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Partitive and quotative division
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True or false? Quotative division answers the question: "How much of the dividend will each one in the divisor get, if they are to split it in equal parts?"
Have you ever tried using division? Have you noticed that there are two different kinds of division? You haven’t? Listen to this! We are going to calculate four divisions.
Two of them are of one kind, two of them of another kind. Listen carefully, and see if you can spot which of them belong together! Let’s go! One Michael has 20 rabbits and 5 cages. How many rabbits will have to share each cage?
20 divided by 5 equals 4. There will be four rabbits in each cage. Two Michael has 20 rabbits. Four rabbits can fit in one cage. How many cages does Michael need?
20 divided by 4 equals 5. He will need five cages. Three A rope that is 24 meters long is to be split into 4 equally long sections. How long is each section? 24 divided by 4 equals 6.
Each section of the rope is 6 meters. Four A rope that is 24 meters long is to be split into 6-meter long sections. How many sections will there be? 24 divided by 6 equals 4. The rope is long enough for 4 sections.
Did you catch all of that? Here comes all the division in one image. The first and the third division are calculated like this: You have a number of something (rabbits or meters of rope), and you divide it into a number of equally sized groups. So, you take the total number divided by the number of groups. The answer, the quotient, tells you how many there will be in each group.
This is called partitive division. Partitive division answers the question: how much of the dividend will each one in the divisor get, if they are to split it in equal parts? What about the two other divisions - the second and fourth? There we also have a certain number of things. But, you weren’t told how many groups to part them into.
Instead you were told how many items should go in each group. You take the total number divided by the number in each group. Now, the quotient tells you the number of groups. This is called quotative division. Quotative division answers the question: how many times can I fit the divisor into the dividend?
You calculate both types of division in just the same way, but they mean different things. You might find partitive division easier to understand, but it’s worth practicing both kinds. Take this division: 14 divided by 0.5. Can you think of a tangible example that illustrates this expression? If you read it as partitive division, it might sound like this: you have 14 cookies and share them between 0.5 persons.
But how on earth do you share something with half a person? Let’s try quotative division instead: you have 14 cookies, and each person is to get half a cookie - how far will the cookies go? Now it’s much easier to see why the quotient is 28, right? Here’s another example: 800 divided by 400. Which kind of division do you think makes most sense to understand this particular expression?
Pause and try on your own to think of two meaningful examples fitting this expression, one using partitive division and one using quotative division. When you have your two examples, hit play again and you’ll get two examples to compare yours with. Partitive division: think about money. You have 800 and are to split that amount among 400 friends. How much will each person get?
Quotative division: you still have 800. Each person is to get 400. How many people can get their cut before you run out of money? Division: one basic operation that can be used in two different ways.