# Problem Solving Using Rates, Unit Rates, and Conversions07:19 minutes

Video Transcript

## TranscriptProblem Solving Using Rates, Unit Rates, and Conversions

Sally, Bennie, and Evette are friends who live in the 'burbs. One day on their way home from school, they hear whimpering from a box on the sidewalk. In the box they find 3 abandoned puppies. They bring the puppies home, and try to contact their local animal shelters, but all of them are full. What now? Sally suggests they start their own animal rescue, Ruff Rescue. Starting an animal rescue involves various issues including fundraising rates and pet food comparisons. In order to help the friends solve these issues we can apply problem-solving using rates, unit rates, and conversions. First, Ruff Rescue needs to raise money to pay for all the food and supplies that they want to purchase. The three friends come up with different ideas about how to fundraise. Sally creates an online fitness challenge where she raises money for every mile she runs and posts the video online. Her efforts get some attention, and Sally raises 50 dollars every 48 hours - hashtag puppylove! Bennie likes making treats for our four-legged friends. By selling them he raises 30 dollars in 3 days, what a treat! Evette teaches the puppies some tricks and has the puppies perform the tricks in the park. She earns 350 dollars a week, a spectacular performance! At this rate how much money are the friends earning? We can’t just combined the numbers because each friend is counting the money differently. What rate unit could we use for all of them? Dollars per day is a good option. We already have dollars for each friend.
We just need to change the time to days. Looking at Sally's fundraising efforts, 48 hours is 2 days. So, that’s 50 dollars every 2 days. Can we convert that to a unit rate to get the denominator as one day? Yes, reducing the ratio by dividing by 2 gives us the unit rate of 25 dollars per day, strong work! Bennie's rate unit is already dollars per days. To get his unit rate of dollars per day all we need to do is reduce the fraction, giving us 10 dollars per day--every little bit of kibble helps! Evette’s puppy performance earns 350 dollars per week, which is 350 dollars in 7 days. Reducing gives us 50 dollars per day! Looks like those puppy tricks are raking in the most dough! Nevertheless, each of their fundraising efforts matter. So, to find out what they can make together let's combined their unit rates by adding the fractions. The denominators are all the same so adding the numerators results in 85 dollars per day.
A good start to their fundraising efforts allows the friends to now research different dog food options. There are two brands they are considering Doggy Delight costs 25.49 for 1.3 kilograms, and it's all natural with extra protein and calcium to strengthen muscles and bones. Bennie Bite is 22.79 for 3 lbs. It's Bennie’s homemade, organic, preparation that the puppies just love. At Ruff Rescue the friends only want to give the best quality to their pups. Nevertheless, they've got to keep an eye on their budget. Therefore, the friends would like to know which option is the least expensive. In order for us to see this clearly, let’s convert the units for each price to the unit rate of dollars per pound. How can we convert Doggie Delight to dollars per pound? By converting the units using a conversion factor. Is there a conversion factor which relates pounds to kilograms? Yes! 1 kilogram is about two point two pounds. We put kilograms on top of the conversion factor to cancel with kilograms on the bottom of our original rate unit, leaving us with dollars per pound. Multiplying through the numerator gives us 25 dollars and 49 cents and also across the denominator gives us 2.86 pounds. Then, using our calculator, we calculate the unit rate by simply dividing. This results in a unit rate of approximately 8.91 dollars per pound. Now we can find the unit rate of Bennie Bite. Bennie already sells his dog food in three pound packages. So to get the unit rate of dollars per pound we just divide by 3. That results in about 7.60 per pound. Bennie's dog food is the best value for Ruff Rescue. Helping puppies is a problem we solved with the use of rates, unit rates, and unit conversions. First we converted the different fundraising rates to a common rate unit, dollars per days. Next, in order to combine the rates we simplified each one to a unit rate using division so the denominator would be one. In the second example, we wanted to compare dog food prices. So, we used a conversion factor to represent each brand with a rate unit of dollars per pounds. Then, we simplified them using division to give us the unit prices. All of these steps can be used in various ways to solve problems involving rates and in this case, to help Ruff Rescue get started. The friends make a poster encouraging people to adopt.
But... oh man. Looks like the friends will need to spend some more time training the puppies.
They’ve got their work cut out for them.
Hashtag Hungry Puppies!

## Videos in this Topic

Unit Rates (7 Videos)

## Problem Solving Using Rates, Unit Rates, and Conversions Übung

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• #### Utilize rate to compare and solve a work problem.

##### Tipps

A unit rate is expressed as a quantity of $1$.

If Sally makes ＄$84$ in $2$ hours, divide both numbers by $2$ to find that she makes ＄$42$ in $1$ hour or ＄$42$ per hour.

Bennie makes ＄$192$ in $4$ days.

• To convert to the unit rate dollars per hour, multiply $4$ days by $24$ hours to find that $4$ days equals $96$ hours.
• We can now see that Bennie makes ＄$192$ in $96$ hours.
• Divide both sides by $96$ yields the unit rate ＄$2$ per hour.

##### Lösung

1. Since each friend is counting their money differently, we need to choose a one time period (hour, day, week, month, year) as the unit.

Sally's units are dollars per hour, Bennnie's units are dollars per day, and dollars per week.

The best unit rate to choose for this scenario is dollars per day since the number can be easily converted into dollars per day.

