From Ratios to Rates and Rates to Ratios

From Ratios to Rates and Rates to Ratios exercise

• Discuss ratios, unit rates, and rate units.

  Hints

  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.

  Solution

  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.

• Calculate unit rate from a given ratio.

  Hints

  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.

  Solution

  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 $15$ lamps--how 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.

• Identify the rate, the unit rate, and the rate unit.

  Hints

  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

  Solution

  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.

  Hints

  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.

  Solution

  • 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: $\frac{2800}{20}=140~\text{mph}$

• Represent ratios with a tape diagram.

  Hints

  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 $4:1$.

  Solution

  • 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.

  The ratio 4:5 is already simplified. The ratio 22:11 reduces to 2:1. The ratio 12:9 reduces to 4:3. The ratio 25:10 reduces to 5:2.

• Reflect on ratios, rates, unit rates, and rate units.

  Hints

  • $8:3$ 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 $2:1$.

  Solution

  • 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.