This document discusses improper fractions and mixed fractions. It defines fractions as numbers of the form N/D where N and D are whole numbers and D is not equal to 0. It explains that an improper fraction is one where the numerator is greater than or equal to the denominator, meaning it represents one whole or more. A mixed fraction represents an improper fraction as a whole number plus a proper fraction, such as 1 1/2. The document provides examples of converting between improper and mixed fractions using long division or multiplication.

1. Improper Fractions and Mixed Fractions

2. Improper Fractions and Mixed Fractions
Fractions are numbers of the form N (or N/D) where N, D are
D
whole numbers* where D ≠ 0.
* We will also use other type of numbers to make fractions later.

3. Improper Fractions and Mixed Fractions
Fractions are numbers of the form N (or N/D) where N, D are
D
whole numbers* where D ≠ 0.
3
6
* We will also use other type of numbers to make fractions later.

4. Improper Fractions and Mixed Fractions
Fractions are numbers of the form N (or N/D) where N, D are
D
whole numbers* where D ≠ 0.
Fractions are numbers that measure parts of whole items.
3
6
* We will also use other type of numbers to make fractions later.

5. Improper Fractions and Mixed Fractions
Fractions are numbers of the form N (or N/D) where N, D are
D
whole numbers* where D ≠ 0.
Fractions are numbers that measure parts of whole items.
Suppose a pizza is cut into 6 equal slices and we have 3 of
of the pieces, the fraction that represents this quantity is 3 .
6
3
6
* We will also use other type of numbers to make fractions later.

6. Improper Fractions and Mixed Fractions
Fractions are numbers of the form N (or N/D) where N, D are
D
whole numbers* where D ≠ 0.
Fractions are numbers that measure parts of whole items.
Suppose a pizza is cut into 6 equal slices and we have 3 of
of the pieces, the fraction that represents this quantity is 3 .
6
3
6
The bottom number is the
number of equal parts in the
division and it is called the
denominator.
* We will also use other type of numbers to make fractions later.

7. Improper Fractions and Mixed Fractions
Fractions are numbers of the form N (or N/D) where N, D are
D
whole numbers* where D ≠ 0.
Fractions are numbers that measure parts of whole items.
Suppose a pizza is cut into 6 equal slices and we have 3 of
of the pieces, the fraction that represents this quantity is 3 .
6
3
6
The bottom number is the
number of equal parts in the
division and it is called the
denominator.
* We will also use other type of numbers to make fractions later.

8. Improper Fractions and Mixed Fractions
Fractions are numbers of the form N (or N/D) where N, D are
D
whole numbers* where D ≠ 0.
Fractions are numbers that measure parts of whole items.
Suppose a pizza is cut into 6 equal slices and we have 3 of
of the pieces, the fraction that represents this quantity is 3 .
6
3
6
The top number “3” is the
number of parts that we
have and it is called the
numerator.
The bottom number is the
number of equal parts in the
division and it is called the
denominator.
* We will also use other type of numbers to make fractions later.

9. Improper Fractions and Mixed Fractions
Fractions are numbers of the form N (or N/D) where N, D are
D
whole numbers.
Fractions are numbers that measure parts of whole items.
Suppose a pizza is cut into 6 equal slices and we have 3 of
of the pieces, the fraction that represents this quantity is 3 .
6
3
6
The top number “3” is the
number of parts that we
have and it is called the
numerator.
The bottom number is the
number of equal parts in the
division and it is called the
denominator.
3/6 of a pizza
* We will also use other type of numbers to make fractions later.

10. Improper Fractions and Mixed Fractions
The numerator of
7 (pizza)
6
is larger than the denominator,
means that we are taking 7 slices of 1/6-pizza,
hence we have more than one whole pizza.
7/6 pizzas
We say the fraction N is improper
D
if the numerator N is not less than the denominator D,
so both 6 , 7 are improper fractions.
6
6
An improper fraction is at least 1 or it’s always 1 or more.

11. Improper Fractions and Mixed Fractions
7/6 pizzas
Since 7/6 pizza consists of one whole pizza and a 1/6-pizza,
therefore we also write 7 as 1 1
6
6
and it’s read as “one and one sixth”.
This is called the mixed–form of an improper fraction
which gives the number of whole one(s) and the extra
fractional bits, if any.
So 6 = 1,
6
7
1
=1 ,
6
6
8
= 1 2 = 1 1 , and 12 = 2, etc..
6
6
3
6

12. Improper Fractions and Mixed Fractions
We convert an improper fraction into the mixed form by long
division.
* The quotient is the whole number part of the mixed form.
* The remainder is the number of slices.
Example A. a. Convert 23 in the mixed form.
4
23
3 = 5 3
23 ÷ 4 = 5 with remainder 3. Hence, 4 = 5 +
4
4 .
In general, its easier to execute the long division if the
improper fraction is simplified first.
b. Reduce 505 first, then convert it to the mixed form.
15
Reduce 505 = 505/5 = 101 . By long divisor, we obtain that
15/5
3
15
505 101
2
Hence,
101 ÷ 3 = 33 with remainder 2.
15 = 3 = 33 3 .
(Verify that we get the same answer without reducing it first.)

13. Improper Fractions and Mixed Fractions
1
Conversely, to convert 1 6 pizzas to the improper form
means to count the total number of 1/6-slices.
and
1 pizza
6
The whole pizza may be counted as 6 slices,
7
1
so all together we have 7 slices or that 1 1 = 6 * 6 + 1 = 6 .
6
If we have 2 1 pizzas, the 2 whole pizzas may be counted as
6
1 pizza
2 x 6 = 12 slices, so in total we have 12 + 1 = 13 slices,
or that 2 1
6
6*2+1
=
6
= 13 .
6

14. Improper Fractions and Mixed Fractions
Conversely, the improper form of the mixed fraction A N is
D
then add
N
D*A+N
=
D
D
A
.
multiply
Example C: Put 5 3 into improper form.
4
then add
5 3 = 5
4
3
4
multiply
=
4*5 + 3
23
=
4
4

15. Improper Fractions and Mixed Fractions
Exercise. A. Reduce the following fractions.
4 8 15 24 30 54 60 72
6 , 12 , 9 , 18 , 42 , 36 , 48 , 108
B. Convert the following improper fractions into Mixed
Fractions then convert the Mixed Fractions back to the
improper form.
1. 9
2. 11 3. 9
4. 13
5. 37 6. 86 7. 121
3
4
5
12
11
2
17