Rate, Ratio, and Proportional Reasoning

A rate is a ratio.   A ratio is a comparison of two numbers.   A ratio can be expressed three ways:

  1. Using the fraction bar as in 2/3.
  2. Using a colon symbol as in 2:3 .
  3. Using the word “to” as in 2 to 3.

When the denominator of a rate is 1, we call the rate a unit rate. We usually use the key word “per” or the division symbol ( /) to indicate a unit rate.   A ratio table is another way of finding answers to ratio problems.   With a ratio table, you start with a known ratio and use it to find other numbers with the same ratio.   You can keep using the numbers you find to discover more numbers until you solve the problem.

A proportion is an equation which states that two ratios are equal. To solve a proportion, you simply cross multiply to find the missing value.   There are three parts to every proportion:  2 fractions and an equal sign. Each of the fractions by itself is a ratio. The numerator of the first ratio and the denominator of the second ratio are called the extremes. The denominator of the first ratio and the numerator of the second ratio are called the means.

We can designate equivalent amounts of a whole by using a fraction, a decimal number, or by using a percent.   To change a fraction to a decimal, divide the numerator by the denominator.   To change a decimal to a percent, we move the decimal point two places to the right.   To change a percent to a decimal, move the decimal two places to the left.   Remember, percent means  out of 100, so the denominator of the percent ratio will always be 100.


1)  Write the ratio in another form.  9:6

2)  Write the ratio of ones to zeros.   1110000

3)  There are 145 students in your school chorus program.  25 students sing soprano and 15 sing alto.  Write a ratio of the number of altos to all the students in the chorus program.  Write your answer in lowest terms.

4)  Make the ratios equal.  9/25 = 36/?

5)  The ratio of apples to oranges in a fruit bowl is 1 to 4.   If there are 8 apples in the bowl, how many oranges are there?

6)  2 pounds of apples cost $0.76.  How much would 5.5 pounds cost?

7)  If it is $55 for 20 people, how much for one?

8)  Which is the better deal: 5 cans of soda for $2.27 or 7 cans of soda for $2.89?

9)  Which is traveling faster?  Traveling 350 miles in 5 hours or traveling 215 miles in 3 hours?

10)  Which is traveling faster? Traveling 170 yards in 45 seconds or traveling 416 feet in 35 seconds