Ratio and Proportions

A  ratio provides a way of comparing two numbers or quantities.  A ratio is a quotient of two quantities with the same units.  The ratio of the number a to the number b may be written in three ways:
                                      a to b               a : b          or                .

The last way (as a fraction) is the most common way of writing a ratio in algebra.

Note that the order must be      not   .  Ratios are written as read from left (numerator) to right (denominator).

When ratios are used in comparing units of measure, the units should be the same.

For example, the ratio of 5 hours to 3 hours would be written as .

To write the ratio of 6 hours to 3 days, first convert 3 days to hours since the units must be the same [in other words each day has 24 hours so that 3 days would have 3 · 24 = 72 hours.]

The ratio would then be .

Because the units of the quantities in a ratio must be the same, ratios do not have units in their final forms.

A proportion is a statement that two ratios are equal.  For example, .

To show that they are equal we must prove that the numerator of the first fraction times the denominator of the second fraction must be equal to the denominator of the first fraction times the numerator of the second fraction.  In other words,  2  ·  12  =  3  ·  8.

This is obviously true since both sides of the equation are equal; therefore, the ratios are equal.  This method of checking the equality of proportions is called cross multiplication.  Simply stated if given the proportion   then  .

Check your understanding by determining if the following proportions are equivalent.

(a)                                [ this is true since  and  ]

(b)                               [ this is not true since  and  ]

Four numbers are given in a proportion.  If any three numbers are given then the fourth can be found.  The fourth unknown number will be represented by a variable.  To solve for the unknown variable, use cross multiplication.

For example, given that            then       .

Using cross multiplication we would obtain .

Then dividing both sides by 9, we would determine that .

Check your understanding by solving for the variable.

(a)     

... using cross multiplication, .  Which means that .  Now, dividing both
sides by 100, we would solve:  

(b)    

... using cross multiplication, .  Which means that .  Now, dividing both sides
by 4, we would solve:    ] 

Using proportions to solve application problems:

A local store is offering 3 packs of toothpicks for $0.87.  How much would it charge for 10 packs?

Let x = the cost of the 10 packs of toothpicks.  Set up the proportion by writing a ratio on the left hand side of the equation of 3 packs to 10 packs.  This would look like .  For the right hand side of the proportion, set up a ratio of the cost of the 3 packs to the cost of the 10 packs (which is unknown).  This would look like   .  Since the ratio would be equal, we would have the equation .

Solving for x by cross multiplication we would obtain               and      .

Check your understanding by setting up a proportion and solving for the unknown.

(a)  If 5 cans of cherries cost $20.50, how many cans of cherries can be purchased with
      $36.90?

(b)  If 3 boxes of raisins cost $8.70, how much do 5 boxes cost?

Answers:  (a)  9 cans    (b)  $14.50

General Algebra Tips

The views and opinions expressed in this page are strictly those of Mary Lou Baker.
The contents of this page have not been reviewed or approved by Columbia State Community College.

This page was edited on 15-Sep-2007