Algebra: Ratios

On this page, we hope to clear up problems that you might have with fractions and their uses in Algebra. Ratios are continually being utilized in math and make many things much easier to do.

Follow any of the links below to be on your way to a better understanding of ratios, and good luck!

Solving proportions
Advanced ratio problems (inferring)
Quiz on ratios

Solving Proportions

In this section we'll help you understand how to deal with ratios.

A fraction is also known as a ratio. For example, 3/4 is also the ratio of 3 to 4. Any statement (or equation) that says two ratios are equal is called a proportion. An important thing to remember when dealing with equal ratios is illustrated below.

     3   15                   4 * 15
     - = --  Cross multiply.  
     4   20                   3 * 20

     4 * 15 = 3 * 20
         60 = 60

You can solve for unknowns in proportions by using that process.

Example:

             4   21
1.   Solve:  - = --
             m   5
  Solution:  Set the cross products equal.

             4 * 5 = 21m

             Finish the problem by dividing both sides
             by 21.

             20
             -- = m
             21

Advanced Ratios

Many times, you use ratios to make solving word problems easier. Many times, you are expected to gather information from the problem that is not directly stated in words. Gathering this information is called inferring. This section will help you better understand how to infer information from a word problem and solve using ratios.

If you are told there is a ratio of red marbles to blue marbles that equals 5:7 you are also told about two other ratios! You know there are 12 total marbles (5 + 7 = 12), so you know the ratio of red marbles to total marbles must be 5:12 and the ratio of blue marbles to total marbles must be 7:12. Also, it is usually helpful to write all these statements and then reread the problem to find any more information and to better understand the problem.

Example:

1.  Problem: The ratio of red marbles to blue marbles is 5 to 7.
             If there are 156 marbles total, how many red marbles
             are there?
   Solution: Write down the information given and inferred.

             R = 5
             B = 7
             T = 12
 
             With this information, there are three different
             proportions you can write.

             R   5   R   5    B   7
             - = -   - = --   - = --
             B   7   T   12   T   12

             The problem asks about red marbles and total marbles,
             so we will use the second proportion and replace the
             variable T with 156.

              R    5
             --- = -- -> 12 * R = 5 * 156 -> R = 65
             156   12

Take the quiz on ratios. The quiz is very useful for either review or to see if you've really got the topic down.

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