Let’s remember what equivalent ratios are.
Complete the double number line diagram with the missing numbers.
Your teacher will show you three mixtures. Two taste the same, and one is different.
Here are the recipes that were used to make the three mixtures:
Which of these recipes is for the stronger tasting mixture? Explain how you know.
Will any of these mixtures taste exactly the same?
Can you make one moon cover another by changing its size? What does that tell you about its dimensions?
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.
| row 1 | water (cups) | drink mix (scoops) |
|---|---|---|
| row 2 | 3 | 1 |
| row 3 | 12 | 4 |
| row 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.
Two ratios $a:b$ and $c:d$ are equivalent ratios if there is a number $s$ that you can multiply both $a$ and $b$ by to get $c$ and $d$ (respectively). In other words, $a\boldcdot s = c$ and $b \boldcdot s = d$.
