Trouble understanding basic proportions

Ok, so I'm in algebra 1 (7th grade) and I'm having some trouble understanding why solving basic proportions works. I understand that a proportion is two equal ratios. So when one of the numbers is a variable, I'm able to solve for the variable, but I don't understand why it works. So when you start to cross-multiply, my question is why do you multiply the numerator of one ratio by the denominator of the other? I mean, the two ratios are totally separate, except for the fact that they are equal. So, why do you multiply them as if they were the same ratio. Sorry if my question was confusing, I just need someone to explain why solving proportions works the way it does.

Re: Trouble understanding basic proportions

The two ratios are not separate. If you have two fractions $\displaystyle \frac{a}{b} = \frac{a_1}{b_1} $ then the fractions are multiples of each other. They will both reduce to the same numerator and denominator.

For example $\displaystyle \frac{5}{15} = \frac{20}{60} $.

$\displaystyle \frac{5}{15} = \frac{1}{3} $ and

$\displaystyle \frac{20}{60} = \frac{4}{12} = \frac{1}{3} $

Usually when you are doing proportions you are given 3 things right? and you have to find one.

So say you have $\displaystyle \frac{a}{5} = \frac{3}{5} $, you want to know what to put in place of a so that $\displaystyle \frac{a}{5} $ becomes $\displaystyle \frac{3}{5}$.

Observe, the algebra is not just nonense, so multiply both sides by 5, you get $\displaystyle a = \frac{3*5}{5} $ .

Now do $\displaystyle \frac{a}{5} = \frac{\frac{3*5}{5}}{5} $ bring the 5 down, and you get $\displaystyle \frac{3*5}{5*5} $ knock out the 5 in num and 5 in denom, you get $\displaystyle \frac{3}{5}$, just what we wanted out of the equality.

Also the problem lies with using the = sign, when infact they are equivalent .

Meaning, $\displaystyle \frac{1}{5} $ is equivalent to $\displaystyle \frac{2}{10} $, you get what i'm saying? $\displaystyle \frac{1}{5} $ is equal to only $\displaystyle \frac{1}{5} $

Also i admire you for asking such questions, they are of fundamental importance, nothing to be trivialized. Keep thinking this over (Asking questions like what does it mean for 2 things to be equivalent rather than equal ).

Re: Trouble understanding basic proportions

Hello, legotboy!

Quote:

Ok, so I'm in algebra 1 (7th grade) and I'm having some trouble understanding why solving basic proportions works.

I understand that a proportion is two equal ratios.

So when one of the numbers is a variable, I'm able to solve for the variable.

But I don't understand why it works.

So when you start to cross-multiply, my question is:

why do you multiply the numerator of one ratio by the denominator of the other?

I mean, the two ratios are totally separate, except for the fact that they are equal.

So, why do you multiply them as if they were the same ratio?

Most students accept cross-multiplication as a welcome shortcut.

. . So I'm impressed that you asked "Why?"

Your teacher should have explained why it works.

Suppose we have the proportion: .$\displaystyle \frac{a}{b} \:=\:\frac{c}{d}$

Multiply both sides by b: .$\displaystyle b\left(\frac{a}{b}\right) \:=\:b\left(\frac{c}{d}\right) $

Multiply both sides by d: .$\displaystyle d\cdot{\color{red}\rlap{/}}b\left(\frac{a}{{\color{red}\rlap{/}}b}\right) \:=\:{\color{red}\rlap{/}}d\cdot b\left(\frac{c}{{\color{red}\rlap{/}}d}\right) $

. . And we have: .$\displaystyle a\cdot d \:=\:b\cdot c$

which is the same result as we'd get by "cross-multiplying".