Sep09

Division Problems: Different Models and Examples

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Today we are going to look at division problems: How do we recognize them? What different models are there? What do we do to solve them?

1. Division Problems: Repetition

This is the first type of division problem you are going to learn to do. For example:

In my living room, there are 120 books in total, placed on 6 shelves. Knowing that each shelf has the same number of books, calculate how many books there are on each shelf.

Find:

A total number of objects: there are 120 books in total.
A number of sets: placed on 6 shelves.
The question about the number of things that there are in each set: How many books are on each shelf?

In order to solve this problem, we must think: if there are 120 books in total, spread equally on 6 shelves, to determine how many books there are on each shelf we divide 120 by 6.

Another example of this type of problem is the following:

On a trip through the woods, we picked 80 blackberries, which we used completely making cakes. If we put 4 blackberries in each cake, how many blackberry cakes did we make?

Find:

A total number of objects: we picked 80 blackberries.
A number of sets: we put 4 blackberries in each cake.
The question about the number of things that there are in each set: How many blackberry cakes did we make?

We must think: if we arrange all of the blackberries in groups of 4 across all the cakes, by dividing the 80 blackberries by the 4 that go in each cake, we will obtain the number of cakes.

2. Division Problems: Comparison Scale

In this type of division problem, we compare an amount with another that is greater or less than.

From NYC, a bus going to Louis’ town costs $12, which is 3 times more than it costs to go to Martha’s town. How much does the bus cost to go to Martha’s town?

Find:

A total number of objects: The bus going to Louis’ town costs $12.
A number that expresses the comparison between the second amount and the first: 3 times more than it costs to go to Martha’s town.
The question about the second amount: How much does the bus cost to go to Martha’s town?

In order to solve the problem, we must think: If it costs three times the amount to go to Louis’ town than to Martha’s town, that is to say, that to Martha’s town, it will cost 3 times less. As a result, we will divide 12 ÷ 3 to get the cost to Martha’s town.

3. Division Problems: Formulas Scale

In this type of division problem, we are given formulas, such as speed. For example:

Paul is a bus driver. He told me that every trip he takes is 240 miles and that he travels at an average speed of 60 miles per hour. How long does it take to complete his journey?

Find:

A total distance: every trip he takes is 240 miles.
A speed: he travels at an average speed of 60 miles per hour.
A question about time: How long does it take to make his journey?

In order to solve the problem, we must think: if he maintains a speed of 60 miles per hour, it is to say that every hour that he drives, he covers 60 miles. Also, we know that in total he covers 240 miles. Therefore, in order to know the time it will take, we must divide 240 by 60: His journey lasts 4 hours.

4. Division Problems: Combination or Cartesian Product

In this type of division problems, we will find two or more sets of things or people. Theses sets are combined together to form possible pairs:

In a café, they offer a breakfast combo every Sunday that allows each customer to choose a combination of a drink and an item from the bakery. Knowing that in this café they offer a total of 5 different drinks, and with the various items from the bakery that can make 40 different breakfast combos…

Find:

The number of elements that has the first set: they offer a total of 5 different drinks.
The number of possible combinations between both sets: can make 40 different breakfast combos.
The question, that refers to the number of elements that the second set has: From how many bakery items can they choose?

In order to solve the problem, we must think: each item from the bakery can be combined with each of the 5 different drinks. Therefore, with each item from the bakery can make 5 different breakfast combos. Knowing that in total there will be 40 different breakfast combos, we are able to find out the number of items from the bakery by dividing 40 ÷ 5: They can choose between 8 items from the bakery.

These have been the four principle models of solving division problems.

And this is everything for today. What did you think of this post? Has it helped you to better understand division problems? If you liked it, remember that you can try a free trial of Smartick to learn much more math.

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