# 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?

Jun 27, 2018

see below

#### Explanation:

Remember $1$ pint $= 2$ cups
$\Rightarrow$
$\textcolor{w h i t e}{\text{XXX}} 8$ pints $= 16$ cups
or (equivalently)
$\textcolor{w h i t e}{\text{XXX}} 2$ pints $= 4$ cups

So the given ratio $10 \text{ servings" : 4" cups}$ could be expressed as $10 \text{ servings" : 2" pints}$;
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) :}

Jun 27, 2018

$40 \text{ servings for 8 pints}$

#### 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.

$\textcolor{b l u e}{\text{Converting cups to pints}}$

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

So by ratio we have: $\left(\text{pints")/("cups}\right) \to \frac{1}{2}$

So for 4 cups: $\left(\text{pints")/("cups}\right) \to \frac{1}{2} \times 1 \to \frac{1}{2} \times \frac{2}{2} = \frac{2}{4}$

Thus 4 cups is the same volume as 2 pints:

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Answering the question}}$

Let the unknown count of servings be $x$

$\left(\text{servings")/("pints}\right) \to \frac{10}{2} = \frac{x}{8}$

Multiply both sides by 8

$x = \frac{10 \times 8}{2} = 40 \text{ servings for 8 pints}$