Assume that $a < b$ , and both are natural numbers (like 2 or 4).
- Is $a/b$ * $a/b$ more, less or equal to $a/b$ ?
- Is $a/b$ / $a/b$ more, less or equal to $a/b$ ?
- Is $ab/a$ - $ab/b$ more, less, or equal to $a/b$ ?
What I'm looking for is advanced answers (well, I'm 15 so maybe not extreme answers), that can showcase that I know if it's more, less or equal to $a/b$.
Here are my thoughts;
Let a = 2, b = 4.
$2/4$ * $2/4$ = $4/16$ , $4/16$ is not equal to $8/16$. Which means that $a^2/b^2$ is not equal to $a/b$, it's lower than it. (Can anyone please show me how I can explain this further ? I know that I can say that because $a < b$, the fraction ($a/b$) < $1$ ).
$2/4$ / $2/4$ = $1$ , $2/4 < 1$, which means that $a/b$ / $a/b$ is not equal to $a/b$, it's more than it. (Same thing here, how do I advance this answer by showcasing it more by fractions? I know that I can write that $a/b$ / $a/b$ = $a*b$ / $a*b$ = 1. But further advancement would be great!).
$8/2$ - $8/4$ = $4 - 2$ = $2$, Which means that it's more than $a/b$. (Could anyone help me here, should I multiply the fractions which a and then b -- if that works could anyone here show it?).
I'm really thankful for any answers that could advance my solutions!
$$a<b$$ $$\frac ab<1$$ $$\frac{a^2}{b^2}<\frac ab$$
As $\frac ab<1$ and $\frac{\frac ab}{\frac ab}=1$, $\frac ab<\frac{\frac ab}{\frac ab}$
$$\frac{ab}a-\frac{ab}b=\frac{ab^2-a^2b}{ab}=\frac{ab(b-a)}{ab}=b-a$$
As b and a are natural nos. with $a<b$, $$b-a>=1>\frac ab$$ $$\frac{ab}a-\frac{ab}b>\frac ab$$
Just remember that most rules for equation solving applying to inequalities as well, so you can transpose and add/delete terms in the same manner.
P.S. I'm only 13 and I'm answering your question.