2. Sally walks dogs and makes＄$75$ in $72$ hours. This is the same as＄$75$ in $3$ days because $72$ hours divided by $24$ hours equals $3$ days. Now we can convert to a unit rate to find how much Sally makes in one day. We can divide both sides by $3$ and find that Sally makes＄$25$ per day.

3. Bennie works at the local pet store and makes＄$200$ in four days. To find the unit rate divide both sides by $4$ and find that Bennie makes＄$50$ per day.

4. Evette teaches puppy school and makes＄$420$ in one week. This is the same as＄$420$ in $7$ days. The unit rate is found by dividing both sides by $7$. Evette makes＄$60$ per day.

5. Now that everything is represented in the same unit, we can add them together:

• ＄$25$ per day $+$＄$50$ per day $+$＄$60$ per day $=$＄$135$ per day.

• #### Convert simple units.

##### Tipps

There are $12$ inches in $1$ foot and $2.54$ centimeters in $1$ inch.

$7$ feet are $84$ inches.

$7$ feet are $213.36$ centimeters.

##### Lösung

There are 12 inches in 1 foot and 2.54 centimeters in 1inch. To convert feet to inches, multiply by 12. To convert feet to centimeters, first multiply by 12 then by 2.54.

5 feet

• There are 60 inches in 5 feet: $5(12)=60$
• There are 152.4 centimeters in 5 feet: $5(12)(2.54)=152.4$
5.75 feet
• There are 69 inches in 5.75 feet: $5.75(12)=69$
• There are 175.26 centimeters in 5.75 feet: $5.75(12)(2.54)=175.26$
6 feet
• There are 72 inches in 6 feet: $6(12)=72$
• There are 182.88 centimeters in 6 feet: $6(12)(2.54)=182.88$
6.5 feet
• There are 78 inches in 6.5 feet: $6.5(12)=78$
• There are 198.12 centimeters in 6.5 feet: $6.5(12)(2.54)=198.12$

• #### Compare using a unit rate.

##### Tipps

A unit rate is expressed as a quantity of $1$.

To find a unit rate, divide both numbers in the ratio by the second number.

It costs $\$16.50$for$5$dog bones. Find the unit rate by dividing both numbers by$5$. It costs$\$3.30$ per dog bone.

##### Lösung

To find a unit rate, divide by the dollar amount by the number of items.

1.$\$29.49$for$95$dog treats • Divide both numbers by$95$. •$\$0.31$ per dog treat
2. $\$18$for$9$rubber balls • Divide both numbers by$9$. •$\$2$ per rubber ball
3. $\$32.50$for$8$dog bones • Divide both numbers by$8$. •$\$4.06$ per dog bone
4. $\$19.99$for$4$dog brushes • Divide both numbers by$4$. •$\$4.99$ per dog brush
5. $\$50.99$for$6$teddy bears • Divide both numbers by$6$. •$\$8.49$ per teddy bear
6. $\$52.75$for$5$bags of dog food • Divide both numbers by$5$. •$\$10.55$ per bag of dog food

• #### Identify the correct solutions to the word problem.

##### Tipps

To find the unit rate, set up a fraction to represent the situation then divide the numerator by the denominator.

There are $36$ inches in $1$ yard.

There are $3$ feet in $1$ yard.

##### Lösung

To find the unit rate, set up a fraction to represent the situation then divide both the numerator and denominator by the denominator.

Since all of the answer choices are represented in yards, convert the fabrics that are listed in inches and feet to yards.

There are $3$ feet in $1$ yard and $36$ inches in $1$ yard.

The unit rates are as follows:

1. Flannel Fabric: $\$27.96$for$4$yards •$\frac{\$27.96}{4\text{ yards}}=\frac{\$6.99}{1\text{ yard}}$• The unit rate is$\$6.99$ per yard.
2. Quilt Fabric: $\$24.97$for$90$inches •$\frac{\$24.97}{90\text{ inches}}\cdot \frac{36\text{ inches}}{1\text{ yard}} =\frac{\$898.2}{90\text{ yards}}=\frac{\$9.99}{1\text{ yard}}$
• The unit rate is $\$9.99$per yard. 3. Satin Fabric:$\$30.31$ for $7$ feet
• $\frac{\$30.31}{7\text{ feet}}\cdot \frac{3\text{ feet}}{1\text{ yard}} =\frac{\$90.93}{7\text{ yards}}=\frac{\$12.99}{1\text{ yard}}$• The unit rate is$\$12.99$ per yard.
4. Fleece Fabric: $\$29.96$for$3.75$yards •$\frac{\$29.96}{3.75\text{ yards}}=\frac{\$7.99}{1\text{ yard}}$• The unit rate is$\$7.99$ per yard.
Correct
• Fleece Fabric costs $\$1$more per yard than Flannel Fabric. This is correct because$\$7.99-\$6.99=\$1$.
• The unit rate for Quilt Fabric is $\$9.99$per yard. • The total cost for$1$yard of Satin Fabric and$1$yard of Fleece Fabric is$\$20.98$. This is correct because $\$12.99+\$7.99=\$20.98$. • The total cost for 1 yard of all 4 fabrics is$\$37.96$. This is correct because $\$6.99+\$9.99+\$12.99+\$7.99=\$37.96$. Incorrect • The unit rate for Flannel Fabric is$\$7.99$ per yard.