In this explainer, we will learn how to use equivalent ratios to complete ratio tables and solve problems in real-life situations.
First, let’s learn some vocabulary associated with ratios.
Definition: Antecedent and Consequent
A ratio compares two numbers. Let’s call these numbers and . The ratio can be written as or . Here, is called the first term, or antecedent of the ratio, and the second term, or consequent.
Two ratios are equivalent when they can be simplified to the same ratio. Tables of equivalent ratios are very useful to find values of two quantities knowing that their ratios are equivalent.
For instance, let’s consider a typical example of equivalent ratios when we buy a different number of items and each item has the same price. Let’s imagine that one bag of apples costs $1.50. Two bags of apples, then, cost $3, three bags cost $4.50, and so on. We can put these values in a table, as shown here.
|Number of Bags||1||2||3||4||5|
All the ratios of the price to the number of bags are equivalent:
This means that, in the table, the numbers in the same column are obtained, one from the other, by multiplying or dividing by 1.50.
This also means that, given one pair of numbers in one column of the table, any other pair can be found by multiplying both numbers by the same number. For instance, we see that we can go from “2 bags cost $3” to “4 bags cost $6” by multiplying the 2 and the 3 by 2.
Note that the price of 5 bags is given by the price of 3 bags plus the price of 2 bags.
Let’s check our understanding.
Example 1: Identifying the Antecedent and Consequent of a Ratio
Complete the following table:
|First Number (Antecedent)||Second Number (Consequent)||Forms of Expressing the Ratio|
Here, we are given a ratio, , written also as . We need to identify the antecedent and the consequent of the ratio. As we know that the antecedent is the first term, it is 31. The consequent is the second term, 35.
Example 2: Finding a Ratio Using a Ratio Table
Using the ratio table, determine how many people 8 pizzas would serve.
|Number of Pizzas||3||4||7||8|
We can check that the ratios of the number of pizzas to the number of people served, as given in the table, are equivalent, by looking at the ratios , , and . They are indeed all equal to 3 (which means that one pizza serves 3 people).
Having found that one pizza serves 3 people, we can find that 8 pizzas serve people. We could also say that 8 is double 4, and so the number of people served is double 12, that is, 24.
Example 3: Finding the Consequent Having the Antecedent and One Equivalent Ratio
Matthew bought 40 packs of baseball cards for a discounted price of $96. If he sells 10 packs to a friend at the same cost, how much should he charge?
|Number of Baseball Card Packs||10||40|
|Cost in Dollars||96|
Here, we know that 40 cards have cost Matthew $96, and we want to find the price of 10 cards at the same cost, meaning when the ratio of the price to the number of cards is the same in both cases (which means that the price per card is the same).
Since 10 is one quarter of 40, that is, or , the price of 10 cards is obtained by dividing the price of 40 cards by 4 as well. The price of 10 cards is dollars.
We could also have calculated first the cost of one card and then multiplied this by 10 to get the cost of 10 cards: . This is of course exactly the same as .