4.1.3: Revisiting Proportional Relationships
Lesson
Let's use constants of proportionality to solve more problems.
Exercise \(\PageIndex{1}\): Recipe Ratios
A recipe calls for \(\frac{1}{2}\) cup sugar and 1 cup flour. Complete the table to show how much sugar and flour to use in different numbers of batches of the recipe.
| sugar (cups) | flour (cups) |
|---|---|
| \(\frac{1}{2}\) | \(1\) |
| \(\frac{3}{4}\) | |
| \(1\frac{3}{4}\) | |
| \(1\) | |
| \(2\frac{1}{2}\) |
Exercise \(\PageIndex{2}\): The Price of Rope
Two students are solving the same problem: At a hardware store, they can cut a length of rope off of a big roll, so you can buy any length you like. The cost for 6 feet of rope is $7.50. How much would you pay for 50 feet of rope, at this rate?
1. Kiran knows he can solve the problem this way.
What would be Kiran's answer?
2. Kiran wants to know if there is a more efficient way of solving the problem. Priya says she can solve the problem with only 2 rows in the table.
| length of rope (feet) | price of rope (dollars) |
|---|---|
| \(6\) | \(7.50\) |
| \(50\) |
What do you think Priya's method is?
Exercise \(\PageIndex{3}\): Swimming, Manufacturing, and Painting
-
Tyler swims at a constant speed, 5 meters every 4 seconds. How long does it take him to swim 114 meters?
distance (meters) time (seconds) \(5\) \(4\) \(114\) Table \(\PageIndex{3}\) -
A factory produces 3 bottles of sparkling water for every 8 bottles of plain water. How many bottles of sparkling water does the company produce when it produces 600 bottles of plain water?
number of bottles of sparkling water number of bottles of plain water Table \(\PageIndex{4}\) - A certain shade of light blue paint is made by mixing \(1\frac{1}{2}\) quarts of blue paint with 5 quarts of white paint. How much white paint would you need to mix with 4 quarts of blue paint?
- For each of the previous three situations, write an equation to represent the proportional relationship.
Are you ready for more?
Different nerve signals travel at different speeds.
- Pressure and touch signals travel about 250 feet per second.
- Dull pain signals travel about 2 feet per second.
- How long does it take you to feel an ant crawling on your foot?
- How much longer does it take to feel a dull ache in your foot?
Exercise \(\PageIndex{4}\): Finishing the Race and More Orange Juice
- Lin runs \(2\frac{3}{4}\) miles in \(\frac{2}{5}\) of an hour. Tyler runs \(8\frac{2}{3}\) miles in \(\frac{4}{3}\) of an hour. How long does it take each of them to run 10 miles at that rate?
- Priya mixes \(2\frac{1}{2}\) cups of water with \(\frac{1}{3}\) cup of orange juice concentrate. Diego mixes \(1\frac{2}{3}\) cups of water with \(\frac{1}{4}\) cup orange juice concentrate. How much concentrate should each of them mix with 100 cups of water to make juice that tastes the same as their original recipe? Explain or show your reasoning.
Summary
If we identify two quantities in a problem and one is proportional to the other, then we can calculate the constant of proportionality and use it to answer other questions about the situation. For example, Andre runs at a constant speed, 5 meters every 2 seconds. How long does it take him to run 91 meters at this rate?
In this problem there are two quantities, time (in seconds) and distance (in meters). Since Andre is running at a constant speed, time is proportional to distance. We can make a table with distance and time as column headers and fill in the given information.
| distance (meters) | time (seconds) |
|---|---|
| \(5\) | \(2\) |
| \(91\) |
To find the value in the right column, we multiply the value in the left column by \(\frac{2}{5}\) because \(\frac{2}{5}\cdot 5=2\). This means that it takes Andre \(\frac{2}{5}\) seconds to run one meter.
At this rate, it would take Andre \(\frac{2}{5}\cdot 91=\frac{182}{5}\), or 36.4 seconds to walk 91 meters. In general, if \(t\) is the time it takes to walk \(d\) meters at that pace, then \(t=\frac{2}{5}d\).
Glossary Entries
Definition: Percentage
A percentage is a rate per 100.
For example, a fish tank can hold 36 liters. Right now there is 27 liters of water in the tank. The percentage of the tank that is full is 75%.
Definition: Unit Rate
A unit rate is a rate per 1.
For example, 12 people share 2 pies equally. One unit rate is 6 people per pie, because \(12\div 2=6\). The other unit rate is \(\frac{1}{6}\) of a pie per person, because \(2\div 12=\frac{1}{6}\).
Practice
Exercise \(\PageIndex{5}\)
It takes an ant farm 3 days to consume \(\frac{1}{2}\) of an apple. At that rate, in how many days will the ant farm consume 3 apples?
Exercise \(\PageIndex{6}\)
To make a shade of paint called jasper green, mix 4 quarts of green paint with \(\frac{2}{3}\) cups of black paint. How much green paint should be mixed with 4 cups of black paint to make jasper green?
Exercise \(\PageIndex{7}\)
An airplane is flying from New York City to Los Angeles. The distance it travels in miles, \(d\), is related to the time in seconds, \(t\), by the equation \(d=0.15t\).
- How fast is it flying? Be sure to include the units.
- How far will it travel in 30 seconds?
- How long will it take to go 12.75 miles?
Exercise \(\PageIndex{8}\)
A grocer can buy strawberries for $1.38 per pound.
- Write an equation relating \(c\), the cost, and \(p\), the pounds of strawberries.
- A strawberry order cost $241.50. How many pounds did the grocer order?
Exercise \(\PageIndex{9}\)
Crater Lake in Oregon is shaped like a circle with a diameter of about 5.5 miles.
- How far is it around the perimeter of Crater Lake?
- What is the area of the surface of Crater Lake?
(From Unit 3.3.1)
Exercise \(\PageIndex{10}\)
A 50-centimeter piece of wire is bent into a circle. What is the area of this circle?
(From Unit 3.2.3)
Exercise \(\PageIndex{11}\)
Suppose Quadrilaterals A and B are both squares. Are A and B necessarily scaled copies of one another? Explain.
(From Unit 1.1.2)