what is 15/2 divided by 69/8?

Question

hero · Answer

You are tasked with simplifying $\dfrac{15}{2} \div \dfrac{69}{8}$ right?

anonymous · Answer

I believe so yes

hero · Answer

Are you familiar with the reciprocal rule of dividing fraction where if have $\dfrac{a}{c} \div \dfrac{d}{b}$ then it is the same as $\dfrac{a}{c} \times \dfrac{b}{d}$

anonymous · Answer

yes

hero · Answer

So according to this rule, 

$\dfrac{15}{2} \div \dfrac{69}{8} = \dfrac{15}{2} \times \dfrac{8}{69}$ right?

anonymous · Answer

yes so would you just divide 15 by 8 and 2 by 69 then simplify?

hero · Answer

At every step, you want to simplify first before attempting to perform any operation. There's another rule you should be aware of and that is cross cancellation. But just to make the rule clearer, if we have $\dfrac{a}{c} \times  \dfrac{b}{d}$, we can write it as one fraction $\dfrac{ab}{cd}$ Furthermore, by the associative property, the values being multiplied are interchangable, therefore if we want we can write $\dfrac{ab}{cd} $ as  $\dfrac{ba}{dc}$

hero · Answer

In your case, to help simplify things easier, we can write 

$\dfrac{15}{2} \times \dfrac{8}{69} = \dfrac{(15)(8)}{(2)(69)}$ And then swap the values of the numerator to get $\dfrac{(8)(15)}{(2)(69)}$ 

Do you follow any of this so far?

anonymous · Answer

then you just divide?

anonymous · Answer

@Hero  then you just divide?

hero · Answer

Well again, we want to simplify if possible before dividing and if the numerator is smaller than the denominator, then it is appropriate to simplify as much as possible and then leave the the result in fraction form.

hero · Answer

Now, swapping the values has a valid purpose because we can now write

$\dfrac{(8)(15)}{(2)(69)}$ as $\dfrac{8}{2} \times \dfrac{15}{69}$

hero · Answer

And in this case we can divide 8/2, but we are limited to simplifying 15/69. Do you see this?

anonymous · Answer

yeah cuz 15/69 is a decimal and you cant have a decimal in a fraction

hero · Answer

It is not necessary to simplify it down to decimals. We want to reduce it to the point where we can just leave it in fraction form.

hero · Answer

Obviously 8/2 = 4 so what we have left is 

$4 \times \dfrac{15}{69}$ 

Do you get that?

anonymous · Answer

I get how you got 4 but what do you do with the 15/69?

hero · Answer

We simplify it by cancelling "factors of one". Notice that 

$\dfrac{15}{69} = \dfrac{(3)(5)}{(3)(23)}$ Do you get that part?

anonymous · Answer

yes

hero · Answer

A particular thing to notice here is that 

$\dfrac{(3)(5)}{(3)(23)} = \dfrac{3}{3} \times \dfrac{5}{23}$

hero · Answer

And obviously $\frac{3}{3} = 1$

anonymous · Answer

so $$1 \times 5/23$$

hero · Answer

But let's not forget the 4 because what remains is:

$4 \times \dfrac{5}{23}$

anonymous · Answer

oh so would you do this: $$4/1 \times 5/23?$$

hero · Answer

Correct, so we can multiply that together since we are done simplifying. Can you compute the final result here?

anonymous · Answer

20/23?

hero · Answer

Great job. So to summarize, the process of multiplying and dividing two fractions, is actually the art of reducing those fractions to their most simplest forms by cancelling factors of one first before performing the appropriate operations on them. Once you have mastered this, then you'll be able to multiply or divide any fractions with confidence.