1. Multiplication of Fractions
Review:
Multiplication of whole numbers means repeated addition.
This is also true for multiplying a whole number to a fraction.
e.g.
This shows that in general, we should have

2. Multiplication of Fractions
When we multiply a fraction to a fraction, the meaning is rather different because we cannot repeat a process a fraction of times.
Definition:
means of
We are going to see several examples that can help you understand further.

3. the whole piece of chocolate
Example 1
Jordan got 13 of a chocolate bar from his sister. He ate ½ of it during lunch break and saved the rest for the evening. How much of a chocolate bar did he eat during lunch break?
Let us answer this question by drawing diagrams. (click)
13 of the whole
the whole piece of chocolate
½ of 13
From the last diagram, we see that the yellow piece he ate is equal to 1 out of 6 from the whole bar. Therefore ½ of 13 is 16 .

4. the whole piece of chocolate
Example 1
Jordan got 13 of a chocolate bar from his sister. He ate ½ of it during lunch break and saved the rest for the evening. How much of a chocolate bar did he eat during lunch break?
Let us answer this question by drawing diagrams. (click)
13 of the whole
the whole piece of chocolate
½ of 13
From the last diagram, we see that the yellow piece he ate is equal to 1 out of 6 from the whole bar. Therefore ½ of 13 is 16 .

5. the whole piece of chocolate
Example 1
Jordan got 13 of a chocolate bar from his sister. He ate ½ of it during lunch break and saved the rest for the evening. How much of a chocolate bar did he eat during lunch break?
Let us answer this question by drawing diagrams. (click)
13 of the whole
the whole piece of chocolate
½ of 13
From the last diagram, we see that the yellow piece he ate is equal to 1 out of 6 from the whole bar. Therefore ½ of 13 is 16 .

6. the whole piece of chocolate
Example 1
Jordan got 13 of a chocolate bar from his sister. He ate ½ of it during lunch break and saved the rest for the evening. How much of a chocolate bar did he eat during lunch break?
Let us answer this question by drawing diagrams. (click)
13 of the whole
the whole piece of chocolate
½ of 13
From the last diagram, we see that the yellow piece he ate is equal to 1 out of 6 from the whole bar. Therefore ½ of 13 is 16 .

7. the whole piece of chocolate
Example 1
Jordan got 13 of a chocolate bar from his sister. He ate ½ of it during lunch break and saved the rest for the evening. How much of a chocolate bar did he eat during lunch break?
Let us answer this question by drawing diagrams. (click)
13 of the whole
the whole piece of chocolate
½ of 13
From the last diagram, we see that the yellow piece he ate is equal to 1 out of 6 from the whole bar. Therefore ½ of 13 is 16 .

8. the whole piece of chocolate
Example 1
Jordan got 13 of a chocolate bar from his sister. He ate ½ of it during lunch break and saved the rest for the evening. How much of a chocolate bar did he eat during lunch break?
Let us answer this question by drawing diagrams. (click)
13 of the whole
the whole piece of chocolate
½ of 13
From the last diagram, we see that the yellow piece he ate is equal to 1 out of 6 from the whole bar. Therefore ½ of 13 is 16 .

9. the whole piece of chocolate
Example 1
Jordan got 13 of a chocolate bar from his sister. He ate ½ of it during lunch break and saved the rest for the evening. How much of a chocolate bar did he eat during lunch break?
Let us answer this question by drawing diagrams. (click)
13 of the whole
the whole piece of chocolate
½ of 13
From the last diagram, we see that the yellow piece he ate is equal to 1 out of 6 from the whole bar. Therefore ½ of 13 is 16 .

10. the whole piece of chocolate
Example 1
Jordan got 13 of a chocolate bar from his sister. He ate ½ of it during lunch break and saved the rest for the evening. How much of a chocolate bar did he eat during lunch break?
Let us answer this question by drawing diagrams. (click)
13 of the whole
the whole piece of chocolate
½ of 13
From the last diagram, we see that the yellow piece he ate is equal to 1 out of 6 from the whole bar. Therefore ½ of 13 is 16 .

11. the whole piece of chocolate
Let us answer this question by drawing diagrams.
13 of the whole
the whole piece of chocolate
½ of 13
From the last diagram, we see that the yellow piece he ate is equal to 1 out of 6 from the whole bar. Therefore ½ of 13 is 16 .
According to our definition, ½ of 13 is
Hence

12. whole piece of chocolate
Multiplication of Fractions
Examples 2
On the other day, Jordan got 57 of a chocolate bar from his mom and he gave 23 of that to his younger brother.
How much of the original bar did he give away?
(click)
57 of the whole
whole piece of chocolate
23 of 57

13. 23 of 57 Multiplication of Fractions
In the diagram we see that the whole is cut into 3 columns and 7 rows, hence there should be × 7 = 21 equal pieces in the whole.
The yellow portion has 2 columns and 5 rows, hence it has 2 × 5 = 10 such pieces.
23 of 57
This shows that the yellow portions takes up 1021 of the whole.
i.e.

14. Multiplication of Fractions
Conclusion:
Exercises:
1. Calculate
(after dividing top and bottom by 2)
2. Calculate
3. Calculate

15. Division of Fractions Review: there are two approaches of division
Repeated subtraction: If there are 12 cookies and we want to take away 3 cookies at a time, how many times can we do this until there is none left?
Partition approach: If there are 12 cookies, and we want to separate them into 3 equal groups, how many cookies will be in each group?
When we are dividing by whole numbers, both approaches work fine, but when we are dividing by fractions, the first will be more practical.

