eli5-Dividing fractions by other fractions-why?

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I understand how to divide fractions by other fractions but want to know why this is a useful mathematical technique. When do you need/want to divide a fraction by another fraction? What’s it useful for? Is it a specialist thing or am I missing an obvious application?

In: Mathematics

I’m not even sure if this question is serious…

US recipes are an obvious use case: you need 1/4 cup but want to cut the recipe by 2/3.

I can eat half a pizza in a third of an hour. How long does it take me to eat a whole pizza?

1/2 pizza / 1/3 hour = 3/2 pizza/hour = 2/3 hour/pizza

I mean it may sound trite but you need/want to divide a fraction by another fraction whenever… you would need to divide a fraction by another fraction. Any time you’re dealing with something where you are dividing fractions.

Division is useful anytime you need to know how many of one number makes up another number. And sometimes those numbers are fractions, which is when dividing fractions by other fractions would be necessary.

An example: let’s say you only have a half-teaspoon measure and you need to put 3 and a half teaspoons of salt in your recipe. How many scoops of salt would you need? Well… 3.5 = 7/2. So…

7/2 ÷ 1/2 = 14/2 = 7 scoops

As a side note, dividing fractions by other fractions is pretty much the same as multiplying them, which use useful to know because multiplying is usually easier. This is true because:

*a*/*b* ÷ *c*/*d* = *a*/*b* × *d*/*c*

(you just need to flip the divisor fraction)

Why is it useful? Sometimes it’s just something that comes up. If I have a cup that holds 1/8 of a gallon of liquid, but a recipe asks for 3/4 of a gallon of a liquid, how many times do I need to fill up the cup to fulfill the requirement?

Divide 3/4 by 1/8.

Its just a thing you need to do sometimes.

Fundamentally it answers the exact same questions you might have about dividing whole numbers.

E.g. how many quarters go into one half? 1/2 ÷ 1/4 = 2

Sometimes you get into a situation where you need to divide one number by another number and both those numbers happen to be fractions. It would be unfortunate if you didn’t know what to do when that happened, wouldn’t it? To give a more tangible example, let’s say I’m making a video game, and in this video game I have a spell that I want to deal more damage the lower the proportion of HP the enemy has, like a big finisher move. To do that, I might write its damage formula as:

Damage = (ATK * 2.5) / X, where ATK is the attack power of my character and X is the percentage HP the enemy has left – if it has 100% HP left, it’s 1, if it has 20% HP left, it’s one fifth (or in decimal form, 0.2). Now, since my character’s ATK value must be a whole number, 2.5x that number will often contain a fraction. If I have 17 ATK, then the number I’m dividing by X is going to be 42.5. Then to get my damage dealt, I can do 42 / 0.2 nice and easy, but then I’ll have to do 0.5 / 0.2. That’s a fraction divided by a fraction.

You’re missing that those rules also apply to units and variables, not just numbers. Can let you simplify an equation into something much easier to work with.

Lets say you have a speed, in meters per second, a fuel burn rate, in liters per second, and want to know how much fuel you’re burning in liters per kilometer. You would divide liters per second by meters per second to get liters per meter, then multiply by 1000 meters per kilometer to get your result in the desired units.

I have 1/2 of a cake, I want to divide it into 1/6 slices. How many slices do I get?