Grade 7 · Unit 1 · Lesson 8 · part 2 of 5
Two ways that always agree
Everything in this unit comes down to one sentence.
Two ways that always agree
Everything in this unit comes down to one sentence.
A proportional relationship always multiplies by the same number, the unit rate.
Real problems give you one pair of amounts and ask about another. There are two ways to get there, and they always agree.
The unit rate way
Find how much for one. Then multiply by how many you want.
The scale factor way
Find how many times bigger the new amount is. Then multiply both amounts by that.
A cookie recipe uses 112 cups of flour for 12 cookies. How much flour for 30 cookies?
| Flour for one cookie: make the cookies 1 | 12 × 112 = 1, so 112 × 112 = 32 × 112 = 324 = 18 cup |
| Times 30 cookies | 18 × 30 = 308 = 334 cups |
| How many times bigger? | 30 ÷ 12 = 2.5 |
| Scale the flour too | 112 × 2.5 = 1.5 × 2.5 = 3.75 = 334 cups |
Both ways give 334 cups. The unit rate way finds the number the relationship multiplies by. The scale factor way says: 30 cookies is 2.5 batches of 12, so you need 2.5 times the flour. Use whichever is friendlier for the numbers, and use the other one to check.