Algebra:

From Ratios to Algebraic Equations

Suppose that a Mathcounts problem tells you, "The ratio of girls to boys in Mrs. Smith's class is 3:2." That's great for the boys if they're looking for girlfriends...but how can you use that ratio to set up algebraic equations? There are several ways in which you can interpret the ratio. Different problems call for different methods. Usually, for a single problem, all of these methods will work, but some will be easier than others.

• Since the girls:boys ratio is 3:2, the number of girls is three times some number, and the number of boys is three times some number. So let the number of girls equal 3x and the number of boys equal 2x.

  Then, if the problem says something like, "Mrs. Smith has 30 students in all," you can write an equation like:
  3x + 2x = 30.

  This approach is especially useful if you see a ratio chain like this: "The ratio A:B:C:D:E is 1:2:3:4:5." Then, instead of having five variables (A, B, C, D, and E), you can express everything using just one variable (x, 2x, 3x, 4x, 5x).

• Another way to view this problem is by creating two fractions. 3 and 2 will be the numerators. To find the denominator, add 3 and 2 to get 5. Now, you can say that 3/5 of the students are girls, and 2/5 of the students are boys. Letting x represent the total number of students, the number of girls is (3/5)x and the number of boys is (2/5)x.

  Then, if the problem asks something like, "Mrs. Smith has 18 girls in her class. How many students does she have in all?" you can write and solve an equation like:
  (3/5)x = 18

  This approach is also useful in probability problems. You know that if you choose a random student from Mrs. Smith's class, there is a 3/5 probability that you'll choose a girl, and there is a 2/5 probability that you'll choose a boy.

• Finally, you can take the ratios you're given, and simply rewrite them in fractional form. All you have to do is change the colons into fraction bars!

  Let g represent the number of girls, and b represent the number of boys.
  Since girls:boys = 3:2,
  g/b = 3/2

  Notice that this method will leave you with two variables, whereas the previous methods only left you with one.