2.1.1: One of These Things is Not Like the Others
Lesson
Let's remember what equivalent ratios are.
Exercise \(\PageIndex{1}\): Remembering Double Number Lines
1. Complete the double number line diagram with the missing numbers.
2. What could each of the number lines represent? Invent a situation and label the diagram.
3. Make sure your labels include appropriate units of measure.
Exercise \(\PageIndex{2}\): Mystery Mixtures
Your teacher will show you three mixtures. Two taste the same, and one is different.
- Which mixture tastes different? Describe how it is different.
-
Here are the recipes that were used to make the three mixtures:
- 1 cup of water with teaspoons of powdered drink mix
- 2 cups of water with teaspoon of powdered drink mix
- 1 cup of water with teaspoon of powdered drink mix
Which of these recipes is for the stronger tasting mixture? Explain how you know.
Are you ready for more?
Salt and sugar give two distinctly different tastes, one salty and the other sweet. In a mixture of salt and sugar, it is possible for the mixture to be salty, sweet or both. Will any of these mixtures taste exactly the same?
- Mixture A: 2 cups water, 4 teaspoons salt, 0.25 cup sugar
- Mixture B: 1.5 cups water, 3 teaspoons salt, 0.2 cup sugar
- Mixture C: 1 cup water, 2 teaspoons salt, 0.125 cup sugar
Exercise \(\PageIndex{3}\): Crescent Moons
Here are four different crescent moon shapes.
- What do Moons A, B, and C all have in common that Moon D doesn't?
- Use numbers to describe how Moons A, B, and C are differnt from Moon D.
- Use a table or a double number line to show how Moons A, B, and C are different from Moon D.
Are you ready for more?
Can you make one moon cover another by changing its size? What does that tell you about its dimensions?
GeoGebra Applet tbmsMsJZ
Summary
When two different situations can be described by equivalent ratios , that means they are alike in some important way.
An example is a recipe. If two people make something to eat or drink, the taste will only be the same as long as the ratios of the ingredients are equivalent. For example, all of the mixtures of water and drink mix in this table taste the same, because the ratios of cups of water to scoops of drink mix are all equivalent ratios.
| water (cups) | drink mix (scoops) |
|---|---|
| 3 | 1 |
| 12 | 4 |
| 1.5 | 0.5 |
If a mixture were not equivalent to these, for example, if the ratio of cups of water to scoops of drink mix were \(6:4\), then the mixture would taste different.
Notice that the ratios of pairs of corresponding side lengths are equivalent in figures A, B, and C. For example, the ratios of the length of the top side to the length of the left side for figures A, B, and C are equivalent ratios. Figures A, B, and C are scaled copies of each other; this is the important way in which they are alike.
If a figure has corresponding sides that are not in a ratio equivalent to these, like figure D, then it’s not a scaled copy. In this unit, you will study relationships like these that can be described by a set of equivalent ratios.
Glossary Entries
Definition: Equivalent Ratios
Two ratios are equivalent if you can multiply each of the numbers in the first ratio by the same factor to get the numbers in the second ratio. For example, \(8:6\) is equivalent to \(4:3\), because \(8\cdot\frac{1}{2}=4\) and \(6\cdot\frac{1}{2}=3\).
A recipe for lemonade says to use 8 cups of water and 6 lemons. If we use 4 cups of water and 3 lemons, it will make half as much lemonade. Both recipes taste the same, because and are equivalent ratios.
| cups of water | number of lemons |
|---|---|
| 8 | 6 |
| 4 | 3 |
Practice
Exercise \(\PageIndex{4}\)
Which one of these shapes is not like the others? Explain what makes it different by representing each width and height pair with a ratio.
Exercise \(\PageIndex{5}\)
In one version of a trail mix, there are 3 cups of peanuts mixed with 2 cups of raisins. In another version of trail mix, there are 4.5 cups of peanuts mixed with 3 cups of raisins. Are the ratios equivalent for the two mixes? Explain your reasoning.
Exercise \(\PageIndex{6}\)
For each object, choose an appropriate scale for a drawing that fits on a regular sheet of paper. Not all of the scales on the list will be used.
Objects
- A person
- A football field (120 yards by 53\(\frac{1}{3}\) yards)
- The state of Washington (about 240 miles by 360 miles)
- The floor plan of a house
- A rectangular farm (6 miles by 2 mile)
Scales
- 1 in : 1 ft
- 1 cm : 1 m
- 1: 1000
- 1 ft: 1 mile
- 1: 100,000
- 1 mm: 1 km
- 1: 10,000,000
(From Unit 1.2.6)
Exercise \(\PageIndex{7}\)
Which scale is equivalent to 1 cm to 1 km?
- 1 to 1000
- 10,000 to 1
- 1 to 100,000
- 100,000 to 1
- 1 to 1,000,000
(From Unit 1.2.5)
Exercise \(\PageIndex{8}\)
- Find 3 different ratios that are equivalent to \(7:3\).
- Explain why these ratios are equivalent.
(From Unit 1.2.4)