From Ratios to Rates and Rates to Ratios 04:21 minutes

Video Transcript

Transcript From Ratios to Rates and Rates to Ratios

Jilll lives on a remote farm with her chickens. One cold day in January, a giant snowfall causes her chicken coop to collapse. Her chickens are freezing and, worse, Jill has run out of some of the supplies she needs to repair the chicken coop. Before she runs to the store, she figures out what she will need and what it will cost using ratios and rates. Let's take a closer look at what she will get. First, the nails. The nails cost 10 dollars for a 5 pound bag of nails. This information can be thought of as a ratio. Ratios tells us how much of one quantity there is compared to another. We write the RATIO of cost to weight of nails as 10 dollars to 5 pounds. She wants to buy one pound of nails. How much do nails cost per pound? We can use a tape diagram to answer this question. Let’s let one rectangle represent 5 pounds of nails. If each rectangle represents 5, how many rectangles do we need to represent 10 dollars? Two rectangles go on top, because 10 divided by 5 is 2. If we want to figure out how much one pound of nails costs, we can use the tape diagram to help us, by substituting in ones for the fives. We can then see that one pound of nails costs two dollars. This is a unit rate. The UNIT RATE is the ratio of how much of one quantity there is compared to ONE unit of another. In this case, we have two dollars to ONE pound of nails. The RATE UNIT tells us how we are measuring our quantities. In this case, we are measuring in dollars per pound. But what are nails without something to hammer them into? Let’s shop for the next item on the list, boards. The boards cost 2 dollars and 80 cents each. That's a unit rate of 2 dollars and 80 cents per one board. Jilll needs 10 boards. Let's figure out how much 10 boards will cost. Instead of using a tape diagram, we can just multiply both sides of the ratio by 10 so the price for 10 boards is 28 dollars. Let's look at the final item on Jilll's list, yarn. 350 yards of yarn costs 7 dollars. Which is a ratio of 350 yards to 7 dollars. Let's calculate the unit rate, which tells us how much yarn we get per dollar spent. Dividing both sides of the ratio by 7 gives us 50 yards per dollar. 50 is the unit rate, and the rate units are yards per dollar. In a similar fashion to the tape diagram, we can use a double number line to represent this unit rate. With a double number line, we have one number line representing the cost in dollars, and another number line representing the length of yarn in yards. Each tick on the number line representing the cost corresponds to the amount of yarn which can be bought for that price. So we can see from the double number line that 1 dollar gets us 50 yards of yarn 2 dollars gets us 100 yards of yarn and so on. The chickens are freezing. Jilll quickly goes shopping and then races home! She realized once she got back that, in her rush, she left the nails at the check out counter but she was able to find a solution using all the yarn she bought. Now the chickens are happily pecking in their winter wonderland!