How many ounces are in 1/4 of a gallon?

YES I KNOW THIS IS PREALGEBRA!!

16 oz = 1 pint
8 pints = 1 gallon

Let the word ounces be represented by `z`
Let the word pints be represented by `p`
Let the word gallons be represented by `g`

So every time you see `z` it means the word ounces
So every time you see `p` it means the word pint
So every time you see `g` it means the word gallon
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`color(blue)("Answering the question")`

Using ratio

We can write the ounces to pints ratio as `z:p`

I much prefer to use the fraction format and you will see why.

We can write the ounces to gallons ratio as `z/p ........ Ratio(1)`
We can write the pints to gallons ratio as `p/g........... Ratio(2)`

`color(brown)("Our target is to get the ratio of ounces to gallons "->z/g)`

Look what happens when we have `Ratio(1)xxRatio(2)`

`z/pxxp/g color(white)("ddd")->color(white)("ddd") (zxxp)/(pxxg)larr("ounces "xx" pints")/("pints "xx" gallons")`

Using an example: `2xx3 " gives the same answer as "3xx2`
So we can swap things round when multiplying

Thus we have:

`(zxxp)/(pxxg)color(white)("d")->color(white)("d")(zxxp)/(gxxpcolor(white)("d")) ->color(white)("d")z/gxxp/p = z/g`

The `p/p` is the equivalent of 1 and anything multiplied by 1 does not change in value.

Use the same approach as above on the numbers:

`z/pxxp/g color(white)("dddd") ->color(white)("dddd")(16z)/(1p)xx(8p)/(1g)`

`color(white)("ddddddddd") ->color(white)("dddd")(16z)/(1cancel(p))xx(8cancel(p))/(1g)`

`color(white)("ddddddddd") ->color(white)("dddd")(16zxx8)/(1xx1g)`

`("ounces")/("gallons")color(white)("dd") ->color(white)("dddd")(128z)/(1g)`

So for a quater of a gallon we have:

`("ounces")/("gallons")color(white)("dd") ->color(white)("dddd")(128zxx1/4)/(1gxx1/4) = (32z)/(color(white)("d") 1/4gcolor(white)("d"))`
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`color(blue)("Check")`

`1/4(16xx8)=16xx2=32`