Why does 10/35 = 2/7?

Question

My question is not meant to be dumb. I'm working on a much more complicated problem, but I forgot some of the basics.

Often we perform steps in math without really knowing why we can do a certain step. Why does $\frac{10}{35} = \frac{2}{7}$? Obviously it simply reduces to $\frac{2}{7}$, but exactly why ? We simply divide both numerator and denominator by $5$ but isn't this really multiplying by $5$? I'm wondering operationally how we do this.

Is it simple $\frac{10}{35} \cdot \frac{5}{5}$? Then we cross reduce? If we think of it $\frac{10}{35}$ divided by $5$ then I think of this as $\frac{10}{35} \cdot \frac{1}{5}$ (the reciprocal), BUT $\frac{10}{35} \cdot \frac{1}{5} = \frac{2}{35}$ (dividing the $5$ by itself and the $10$ by $5$ to get $2$.) Is there an actual proof or law that allows us to do this ?

Answer 1

I think a right thinking here it's the following. $$\frac{10}{35}=\frac{2\cdot5}{7\cdot5}=\frac{2}{7}.$$

Answer 2

Well, I think what you ask has a "double" answer: The intuitive one and the axiomatic one.

For the intuition: It really does make sense to manipulate fractions like that and in fact we do it all the time in every day life without realizing it. Saying that $\frac{10}{35}=\frac{2}{7}$ is like saying that whether you are at a party with 10 pizzas and 35 people or at a party with 7 pizzas and 2 people ,you will eat the same amount of pizza. This is the thinking that one tries to capture in the following axiomatic approach.

Aximoatic approach: As a mathematician you need to know what are the objects you are working with, same goes with rations. What is a fraction of two numbers? Well, to define them you say this: Let's make formal symbol of two natural number $\frac{p}{q}. You define two such symbols to be equal if and only if the obey the simple rule you are describing in your post, namely

$\frac{p_1}{q_1}=\frac{p_2}{q_2}$ if and only if $p_{1}\cdot q_2 =p_2 \cdot q_1$

So a fraction is the set of all satisfying the above condition.

Example: If you want to be extremely formal, when you write $\frac{7}{2}$ you mean all the numbers $\frac{p}{q}$ where $7q=2p$.

Finally let me say that if you understand this concept well you are on your way to understanding a much more abstract and deep mathemacal idea called "Localization of Ring" which generalizes what we did with the natural numbers in more weird "arithmetic systems" called rings.

You can search in wikipedia (here) to see a more rigorous presentation than what I am writing here. Here is about rings(here) and localization (here)

Answer 3

$$\frac{a}{b} = \frac{\frac{a}{\gcd(a,b)}}{\frac{b}{\gcd(a,b)}}$$ when $b \ne 0$ in general. In this case, since $\gcd(10,35) = 5$, we can write $$\frac{10}{35} = \frac{\frac{10}{5}}{\frac{35}{5}} = \frac{2}{7}$$

Answer 4

For an algebraic proof, you can start from the algebraic rule for multiplying any two fractions:

$\dfrac{a}{b} \times \dfrac{c}{d} = \dfrac{a \times c}{b \times d}$

so

$\dfrac{2}{7} = \dfrac{2}{7} \times 1 = \dfrac{2}{7} \times \dfrac{5}{5} = \dfrac{2 \times 5}{7 \times 5} = \dfrac{10}{35}$

For a more intuitive argument, you can say that $5 \times 7 = 35$ and thirty five thirty-fifths is one, so five thirty-fifths must be the same as one seventh, and so ten thirty-fifths is the same as two sevenths.

Answer 5

We are not dividing nore multiplying the fraction by 5. We are dividing both the top and the bottom of $10/35$ by 5. To make sense out of this we need a definition for equality of fractions. We define $$ a/b=c/d$$ if and only if $$ad=bc$$ Therefore $$10/35=2/7$$due to the fact that$$10\times7=35\times2$$