Question

How do you get this?

A factory produces 4 gallons of orange juice for every 24 gallons of apple juice produced. Use this ratio to answer the question.

The factory produced 6,260 gallons of orange juice during the rest of the week. How many gallons of apple juice did they produce during the rest of the week?
gallon(s)

Answer 1

In the first paragraph it states a ratio. Keep in mind a key word to look out for when trying to identify if something is a ratio is the word every. Also the one that pops up first is almost always going to be in the numerator, and the other number will be in the denominator. 

Keep in mind: 

$\frac{numerator}{denominator}$

$\frac{4galOJ}{24galAJ}$

You can reduce that ratio to the following:
$\frac{1galOJ}{6galAJ}$

OJ = Orange Juice
AJ = Apple Juice
gal = gallon
I recommend setting up this conversion like I'm about to. In fact, I recommend doing this for every conversion you need to do. 
$\frac{6260galOJ}{1} * \frac{6galAJ}{1galOJ}$

Are you familiar with the concept that $\frac{8}{8}$ "cancels" out. That of course is true for any two like numbers you plug in there. Did you also know that this "canceling" out also works for UNITS? Units being what is slapped on the end of your number (Don't confuse these with variables later on... variables being things like "x" or "y"). In this case we have two different variations of the unit gallon. We have gallons of Orange Juice, and gallons of Apple Juice, which are two different units. Our goal is to convert to gallons of apple juice, so we want to cancel out gallons of orange juice. 

To do this I set up two fractions. One with the number we want to convert over one, and another empty fraction like this: 

$\frac{6260galOJ}{1}* \frac{?}{?}$

We need to get it where gal OJ cancels itself out in the problem. So we need to get the unit over itself. Don't forget you can rewrite the multiplication of fractions like this: 

$\frac{6260galOJ*?}{1*?}$

While I won't continue using that form it can be helpful to visualize how you can cancel it out. You simply put the gal OJ unit in the bottom ? slot. With nowhere else to go you put the gal of AJ unit in the top ? slot... so it looks like this: 

$\frac{6260galOJ}{1} * \frac{?galAJ}{?galOJ}$

Now we need to scratch our heads a bit and think... hmmm! What do we put here. If we want to convert we need some type of ratio (which is also sometimes called a conversion value). Luckily we have one. We just need to plug in the correct numbers. Our ratio looks like this: 
$\frac{1galOJ}{6galAJ}$

So, plug in 6 next to the galAJ unit, and 1 next to the galOJ unit. 

$\frac{6260galOJ}{1}* \frac{6galAJ}{1galOJ}$

Now, multiply across. So, 6260galOJ*6galAJ = 37560galAJ

Our final answer is:
37560 gallons of Apple Juice!