Question

To make 10 servings of soup you need 4 cups of broth. You want to know how many servings you can make with 8 pints of broth. What proportion should you use?

Answer 1

see below

Explanation:

Remember `1` pint `= 2` cups
`rArr`
`color(white)("XXX")8` pints `= 16` cups
or (equivalently)
`color(white)("XXX")2` pints `= 4` cups

So the given ratio `10" servings" : 4" cups"` could be expressed as `10" servings" : 2" pints"`;
and the desired `8` pints could be expressed as `16` cups

If the number of servings per cups is denoted as `x_c`
and the number of servings per pint is denoted as `x_p`

The ratio could be expressed
`{: ("either as ","servings : cups"," or ", "servings : pints"), (,10 : 4,,10 : 2), (,x_c : 16,,x_p : 8), ("so",," or ",), (,(x_c)/16=10/4,,(x_p)/8=10/2), (,x_c=40,,x_p=40) :}`

Answer 2

`40" servings for 8 pints"`

Explanation:

We are mixing units of measurement. To make the calculation more straight forward lets use pints through out. This means we have to determine how many pints are in 4 cups.

`color(blue)("Converting cups to pints")`

Assumption: 2 cups have the same volume as 1 pint.

So by ratio we have: `("pints")/("cups") -> 1/2`

So for 4 cups: `("pints")/("cups") -> 1/2xx1 -> 1/2xx2/2=2/4`

Thus 4 cups is the same volume as 2 pints:

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Answering the question")`

Let the unknown count of servings be `x`

`("servings")/("pints") -> 10/2 = x/8`

Multiply both sides by 8

`x=(10xx8)/2 = 40" servings for 8 pints"`