Solving Rate Problems

​Solving Rate Problems

By Ashley Moser

• solve for unit rates that represent speed or pace

• apply unit rates to determine unknown distances or elapsed times and explain the solution method

• create a diagram or table to represent the situation involving two objects moving at constant speeds

• apply reasoning about ratios and rates to justify whether a given price is a good deal

​Learners can...

Vocabulary: pace, speed

​Think about a time you saw a car zoom by on the road or a turtle walk painfully slowly across a street.

​An object moving at a constant rate will consistently show equivalent ratios between distance and time.

The table at the right shows the constant rate at which a moped travels over different periods of time.

Draw

While training for a race, Andre's dad ran 12 miles in 75 minutes on a treadmill.

If he runs at that rate constantly, how long will it take him to run eight miles?

Show all of your work using any strategy you prefer.

​Drew wanted to know how many minutes it took Andre’s dad to run one mile, so she divided 12 miles by 12 and 75 minutes by 12.

Once she knew that it took 6.25 minutes to run one mile, she multiplied the unit rate by eight.

This showed her that Andre’s dad can run 8 miles in 50 minutes.

​Max calculated the unit rate by dividing 75 minutes by 12 miles and found that Andre’s dad ran 6.25 miles per minute.

He multiplied the first column values, distance in miles, by the unit rate, 6.25 miles per minute, to get the value in the second column, time in minutes: 8 x 6.25 = 50 minutes in total.

This showed him that Andre’s dad can run 8 miles in 50 minutes.

Poll

Did you use a similar strategy?

• My strategy was similar to Drew's

• My strategy was similar to Max's

• My strategy was totally different

• I sort of combined the strategies.

Hotspot

Staying active and spending time outside is great for overall health. Rail trails are a popular and safe location for people to spend time outside.

The tables below show the distance traveled by different people on a rail trail. The people were walking, running, rollerblading, and biking. All of them traveled at a constant speed except one.

Which table shows a person not traveling at a constant speed?

You learned that all rates have two unit rates.









Sometimes, one unit rate is more helpful than the other, depending on the scenario and the question being asked.

​​An electric scooter travels 5 miles in 15 minutes.

When you find the number of miles per minute or meters per second an object is moving, you are finding the object's speed.

Speed is a rate that tells how much distance an object travels in a certain amount of time.

For example, this bike's speed is 23 kilometers per hour.

​​Speed

When you find the number of minutes per mile or seconds per meter, you are finding the pace of the object.

Pace is a rate that tells how much time it takes to travel a certain distance.

For example, this runner's pace is 8 minutes and 24 seconds per mile.

​​Pace

​When describing how fast something is moving, you can use either of the two unit rates.

Draw

Kiran and Clare live 24 miles away from each other along a rail trail. One Saturday, the two friends started walking toward each other along the trail at 8:00 am with a plan to have a picnic where they meet.

Kiran walks at a speed of 3 miles per hour. Claire walks 3.4 miles per hour. After 1 hour, how far apart will they be?

Draw

The friends continue to walk at a constant rate. Kiran walks 3 miles per hour. Clair walks 3.4 miles per hour.

Complete the table.

Hotspot

If the two friends begin walking at :00 a.m., at what time will the two friends meet and have their picnic?

Poll

If Kiran walked three miles per hour toward Clare, and she walked 3.4 miles per hour toward Kiran, is that the same as one friend staying in place and the other jogging 6.4 miles per hour toward the other?

• Yes

• No

​Max knows that each hour, the total distance between Kiran and Clare is decreasing by 6.4 miles, so the amount of time it will take them to meet is the same as if one person stays put and the other travels at 6.4 miles per hour.

In a similar scenario, you may be given the time it takes for the two friends to meet, but only one of their speeds.

The next weekend, both friends set out to meet along the same 24 mile trip.

They left at 9:00 a.m., this time with Clare jogging and Kiran walking at three miles per hour.

They met at 12:30 p.m. How fast was Clare jogging?

Determining a Missing Speed: The Question

At 12:30, they met after 3.5 hours. In 3.5 hours Kiran traveled (3) x (3.5) = 10.5 miles.

Because 24-10.5=13.5, Clare jogged the remaining 13.5 miles in 3.5 hours.

Divide distance by time to find the speed. 13.5 miles in 3.5 hours means Clare jogged about 3.9 miles per hour.

