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

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.

• If one of these answers have helped you, please give it a big $\color{green}{\checkmark}$ :D Commented Oct 7, 2018 at 23:47
• I meant JohnJoy Commented Oct 7, 2018 at 23:50

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

• Why wouldn't $\frac{5}{3x}$ be the same as $\frac{5x}{3}$ if the order wont matter? Wont it be the same thing as $\ 5 / 3 * x$? Commented Nov 29, 2014 at 23:56
• Because in the first one, you are dividing by x and in the second you are multiplying by x. Anything in the denominator can be thought of as "divided by". And similarly, items in the numerator can be thought of as "multiplied by". Commented Nov 29, 2014 at 23:57
• Ah right, I see it now. Thanks Commented Nov 29, 2014 at 23:58
• Choose of estetic Commented Oct 16, 2018 at 13:08

$$\frac{5x}{3}=\frac{5}{3}x\ne\frac{5}{3x}$$ 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.

• Ah thank you for breaking it down. Commented Nov 29, 2014 at 23:48

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

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. $$\frac{5x}{3}=(5x)\cdot\frac{1}{3}=5\cdot(x\cdot\frac{1}{3})=5\cdot(\frac{1}{3}\cdot x)=(5\cdot\frac{1}{3})\cdot x=\frac{5}{3}x$$

I take a numerical approach here. Lets pretend (x) is just a number like 7. You're asking if:

$${5 \times 7 \over 3} = 5 \times {7\over 3}$$

Since the order does not matter here, lets just rewrite the expression on the left; notice that it doesn't matter if you multiply a number before dividing it; or if you divide before multiplying; the answers always the same, so:

$$(5 \times 7) \div 3 = 5 \times (7 \div 3) %$$

Now we can just rewrite these as fractions if we want to:

$${5 \times 7 \over 3} = 5 \times {7 \over 3}$$

So you could repeat this reasoning with an x in the place of the 7, and you'll come to the same conclusion :)