# Thread: Basics of multiplication and Division (fractions)

1. ## Basics of multiplication and Division (fractions)

Hello,

I have a few questions regarding multiplication of and division of fractions of size less than 1.

I understand how to multiply straight across:

ex. -

1/2 x 3/8 = (1*3)/(2*8) = 3/16

But, if I was to explain to say, a third grader, how and why we can multiply across, and why when we multiply these fractions the resultant value gets smaller than its original multiplicants, what would I say?

Similarly, why when we divide, can we flip the second term and multiply across? And why does the division of two fractions of the like produce a value greater than its two original fractions?

How could I explain this in a discrete manner?

Thanks a lot!

2. ## Perhaps some help?

But, if I was to explain to say, a third grader, how and why we can multiply across, and why when we multiply these fractions the resultant value gets smaller than its original multiplicants, what would I say?
I think for the first part it might be best to explain that multiplication and division (by nonzero numbers) is commutative, and i mean a fraction $\displaystyle \frac{a}{b}\in \mathbb{Q}$ a field is really just $\displaystyle a*b^{-1}$ so when you multiply $\displaystyle \frac{a}{b}\frac{c}{d}$ it is like $\displaystyle a*b^{-1}*c*d^{-1}=a*c*b^{-1}d^{-1}=(ac)*(bd)^{-1}=\frac{ac}{bd}$

For the second part, just think of what it means to be multiplying by a fraction less than 1. It would be like a percentage less than 100%. 3/4 is 75% for instance, so obviously if you took a number (the first number doesnt even necessarily need to be less than 1 for this to be true) and multiplied it by 3/4 the result is going to be 75% of the original number.

Let $\displaystyle a,b \in \mathbb{Q}$ such that $\displaystyle 0\leq a,b \leq 1$. In particular $\displaystyle a\leq 1$ and $\displaystyle b\leq 1$ multiply the second equation by a on both sides to see why $\displaystyle ab\leq a$ do the same to the first equation by multiplying both sides by b to see that $\displaystyle ab\leq b$ so clearly their product is smaller than both of them were originally.

Similarly, why when we divide, can we flip the second term and multiply across? And why does the division of two fractions of the like produce a value greater than its two original fractions?
yeah pretty much the exact same arguments above explain this as well. Just note that the inverse of $\displaystyle a\in \mathbb{Z}$ is $\displaystyle \frac{1}{a}$ this logically generalizes to rational numbers because of the first question by how we define multiplication in $\displaystyle \mathbb{Q}$ the inverse in this field is just a number that you multiply by and get 1 it is also clear that inverses are unique, and so this is the one and only. That is all division is, multiplying by the multiplicative inverse.

hope that helps

Oops forgot to mention the very last thing. Notice about what i said before, if a rational number is less than 1 its numerator is smaller than the denominator, so when you invert, the numerator is bigger than the denominator and thus this inverted fraction is greater than 1. Then apply the same argument as before.