Is $\frac{5x}{3}$ The Same As $\frac{5}{3}x$?

Question

I believe they are the same but I'm not sure. Can someone please clarify this for me, and also explain why it would be the same or different.

Answer 1

First lets just consider 5/3. Study the graphic for a moment to convince yourself that $$5\div3$$ $$5\times\frac{1}{3}$$ $$\text{and }\frac{5}{3}$$ are all equivalent expressions.

This idea (of equivalence) combined with the Commutative ($ab=ba$) and Associative [(ab)c=a(bc)] Properties gives us. $$\frac{5x}{3}=(5x)\cdot\frac{1}{3}=5\cdot(x\cdot\frac{1}{3})=5\cdot(\frac{1}{3}\cdot x)=(5\cdot\frac{1}{3})\cdot x=\frac{5}{3}x$$

Answer 2

Depending on your formatting...

$$\frac{5x}{3}=\frac{5}{3}x$$ Since order wouldn't matter you could think of it as $5\times x \div 3$ is equivalent to $5 \div 3 \times x$

Answer 3

$$\frac{5x}{3}=\frac{5}{3}x\ne\frac{5}{3x}$$ I tend to prefer the second over the first.

Both are equivalent to $\frac{1}{3}\cdot5\cdot x$, which can be rearranged in any order: because multiplication is both commutative and associative, you can change the order of a series of multiplications as you see fit.

Answer 4

$\frac{5}{3} x = \frac{5}{3} \frac{x}{1} = \frac{5 \times x}{3 \times 1} = \frac{5 x}{3}$.