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!
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Discuss ratios, unit rates, and rate units.
Tipps
Suppose that the prices of special chicken feed is $＄ 3$ per pound. This can be expressed the ratio $＄ 3:1~\text{lb}$. We know this is a unit rate because the second value in the ratio is $1$. The rate unit is dollars per pound.
Not every ratio is a unit rate. The ratio $6:2$ is not a unit rate, because the second value is $2$. In order to be a unit rate, the second value must be $1$. In this case, there are no units given with the ratio, so we don't know the rate unit.
Suppose the price of chicken fencing is $＄ 9:2~\text{yd}$. This isn't a unit rate, but we know the rate unit is dollars per yard. We also know that if we change the order of the ratio to $2~\text{yd}:＄ 9$, the corresponding rate unit is yards per dollar.
Lösung
The ratio $＄ 10 : 5~\text{lb}$ shows us the cost of nails per 5 pound bag. This means that nails cost $＄ 2$ per pound. $＄ 2$ per pound is the $\text{unit rate}$. The $\text{rate unit}$ is dollars per pound.
The ratio $＄ 2.80 : 1 ~\text{board}$ is already a $\text{unit rate}$ because it shows the cost of $1$ board. The rate unit is $\text{dollars per board}$. $＄ 28.00 : 10 ~\text{boards}$ is an equivalent $\text{ratio}$, but it's not a unit rate. All unit rates feature a second term of $1$.
The ratio $350~\text{yd}:＄ 7$ $\text{is not}$ a unit rate. Even though we don't know the unit rate, we still know the rate unit is $\text{yards per dollar}$. We can change the order of the ratio and write it as $＄ 7 : 350~\text{yd}$. Now the rate unit will be $\text{dollars per yard}$. With ratios, unit rates, and rate units, order matters.

Represent ratios with a tape diagram.
Tipps
Remember that in a tape diagram, each rectangle has the same value, regardless of whether it's on the top or bottom row.
Remember, ratios can be reduced by dividing, or expanded by multiplying.
 $4:7$ is equivalent to $12:21$.
 $16:4$ is equivalent to $4:1$.
Lösung
 Count the number of rectangles on the top and bottom of the tape diagram to determine the ratio.
 Reduce the given ratios by dividing in order to compare them with the ratio of the rectangles.

Calculate unit rate from a given ratio.
Tipps
If we have a ratio like $12:4$, we can divide to determine the unit rate. $\frac{12}{4}=3$, so there are $3$ rectangles on the top and $1$ on the bottom, each with a value of $4$.
If we have a ratio like $100:20$, we divide to determine the unit rate of $5$. That means $5$ rectangles on the top with a value of $20$ each, and $1$ on the bottom, also with a value of $20$.
If we have a ratio of any two numbers and we want to find the unit rate of the first quantity to the second, we divide the first by the second. The result is the unit rate.
Lösung
For $＄42$, Jill can buy $6$ bales of straw to line the her chicken coop. She only needs $1$ bail of straw. To determine the price, we divide $42$ by $6$ to get $7$. We can use a tape diagram to represent the ratio of dollars to bales. Each rectangle has a value of $6$. The unit rate is visible as the ratio of rectangles on top to bottom, giving us a unit rate of $＄7$ per bale.
For $＄75$, Jill can buy $15$ special heating lamps to install throughout the chicken coop. Jill doesn't want to buy $15$ lampshow much does each cost? We can divide $75$ by $15$ to get $5$. We can use a tape diagram to represent the ratio of dollars to lamps. Each rectangle has a value of $15$. The unit rate is the ratio of rectangles on top to bottom, that is, $＄5$ per lamp.
Super special treats forJill's chickens are in packs of $110$ for $＄10$. How many will a dollar buy her? We can divide $110$ by $10$ to get $11$. Our tape diagram represents the ratio of treats to dollars. Each rectangle has a value of $10$. The unit rate is $11$ treats per dollar.

Reflect on ratios, rates, unit rates, and rate units.
Tipps
 $8:3$ is a ratio.
 $21$ miles in $3$ hours is a rate.
 $7$ miles per hour represents a unit rate with rate units of miles per hour.
 $7$ kilometers in $3$ hours is a rate and a ratio.
 It's a rate because kilometers and hours are different units.
 $7:3.5$ is equivalent to $14:7$ because we can multiply each value in the original ratio by $2$.
 $14:7$ is equivalent to $2:1$.
Lösung
 All ratios are rates. False.
 All rates are ratios. True.
 All rates are unit rates. False.
 A unit rate is the same as a rate unit. False.
 Pounds per dollar and dollars per pound are both valid rate units/ True.
 To convert a rate to a unit rate, divide so the second value is $1$. True.
 If a marathoner drinks $12.5$ liters of water every $2.5$ days, then the marathoner drinks $6$ liters per day. False.

Identify the rate, the unit rate, and the rate unit.
Tipps
Not all rates are unit rates. A rate is a ratio of $2$ values, and neither of them have to be $1$.
For example, $\$5$ for $2$ gallons of gas is a rate, but it's not a unit rate. The unit rate would be $\$2.50$ per $1$ gallon. We know it's a unit rate because the second value is $1$.
In the above example:
rate: $\$5:2~\text{gallons}$
unit rate: $2.50$
rate unit: dollars per gallon
Sometimes we want to convert from a given rate to a unit rate:
If Dave drives $100$ miles in $2$ hours, his unit rate is $50$ miles per hour. The rate unit is miles per hour.
Sometimes we want to convert from unit rate to another, equivalent ratio:
If resoling a shoe costs $\$10$ per shoe, that's $\$ 20$ for $2$ shoes. The cost for $2$ shoes is probably more interesting to us than the cost per shoe!
The unit rate is always a number.
The rate unit is not a number, it's a unit composed of other units.
Some examples of rate units are:
 miles per hour
 dollars per pizza
 candies per dollar
 kilometers per gallon
Lösung
Sally is in first grade, and she's just starting to read. She reads $20$ words in $2$ minutes. $\frac{20}{2}=10$, so Sally reads $10$ words per minute. Yesterday she read $30$ pages!
 rate: $20$ words in $2$ minutes
 unit rate: $10$
 rate unit: words per minute
Ricardo eats $2$ apples per day. That's equivalent to $14$ apples every $7$ days. In two weeks he ate $28$ apples.
 rate: $14$ apples every $7$ days
 unit rate: $2$
 rate unit: apples per day
Janelle is a singer who uses vocal exercises to keep her voice in shape. This week, she exercised twice as many times as she did last week. In the future, she hopes to exercise about $21$ times per week, which means she needs to exercise about $42$ times every $2$ weeks.
 rate: $42$ times every $2$ weeks
 unit rate: $21$
 rate unit: times per week

Compare unit rates.
Tipps
To convert a ratio to a unit rate, divide the first number by the second.
If the second unit is a decimal, we can multiply each number in the ratio in order to eliminate the fraction.
For example:
 $12:1.5$ features a decimal value of $1.5$
 We can multiply $12$ and $1.5$ by $2$, so that the ratio becomes: $24:3$
 $\frac{24}{3}=8$, so $8$ is the unit rate.
The world record for downhill skiing speed is 158.424 miles per hour.
Lösung
 cyclist: $\frac{50}{2}=25~\text{mph}$
 motorcyclist: $\frac{150}{3}=50~\text{mph}$
 general: $90:1.5=180:3=\frac{180}{3}=60~\text{mph}$
 cheetah: $80~\text{mph}$
 skiier: $\frac{2800}{20}=140~\text{mph}$