Find the number of bicycles and tricycles

Question

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?"

Answer 1

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.

Answer 2

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?

Answer 3

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.

Answer 4

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

Answer 5

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:

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

Answer 6

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)

Answer 7

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)

Answer 8

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

Answer 9

\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}