16. Why should 3 ÷ 8 be 38 ?
Before we know the existence of fractions, the problem 3 ÷ 8 cannot be solved, and there will be a remainder of 3.
Now we can use fractions, what would be the appropriate answer to 3 ÷ 8 ?
Let’s consider the following example:
There are 3 cakes to be divided evenly among 8 people, how can this be done fairly and how much of a cake will each person get?

17. Solution:
The most logical way to do this is to cut each cake in to 8 equal pieces, and then let each person take 1 piece from each cake. (click to see animation)

18. Solution:
The most logical way to do this is to cut each cake in to 8 equal pieces, and then let each person take 1 piece from each cake.

19. Solution:
The most logical way to do this is to cut each cake in to 8 equal pieces, and then let each person take 1 piece from each cake.
In the end, each person will have 3 equal pieces, and since each piece is 18 , the total amount each person will get is 38 .
Therefore the answer to 3 ÷ 8 should be 38 .

20. Conclusion:
For any non-zero whole numbers a and b, a ÷ b = ab .
Exercises
1. What is 3 ÷ 4?
2. What is 6 ÷ 5?
3. What is 12 ÷ 9?

21. ¾ as 3 copies of ¼. Division of Fractions with a common denominator
What is equal to?
Solution:
We should think 154 as 15 copies of ¼ , and think
¾ as 3 copies of ¼.
In terms of money, this is the same as thinking 154 as 15 quarters, and ¾ as 3 quarters.

22. ¾ as 3 copies of ¼. Solution:
We should think 154 as 15 copies of ¼ , and think
¾ as 3 copies of ¼.
In terms of money, this is the same thinking 154 as 15 quarters, and ¾ as3 quarters.
Once we change to this setting, it is easy to see that the answer is the same as 15 ÷ 3.
i.e is equal to 15 ÷ 3 = 5.

23. Another Example
What is 2110 ÷ 310 ?
Solution:
Again we are going to use money as an analogy.
2110 of a dollar is the same as 21 dimes, and 310 of a dollar is the same as 3 dimes.
Our question is how many times can we take away 3 dimes from 21 dimes until there is none?
Clearly, the answer is 7 because 21 ÷ 3 = 7.

24. Observations: In the above examples, we see that
the repeated subtraction approach works well when the denominators are the same, otherwise it would not even make sense. For instance, taking away dimes from a collection of quarters is impossible (unless we first trade).
the denominator does not appear in the answer, because we are only interested in how many times we can remove 3 dimes from a collection of dimes, and not the value of each dime. (You don’t even need to know how much a dime is worth!)

25. Division of Fractions Conclusion:
When we divide fractions with equal denominators, we can ignore the denominator and divide just the numerators,
i.e.

26. Division of Fractions
Exercises:
1. What is
2. What is
3. What is

27. Division of Fractions with different denominators
Example:
How can we perform the division
It is clear in the following picture that the pieces are not of the same size, and hence we cannot find a pieces of the size 1/5 in 3/4 to take away.

28. Division of Fractions with different denominators
Example:
How can we perform the division
It is clear in the following picture that the pieces are not of the same size, and hence we cannot find a pieces of the size 1/5 in 3/4 to take away.

29. Division of Fractions with different denominators
Example:
How can we perform the division
It is clear in the following picture that the pieces are not of the same size, and hence we cannot find a pieces of the size 1/5 in 3/4 to take away.

30. Division of Fractions with different denominators
Example:
How can we perform the division
It is clear in the following picture that the pieces are not of the same size, and hence we cannot find a pieces of the size 1/5 in 3/4 to take away.

31. Division of Fractions with different denominators
Example:
How can we perform the division
It is clear in the following picture that the pieces are not of the same size, and hence we cannot find a pieces of the size 1/5 in 3/4 to take away.

32. Division of Fractions with different denominators
Example:
How can we perform the division
It is clear in the following picture that the pieces are not of the same size, and hence we cannot find a pieces of the size 1/5 in 3/4 to take away.

33. Division of Fractions with different denominators
Example:
How can we perform the division
It is clear in the following picture that the pieces are not of the same size, and hence we cannot find a pieces of the size 1/5 in 3/4 to take away.

34. Division of Fractions with different denominators
Example:
How can we perform the division
It is clear in the following picture that the pieces are not of the same size, and hence we cannot find a pieces of the size 1/5 in 3/4 to take away.

35. Division of Fractions with different denominators
Example:
How can we perform the division
It is clear in the following picture that the pieces are not of the same size, and hence we cannot find a pieces of the size 1/5 in 3/4 to take away.

36. Division of Fractions with different denominators
Example:
How can we perform the division
It is clear in the following picture that the pieces are not of the same size, and hence we cannot find a pieces of the size 1/5 in 3/4 to take away.

37. Division of Fractions with different denominators
Example:
How can we perform the division
It is clear in the following picture that the pieces are not of the same size, and hence we cannot find a pieces of the size 1/5 in 3/4 to take away.

38. Example:
How can we perform the division
It is clear in the following picture that the pieces are not of the same size, and hence we cannot find a pieces of the size 1/5 in 3/4 to take away.

39. Example:
How can we perform the division
It is clear in the following picture that the pieces are not of the same size, and hence we cannot find a pieces of the size 1/5 in 3/4 to take away.

40. Example:
How can we perform the division
It is clear in the following picture that the pieces are not of the same size, and hence we cannot find a pieces of the size 1/5 in 3/4 to take away.
The only solution is to cut the pieces into smaller ones such that they are all equal in size.
This implies that we need to find a common denominator.

41. Example:
How can we perform the division
The common denominator in this case is clearly 4 × 5 = 20.
and
i.e.
Therefore

42. Another example:
How can we perform the division
The common denominator in this case is clearly 10 × 9 = 90.
i.e.
and
Therefore

43. The general case
How can we perform the division
The common denominator in this case is clearly b × d .
i.e.
and
Therefore