.

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