# Find the number of bicycles and tricycles [duplicate]

Help for my son. My math is a bit rusty and I'm trying to remember how to go about answering this question: "There are 3 times as many bicycles in the playground as there are tricycles. There is a total of 81 wheels. What is the total number of bicycles and tricycles in the playground?"

## marked as duplicate by user296602, colormegone, Em., user91500, Jyrki LahtonenJan 16 '16 at 6:17

Without using equations and variables:

There are 3 times as many bicycles in the playground as there are tricycles.

Make groups of three bicycles and one tricycle each. Each group consists of 4 toys and has 9 wheels.

There is a total of 81 wheels.

There are 9 groups and thus 36 toys on the playground.

• "Work smarter, not harder" - this answer's motto – corsiKa Jan 15 '16 at 19:41
• Bicycles are toys? Then I ride a toy to work every day. :) – Théophile Jan 15 '16 at 21:07
• thank you so much for this simpler approach - my son is 9 and I was hoping there is an age appropriate methodology; a bit embarrassed I didn't think of it myself – Rpoe Jan 16 '16 at 16:57

Hint: Let there be $b$ bikes and $t$ trikes. Each sentence provides an equation, giving two simultaneous equations in two unknowns. Or group three bikes with a trike (based on the first sentence). How many wheels does it have? How many groups are there?

Denote the number of bicycles by $b$ and the number of tricycles by $t$.

You know that $b = 3t$ from "3 times as many bicycles in the playground as there are tricycles." and you know that $2b + 3t = 81$ as the total number of wheels is the number of bicycles times $2$ (two wheels per bike) plus the number of tricycles times $3$ (three wheels per bike).

Now you can plug in $3t$ for $b$ in the second equation to get $2 (3t) + 3t = 81$ so $9t = 81$. From there you get $t$ and then $b$ by the first equation.

It is possible that your son is not really supposed to use more than one variable. In this case call the number of tricycles $t$ and argue that $2(3t) + 3t =81$ in about the same was as above.

Just solve the following system of equations $$\begin{array}{lll} b-3t&=&0\\ 2b+3t&=&81\\ \end{array}$$

• This would be a better answer if you explained how to get these equations from the question. – DanTheMan Jan 15 '16 at 18:13
• @DanTheMan I appreciate your point of view (which seem to be identical to the point of view of all of the other posters), but my goal was to have the OP try to figure out how to get the equations. From a cognitive point of view, it think, it produces strong skills. With such a point blank answer, staring at it is sufficient to "get it". – John Joy Jan 15 '16 at 18:49

One way to wring the equations out of the word problem is to start with made-up numbers, and then abstract the letters.

"There are three times as many bicycles as tricycles." So if we have $4$ tricycles, then we have three times as many bicycles, which is $12$. Abstracting, if we have $T$ tricycles, then we have $B=3T$ bicycles.

Same with the wheels. If we have $12$ bikes, then we have twice as many wheels, which is $24$ wheels. So the number of bike wheels is $W_B = 2B$.

Likewise, the number of trike wheels is $W_T = 3T$.

The last piece of information we know is that the total number of wheels is $81$: $W_B + W_T = 81$.

Now, we substitute to solve for either the number of bikes $B$ or the number of trikes $T$:

$$W_B + W_T = 2B + 3T = 2(3T) + 3T = 9T = 81,$$

so the number of trikes is $T=9$. Then, the number of bikes is

$$B = 3T = 3(9) = 27.$$

And we can have some assurance we're right by calculating the number of wheels:

$$W_B + W_T = 2B + 3T = 2(27) + 3(9) = 54 + 27 = 81.$$

B = 3T. (B for Bycicle, T for Trycicle).

2B + 3T = 81 (wheels)

Just replace B for 3T: 2(3T) + 3T = 81. That gives you: 9T = 81. So there are 9 tricycles. Therefore B = 3*9 = 27 (bicycles)

1: 3t = b (three times as many bikes as trikes)

2: 2b + 3t = 81 (each bike has two wheels, trike three wheels, totaling 81)

Solving:

3: b - 3t = 0 (from line 1, moving 3t to the other side)

4: 2b - 6t = 0 (multiplying both sides by 2)

5: 9t = 81 (subtracting line 4 from line 2)

6: t = 9 (dividing both sides by 9)

7: b = 27 (substituting 9 into line 1)

\begin{align} &\text{Number of bicycles} =3 \times 2x\text{ wheels}\\ &\text{Number of tricycles} =3x\text{ wheels}\\ &3 \times 2x\text{ wheels}\ +\ 3x\text{ wheels}=81\text{ wheels}\\ &9x=81\\ &x=9\\ &\text{Number of bicycles} =3 \times 9\\ &\text{Number of tricycles} =9\\ \end{align}

There are N tricycles and each tricycle has 3 wheels There are (3 * N) bicycles and each bicycle has 2 wheels Number of Wheels = 81

(Number of tricycles * Number of Wheels per tricycle) + (Number of bicycle * Number of wheels per bicycle) = Total Wheels

(N * 3) + (3N * 2) = 81

3N + 6N = 81

9N=81

N= 81/9 = 9

So total number of bicycles in the playground = 3*N = 29 total number of tricycles in the playground = N = 9