Identifying Proportional Relationships
When quantities are proportional, their ratios are equal. For example, the ratios \(\frac{2}{5}\) and \(\frac{8}{20}\) are proportional. Note that \(\frac{4}{10}\) and \(\frac{12}{30}\) are equivalent fractions because they both simplify to \(\frac{2}{5}.\)
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Proportional Relationships from Verbal Descriptions
When given a verbal description that compares quantities, we can write ratios and then compare them to determine if they are proportional.
Marz makes $15 per hour working at the General Store. Is the amount of money that she makes proportional to the number of hours that she works?
In 0 hours, Marz makes $0. In 2 hours, Marz makes $30. In 3 hours, Marz makes $45.
The ratio of money that Marz makes to the number of hours that she works is always \(15:1.\) We can multiply the number of hours that she works by $15 to determine how much money she makes.
Therefore, the amount of money that she makes is proportional to the number of hours that she works.
Tor's jar of vitamins has 100 vitamins in it. He plans to take two per day. Is the number of vitamins remaining in his bottle proportional to the number of days that have passed?
On day 0, Tor has 50 vitamins. After day 1, Tor has 48 vitamins. After day 2, Tor has 46 vitamins. The ratio of \(1:48\) is not equivalent to the ratio of \(2:46\) so the number of vitamins remaining in his bottle is not proportional to the number of days that have passed.
Kai is given a piggy bank with $96 in it. Each week he saves $6 per week and adds it to his piggy bank. Is the amount of money in his bank proportional to the number of weeks he saves?
Proportional Relationships from Tables
When given a table that compares quantities, we can write ratios and then compare them to determine if they are proportional.
Heather is creating towers of nickels and measuring the height, in millimeters, of the stacks. Her data is shown below.
Number of Nickels Height 0 0 1 2 2 4 5 10 12 24 Is the height of a tower proportional to the number of nickels?
The ratios of \(1:2,\) \(2:4,\) \(5:10\) and \(12:24\) are all equivalent. We can find the height of any tower by multiplying the number of nickels by 2. Therefore, the height of a tower is proportional to the number of nickels.
Given the table below, are \(x\) and \(y\) proportional?
Proportional Relationships from Graphs
Graphs of proportional relationships are lines that travel through the origin.
Which of the following graphs show proportional relationships?
None of the graphs show proportional relationships.
Graph A is linear but does not travel through the origin.
Graph B is linear but does not travel through the origin.
Graph C is not linear.
Which graph shows a proportional relationship?