Determining a Missing Speed: The Solution

Multiple Choice

Several months later, Kiran and Clare both set out at 8:00 a.m. again to meet along their 24-mile journey, this time with Kiran jogging and Clare walking at 3.4 miles per hour.

They met at 10:30 a.m. How fast was Kiran jogging?

• 4.8 mph

• 6.2 mph

• 8.5 mph

​

Code Word:

Gratitude

Multiple Choice

Jada rollerblades 7 miles in 56 minutes at a constant rate. At what pace, in minutes per mile, does she rollerblade?

Hint: Per mile... mile should be the denominator (bottom)

• 0.885 minutes per mile

• 8 minutes per mile

• 9 minutes per mile

​Your teammates realized that unit rates can be used when shopping, too! They can use unit rates to help look for deals that can save them money.

​Your teammates were given a card with an original price and a new price of juice boxes, as shown.






Think about which deal is better: 6 juice boxes for $2.40 or 10 juice boxes for $3.50.

• The original carton contained 10 juice boxes for $3.50.

• Four of the juice boxes were purchased, so now the carton of 6 juice boxes will cost $2.40.

​

Drew found the unit rate for each bottle in the original and new pack.

• She divided the cost of 10 juice boxes by 10 and found the unit rate of $0.35.

• She divided the cost of 6 juice boxes by 6 and found the unit rate of .

The better deal has the lower unit rate.

Find and Compare rates

Remi found the unit rate of the original pack and applied it to the number of items in the new pack.

• She divided the cost of 10 juice boxes by 10 and found the unit rate of $0.35.

• Then, she multiplied the unit rate by 6 juice boxes in the new pack. ($0.35)x6=$2.10

$2.10 for 6 juice boxes is a better deal than $2.40 for 6 juice boxes.

Find the unit rate of the original pack

Using both offers, Alex found the price of 30 juice boxes, a multiple of both 10 and 6.

• He multiplied the cost of 10 juice boxes by 3 and found that juice boxes cost $10.50.

• He multiplied the cost of 6 juice boxes by 5 and found that juice boxes cost $12.00.

The better deal has the lower cost for the same number of juice boxes.

Compare offers for the same number of items

You want to buy a 4-pack of drinks. One pack is left on the shelf, but only 3 bottles remain in that pack.

The clerk offers to sell you the 3-pack for $2.25.

If the cost of the 4-pack was $3.16, is this a better deal?

​

Lets Practice

Multiple Choice

You want to buy a 4-pack of drinks but there is only one pack on the shelf with 3 bottles in it. The clerk offers to sell you a 3-pack of drinks for $2.25.

If the cost of a 4-pack was $3.16, will you take the deal?

• Yes, because the unit price of the 3-pack is less than the unit price of the 4-pack.

• No, because the unit price of the 3-pack is more than the unit price of the 4-pack.

• Either deal works because the unit price is the same for both the 4-pack and 3-pack.

Multiple Choice

A penguin walks ten feet in six seconds.

At this rate, how far does the penguin walk in 45 seconds?

• 60 ft

• 75 ft

• 90 ft

Multiple Choice

A penguin walks ten feet in six seconds. At this rate, how long does it take the penguin to walk 45 feet?

• 27 seconds

• 20 seconds

• 18 seconds

Reorder

A restaurant sells 10 tacos for $8.49, or 6 of the same kind of taco for $4.92. They also sell individual tacos for $0.90 each.

Order the taco offers from worst deal to best deal.

• $0.90 for 1 taco

• 10 tacos for $8.49

• 6 tacos for $4.92

1

2

3

• 1
• 2
• 3

Multiple Choice

This package of sliced cheese costs $2.97. How much would a package with 18 slices cost at the same price per slice?

• $3.78

• $4.56

• $4.86

Today we learned...

• solve for unit rates that represent speed or pace;

• apply unit rates to determine unknown distances or elapsed times and explain the solution method;

• create a diagram or table to represent the situation involving two objects moving at constant speeds;

• apply reasoning about ratios and rates to justify whether a given price is a good deal.

  In a future lesson, you will review what you learned in this topic and take a quiz.

Exit Ticket!

Categorize

Options (3)

Granola Bars

Hummus

Yogurt

Three different offers with new prices are shown.

Which of these offers is a good deal compared to the original price and which are not?

Good Deal

NOT a good deal