The end of the summer is a great time for a ride around the neighborhood. What do you notice in the story below? What are you wondering about? Leave a comment to tell us your thoughts!

## Wheels R Us

One fourth of the vehicles at Danielle’s Cycle Shop are tricycles. The rest are bicycles. Danielle counted a total of 45 wheels in her shop.

Let x be the number of tricycles and y the number of bicycles. The linear equation that represents this problem is: 3x+2y=45. But we know that tricycles are 1/4th of the total vehicles, so there is more I needed to do to solve this problem, so I used excel to calculate faster all the possible combinations of tricycles and bicycles whose wheels add up to 45. There is a total of 8 possibilities. I added up the total vehicles for each possibility and lastly, I looked for the tricycle/total vehicle ratio.

Turns out that Wheels R Us has a total of 5 tricycles and 15 bicycles.

x y 3x+2y=45 (x+y) [x/(x+y)]

tricycles bicycles total wheels total vehicles tricycle:vehicle %

15 0 45 15 100%

13 3 45 16 81%

11 6 45 17 65%

9 9 45 18 50%

7 12 45 19 37%

5 15 45 20 25%

3 18 45 21 14%

1 21 45 22 5%

Fun problem!

capruitt — thanks for sharing your problem solving! It sounds like you were solving for the number of bikes and tricycles at the store. I wonder if that’s what most people are wondering?

Your solution made me wonder too… could she have 45 wheels and 1/3 of the vehicles be bikes? 1/2 the vehicles? Any fraction of the vehicles?

What else could we wonder about this problem?

There are 4 equal parts–> 1 part tricycles and 3 parts bicycles.

So, number of bicycles = 3* (number of tricycles).

Every tricycle contributes three wheels and every bicycle contributes two wheels.

Let number of tricycles = T; and number of bicycles = B.

Then,

total number of wheels = 3 T + 2 B; that is

45 = 3 T + 2 B; that is (by substitution)

45 = 3 T + 2 (3 T )

45 = 3 T + 6 T

45 = 9 T , and we then divide both sides by 9 (or multiply by 1/9) to get

5 = T.

So, there are 5 tricycles, and therefore 3*5 bicycles = 15 bicycles.

Check: 5*3 + 15*2 = 15 + 35 = 45.

(Oh, why do we math people get stuck on using the variables x and y? Are not the first letters of the words, when they make sense, more natural?)

My Question:

What is the total number of tricycles and bicycles that would equal 45 wheels if 1/4 are tricycles and 3/4 are bicycles.

Strategy: Since 45 wheels is a reasonably low number, I will use a guess and check.

I noticed that–

–I need to come up with a number that is divisible by 4 in order to break it into the fraction parts 1/4 and 3/4 for the total number of vehicles.

–I need the total number of vehicles to equal 45 wheels

–the number of bicycles and tricycles must be a fairly small amount since there are only 45 wheels in total. For example, 10 tricycles times 3 wheels equals 30 wheels. If the 10 tricycles were the 1/4 I am already going to exceed the constraint of 45 wheels. The 3/4 bicycles would have to be 30 or 60 wheels which would be 90 wheels total.

I wonder if –

–I just think of numbers that are easily divisible by 4 and calculate the number of wheels will solve the problem

–I wonder if I need an organized list to keep track since clearly it has to be 1/4 and 3/4 of the total vehicles and this total is a fairly low number of vehicles

Solving:

I started with thinking of numbers easily divisible by 4 and that are fairly low. Reasonable guesses of numbers divisible by 4: 12, 16, 20, 24

Total Vehicles 12

1/4 tricycles 3 x 3 = 9 wheels

3/4 bicycles 9 x 2 = 18 wheels

Total wheels 27 Answer is too low

Total Vehicles 20

1/4 tricycles 5 x 3 = 15 wheels

3/4 bicycles 15 x 2 = 30 wheels

Total wheels is 45 is correct

Solution: There are 5 tricycles and 15 bicycles for a total of 20 vehicles in the shop.

My Extra: Can the shop still have 1/4 tricycles and 3/4 bicycles if there are a total of 46 wheels? Explain why or why not?

No, because the total number of vehicles has to be divisible by 4 to have there be 1/4 tricycles and 3/4 bicycles. Since the number of wheels has increased by one there needs to be an adjustment of number of vehicles. In order to maintain the 1:3 ratio of tricycles to bicycles, the next number (after 20) divisible by 4 is 24. This equals a total of 54 wheels. There is no way to get 46 wheels with 1/4 tricycles and 3/4 bicycles for total vehicles. It is possible if the fractions are changed.

Math:

24 Vehicles

¼ tricycle 6×3=18 wheels

¾ bicycles 18×2=36 wheels

Total wheels is 54

Barbara mentions that the numbers aren’t really that big. I agree. I think I’d take out 45 pennies (I have LOTS of pennies) and play around.

Seth,

Ahhhh…..the ever-forgotten manipulatives…..yes, pennies would be a great idea to help students see the patterns and combinations.

Thanks for reminding me!

I tried not to start with the “typical” problem that we see with this scenario, but really focus on just the given information.

I noticed that there are two kinds of vehicles – bikes (2 wheels each) and trikes (three wheels each) in the bike store. I noticed there are 45 wheels in all.

As a math educator, I wondered if students who hadn’t seen the typical problem would see”notice” that you could figure out the typical problem from the given information.

I wondered what fractions and total wheels would work and what had to be true to make them work.

I thought about basketball and how you could have a similar scenario with 1, 2, and three point plays.

I noticed that I like noticing and wondering without the burden or anxiety of immediately having to solve a problem.

There is nothing that says the wheels in the shop are attached to the bikes or trikes. Spare wheels are a thing. More information is needed. As Dan Meyer suggests, let the kids figure out what information is needed.