GED Mathematical Reasoning: Understanding Ratios and Rates

  • A ratio is a comparison of two quantities having the same units. There are a few ways to denote a ratio.
  1. We can state the ratio using the word to”. For example, you can say that the ratio of games won to games lost is “9 to 6”
  2. Or we can use colon to state ratio. For example, 9:6
  3. The third way for writing a ratio is using a fraction. For example, we can state the ratio of games won to games lost as \frac{9}{6}
  • To reduce the ratio to lowest terms, reduce as you would a fraction by dividing both the numerator and denominator by a common factor. For example, \frac{9}{6} = \frac{3}{2}
  • As a general rule, always reduce a ratio to lowest terms. Also, if the ratio represents an improper fraction, leave it be – do not change it into a mixed number or a whole number.
  • Please note that the ordering of the numbers in the ratio is very important.  The order in which the numbers appear must correspond to the order of the words.
  • A rate is similar to a ratio in that it is a comparison of two quantities, but in a rate the units are different. The word “per” is often used with rates. Per means: “for each.” For example, you might use a rate to compare dollars per pound or miles per hour. We refer to this as a unit rate which is the rate for one unit of the quantity.
  • To find the unit rate, simply divide the two numbers given in the rate. (This makes sense since the fraction bar represents division.)

 

Example 1

So far this season, Tim’s basketball team has won 9 games and lost 6 games. Which fraction represents the ratio of games won-to-games lost in lowest terms?

Answer Options are:

  1. \frac{9}{6}
  2. \frac{3}{2}
  3. \frac{6}{9}
  4. \frac{2}{3}

 

Since we were asked to state the ratio of games WON to games LOST, we need to place the number representing games won FIRST or in the numerator and the number representing games lost SECOND or in the denominator. The result is \frac{9}{6}[latex].  Now we will reduce the ratio to lowest terms by dividing both the numerator and denominator by 3. Therefore, the ratio of games won to games lost written in lowest terms is answer (2): [latex]\frac{3}{2}

The ratio: \frac{3}{2} is NOT the same as the ratio \frac{2}{3}.

 

Example 2

A bunch of 6 bananas costs $1.74. What is the cost per banana?

 

To find the unit rate representing the cost per banana first write the rate as: $1.74 over 6 bananas.

cost per banana = \frac{\$1.74}{6\, bananas}

Notice that we wrote the cost in the numerator and the number of bananas in the denominator based on the order the words appear in the phrase: “cost per banana”

cost per banana = \frac{cost}{number\, of\, bananas}

Now, divide 1.74 by 6 to find the unit rate, or unit cost. Using long division or the calculator yields a result of $0.29. So the cost per banana is $0.29.

\$1.74 \div 6 = \$0.29

cost per banana = \frac{\$1.74}{6\, bananas} = \frac{\$0.29}{1\, banana}

Notice that you could use this unit rate to calculate the cost of purchasing more or fewer bananas. For instance, if you wanted to purchase only 5 bananas, the cost would be 5 times $0.29 or $1.45

5 \times \$0.29 = \$1.45

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