I am teaching an elementary student. He has a homework as follows.

There are $16$ students who use either bicycles or tricycles. The total

number of wheels is $38$. Find the number of students using bicycles.I have $3$ solutions as follows.

## Using a single variable.

Let $x$ be the number of students in question. The number of students using tricycles is $16-x$. The total number of wheels is the sum of the total number of bicycles times $2$ and the total number of tricycles times $3$.

$$

2\times x + 3 \times (16-x) = 38

$$The solution is $x=10$.

## Using $2$ variables.

Let $x$ and $y$ be the number of students using bicycles and tricycles, respectively.

It implies that\begin{align}

x+y&=16\\

2x+3y&=38

\end{align}The solution is $x=10$ and $y=6$.

## Using multiples

The multiples of $2$ are $2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22,24, 26,28,30,32,\dotsc$

The multiples of $3$ are $3,6,9,12,15,18,21,24,27,30,33,36,\dotsc$

The possible wheel combinations with format $(\#\text{bicycle wheels}, \#\text{tricycle wheels})$:

$\hspace{6cm} (32,6)$ but there are $18$ students

$\hspace{6cm} (26,12)$ but there are $17$ students

$\hspace{6cm} (20,18)$ there are $16$ students

$\hspace{6cm} (14, 24)$ there are $15$ students

$\hspace{6cm} (8, 30)$ there are $14$ students

$\hspace{6cm} (2,36)$ there are $13$ students

Thus the correct combination is $10$ bicycles and $6$ tricycles.

## My question

Is there any other simpler method?

**Answer**

If everybody had bicycles, there would be $16 \times 2 = 32$ wheels. There are actually $38$, so that’s $6$ additional wheels. Each one must be on a different tricycle.

So there are $6$ tricycles, and the other $10$ students have bicycles.

**Attribution***Source : Link , Question Author : kiss my armpit , Answer Author : Robert Israel*