Download Section 1.1: Fractions

College Mathematics Notes
Section 1.1
Page 1 of 14
Chapter 1: Pre-Algebra
Section 1.1: Fractions
Big Idea for this section: A fraction represents how many parts of a whole you have. Thinking of fractions as
actual pieces of something can really help understand how to do arithmetic (adding, subtracting, multiplying,
and dividing) with them.
Big Skills for this section: You should be able to do arithmetic with fractions. That means be able to compare,
add, subtract multiply, and divide fractions.
Section 1.1.1: Meaning of a Fraction
 A fraction is meant to represent the number of pieces of an object you take or use when you divide it
into equal pieces. So, if you cut a pizza into eight equal-size pieces and then take five of them, we are
5
talking about five of the eight equal-sized pieces. We write this amount of pizza as the fraction (of
8
the pizza).
5 
Numerator

There are three pieces to writing a fraction. Their names are:  

The denominator gets its name from the term “denomination”, which means “a value or size of a series
of values or sizes” ( http://www.merriam-webster.com/dictionary/denomination ). (Think of sub-dollar
coins…)

The numerator gets its name from the fact it is the number of pieces of a given size (or denomination)
you have.

Fractions are usually called “rational numbers” in mathematical circles because a ratio is a name for
when you compare how many pieces you have to a total number of pieces.
Fraction Bar
8  Denominator
Practice:
1. If there are 24 problems in a homework assignment, and you and three friends divide up the
problems evenly, write the fraction of the homework assignment each of you has to do.
2. Suppose there are five people in your family, and you buy three big sub sandwiches for the
whole family to share. Write the fraction of a sub that each person gets.
College Mathematics Notes
Section 1.1
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Section 1.1.2: Equivalent Fractions
 One weird fact about fractions is that you can write the same fractional amount using infinitely many
different-looking fractions.







For instance, if you divide a rectangle into five equal strips and then take two of the strips, you have
2
of
5
the rectangle.
But if you divide it into ten equal strips and then take four of the strips, we are talking about a fraction of
4
4 2
 .
of the rectangle, but you can see from the pictures that we have the same amount…
10
10 5
6 2
 .
15 5
Fractions that have different numbers but that represent the same amount are called “equivalent fractions”.
2
Perhaps you can see mathematically what is going on: to convert from the fraction to the equivalent
5
4
fraction of
, you multiply the numerator by 2 and the denominator by 2. Likewise, to convert from the
10
6
fraction to the equivalent fraction of
, you multiply the numerator by 3 and the denominator by 3.
15
6
2
You can also think about it “in reverse”: to convert from the fraction
to the equivalent fraction of , you
15
5
divide the numerator by 3 and the denominator by 3. Here is the general rule we can draw from these
specific cases:
Likewise if you take six out of fifteen strips:
To convert one fraction to another equivalent fraction, you can either:
 Multiply both the numerator and denominator by the same number, or
 Divide both the numerator and denominator by the same number.
College Mathematics Notes
Section 1.1
Page 3 of 14
Practice:
1. Write
3
as three different equivalent fractions.
8
2. Write
4
as an equivalent fraction with a denominator of 36.
9
3. Write
18
as an equivalent fraction with a denominator of 10.
60
Reducing Fractions: “Reducing a fraction” involves writing the fraction as an equivalent fraction with smaller
numbers in the numerator and denominator. Reducing a fraction until the numbers are as small as possible is
called reducing the fraction to “lowest terms”.
2

One way to reduce a fraction is just to do your best to see what you can divide top and bottom by.

Another method for reducing fractions is to do a prime factorization of the numerator and denominator,
then divide out (or “cancel”) all the common factors.

Here is a list of the first twenty prime numbers to aid your factoring:
3
5
7
4.
15

25
5.
52

60
6.
840

1092
Practice:
11
13
17
19
23
29
31
37
41
43
47
53
59
61
67
71
College Mathematics Notes
Section 1.1
Page 4 of 14
Equivalent fractions are also useful for comparing fractions when you don’t have a calculator.
To compare fractions:
 Write each fraction as an equivalent fraction, but with the same denominator.
 The bigger numerator belongs to the bigger fraction.
Practice:
7. Determine which is greater:
7
23
or
.
12
36
8. Determine which is greater:
17
1
or .
128
9
9. Determine which is greater:
5
7
or .
6
9
Section 1.1.3: Improper Fractions and Mixed Numbers

So far, the fractions we have dealt with had values between zero and one… 0 
3
 1 . When a fraction
8
has the numerator less than the denominator, we call it a “proper fraction”.

To write a fractional amount that is between numbers bigger than one, we use either “improper
fractions” or “mixed numbers”.

