Unit 6: Ratios and Proportions

6a. Use ratio concepts to solve problems

  • What is a ratio?
  • What does a ratio represent?

A ratio is essentially just a fraction. The only difference is how we think about the values of the numerator and denominator. In a fraction, we usually think of the numerator as the part and the denominator as the whole. In a ratio, the numerator and denominator represent a relationship between two quantities. 

For example, let's say that for every $2 you earn, your boss earns $3. That could be represented as a ratio of 2/3, also written as 2:3. In this case, the 3 isn't the total amount earned but rather the amount your boss earns in relation to the amount you earn. 

To review, see:

  • [6.1 Introduction to Ratios]
  • [6.1 Ratios]


6b. Calculate unit prices and rates 

  • How do you calculate a price per unit?
  • How do you calculate a rate per hour/day/etc.?

To calculate unit price (the price per unit), simply divide the total cost by the number of units. When you see the word per, the value that comes after it is always what you divide by, so price per unit means the total price divided by the number of units. For example, if you buy 6 cans of tomatoes for $3, the price per unit is $3/6=$0.50 per can.

To calculate a unit rate, first you must decide the unit that you want to calculate. For example, if a new roof for your house costs $10,000, and will take 4 days to complete, you could calculate the cost per day, by dividing 10,000 by 4. $10,000 ÷ 4=$2,500 per day. However, maybe the total working hours are 100 hours; then, you could calculate the cost per man-hour as $10,000 ÷ 100=$100 per working hour.

To review, see:

  • [6.2 Finding Unit Rates and Prices]


6c. Apply the processes of solving a proportion 

  • What is a proportion?
  • How do you solve a proportion?

A proportion is an equation where two ratios are equal to each other, with one quantity missing. For example, continuing from the previous example, if the ratio of your income to your boss' income is 2:3, how much does your boss make when you earn $400? For a proportion like this, set up the known ratio on one side, and a fraction on the other side where the corresponding values match up. \frac{2}{3}=\frac{400}{x} would be the correct setup. In this case, the x represents the amount your boss makes. To solve, simply cross-multiply to get 2x=1200, and then by dividing, you will find the answer is $600.

To review, see:

  • [6.3 Solving Proportions]


6d. Analyze proportional relationships to solve real-world and mathematical problems 

  • How do you solve word problems with proportions?

When solving a word problem involving proportions, it is important to first understand the problem and then transform the problem correctly into an equation that you can solve. Once you have transformed the problem into an equation, you can solve it just as you would solve any other equation involving proportions.

For example, say you are driving your car at 75mph on the highway. How far will you travel in 5 hours?

First, set up the proportion. 75 mph is a ratio of \dfrac{\text{75 miles}}{\text{1 hour}}. Set up the proportion with 5 hours and the unknown distance, \dfrac{\text{75 miles}}{\text{1 hour}} =\dfrac{\text{x miles}}{\text{5 hours}}. Cross multiply to find x=375 \text{ miles}. 

To review, see:

  • [6.3 Applications of Proportions]


Unit 6 Vocabulary

This vocabulary list includes terms you will need to know to successfully complete the final exam.

  • per
  • proportion
  • rate
  • ratio
  • unit price
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