As an example of a mixed number, suppose I have two pizzas plus
3
more of a pizza. Then I have two
5
3
and three-fifths pizzas, which we should write as 2  pizzas, but people are too lazy to write the plus
5
3
sign, so instead they write 2 .
5
College Mathematics Notes

Section 1.1
Page 5 of 14
An improper fraction is a fraction where the numerator is larger than the denominator. To see how to
3
5
write 2 pizzas as an improper fraction, you can think of the first pizza as being of a pizza (i.e., five
5
5
5
3
of the five equal pieces), the second pizza as more, and then you also have the remaining of a
5
5
pizza. So, you have 5 + 5 + 3 = 13 pieces of pizza, each of which is an equal fifth of a pizza. Since a
fraction is the number of pieces we have over the number of equal pieces in a whole, we can write the
3
13
mixed number 2 as the improper fraction
.
5
5
To convert a mixed number to an improper fraction:
 Multiply the denominator and the whole number part.
 Add that answer to the numerator to get the improper numerator (the denominator stays the same).
Practice:
7
10. Write 3 as an improper fraction.
8
11. Write 5
5
as an improper fraction.
12
To convert an improper fraction to a mixed number:
 Divide the denominator into the numerator.
 The quotient is the whole number part; the remainder is the numerator.
Practice:
12. Write
25
as a mixed number.
6
College Mathematics Notes
13. Write
Section 1.1
Page 6 of 14
320
as a mixed number.
3
Section 1.1.4: Adding and Subtracting Proper Fractions
The key to adding or subtracting fractions is to remember that you can only add together pieces that are the
same size.
 If the denominators are not the same, then you have to change your fractions to equivalent
fractions so that they have the same denominators. Thus you will be adding or subtracting pieces
that are the same size.

You can multiply the denominators together to get a common denominator, but you will do the
least amount of work in the long run if you use the Least Common Denominator (LCD).

The LCD has to have all the factors from every denominator:
College Mathematics Notes
Practice:
7 2
 
1.
9 9
2.
1 1
 
2 4
3.
7 23


12 36
4.
17 1
 
128 9
5.
5 7
 
6 9
6.
17 14
 
20 15
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College Mathematics Notes
Section 1.1
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Section 1.1.5: Adding and Subtracting Mixed Numbers
To add or subtract mixed numbers, first deal with the whole numbers, then deal with the fractions.
1
1
1
1
1  2  1  2 
2
3
2
3
1 1
 3 
2 3
3 2
 3 
6 6
5
 3
6
5
3
6
If the fractions add up to an improper fraction, convert the improper fraction to a mixed number, and then
combine the whole numbers again.
5
11
5
11
2 5  2 5
7
14
7
14
5 11
7 
7 14
10 11
7 
14 14
21
7
14
7
 7 1
14
1
 8
2
1
8
2
College Mathematics Notes
Section 1.1
Page 9 of 14
If subtracting the fractions would result in a negative fraction, borrow from the whole number part to make the
positive fraction bigger.
1
9
1
9
7 5  7 5
2
16
2
16
1 9
2 
2 16
8 9
2 
16 16
16 8 9
1  
16 16 16
24 9
1

16 16
15
1
16
15
1
16
Practice:
3
1
7. 5  3 
4
3
3
1
8. 5  3 
4
3
3
11
9. 5  2 
8
12
College Mathematics Notes
Section 1.1
Page 10 of 14
Section 1.1.6: Multiplying and Dividing Proper and Improper Fractions
Multiplying fractions: One way to remember how to multiply fractions is to think of finding the area of a
rectangle:
The rule for multiplying proper (and improper) fractions is to take numerator times numerator and denominator
times denominator. Example:
5 3 5  3 15
 

8 4 8  4 32
One way to be efficient is to cancel common factors before you multiply:
3 48
 
4 9
3 48
 
4 9
College Mathematics Notes
Section 1.1
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Practice:
3 2
 
1.
7 5
2.
11 7
 
16 2
3.
2 3
 
3 4
4.
4 15
 
5 8
5. 2 
3

4
Dividing fractions: One way to help understand what happens when dividing with fractions:
 If you divide by a number that is bigger than the number one, then you get a smaller number,
because you are dividing something into smaller pieces.

If you divide by a number that is smaller than one, you get a bigger number, because when you
divide something into smaller pieces, then you get more pieces overall.
The rule for dividing proper (and improper) fractions is to take the reciprocal of the second fraction, then
multiply. Example:
5 3 5 4 5  4 5 1 5
   


8 4 8 3 83 23 6
College Mathematics Notes
Practice:
2. 3  4 
3.
3 2
 
7 5
4.
11 33
 
16 2
5.
1
4 
2
6. 4 
1

2
Section 1.1
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College Mathematics Notes
Section 1.1
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Section 1.1.7: Multiplying and Dividing Mixed Numbers
To multiply or divide mixed numbers, you have to convert them to improper fractions, then use the regular rules
for multiplying and dividing fractions just covered.
Practice:
1 2
1. 1  3 
2 3
1 1
2. 4  1 
2 8
College Mathematics Notes
Section 1.1
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Section 1.1.8: Applications of the Arithmetic of Fractions
Practice:
3. If you want to cut a 52 inch board into five equal pieces, find the length of each piece, taking into
account that the saw blade is one-eight of an inch thick.
1
4. A meter is about 1 of a yard. How many yards are there in 100 meters?
4