Fractions in Algebra - Section 1.2
A fraction is a ratio of two numbers. The number above the fraction bar is called the numerator. The number
below the fraction bar is called the denominator.
5
7
numerator
denominator
A fraction really means the numerator is divided by the denominator.
We will first deal with fractions where the numbers are natural numbers (or counting numbers): 1, 2, 3, 4, 5,
A mixed number consists of the sum of a whole number and a fraction.
23
8
2
3
8
An improper fraction is a fraction where the numerator is larger than the denominator.
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8
In algebraic expressions, it is easier to deal with an improper fraction than a mixed number. Thus, if we encounter
a mixed number in algebra, we normally want to convert it to an improper fraction.
Converting a mixed number to an improper fraction:
1. Multiply the denominator of the fraction by the whole number and add the numerator to this product.
2. Place the result in step 1 over the denominator of the original mixed number.
Convert 2 34 to a improper fraction.
A "visual" of why this procedure works:
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4
1
Converting improper fractions to mixed numbers
1. Divide the denominator into the numerator. Record the quotient and the remainder.
2. Write the mixed number in the following form:
quotient
remainder
original denominator
Convert 29 to a mixed number.
5
A prime number is a natural number greater than one that has only itself and 1 as natural number factors.
A composite number is a natural number greater than 1 that is not a prime number.
Every composite number can be written as a product of prime numbers in a unique way. We call this the
prime factorization of the composite number.
Find the prime factorization of the given composite number:
24
120
2
Two fractions are equivalent if they represent the same value.
The following are all equivalent fractions:
1
2
3
6
5
10
20
40
Writing a fraction in an equivalent form with a smaller denominator is called reducing the fraction (or
simplifying the fraction).
A fraction is in lowest terms when the numerator and denominator have no common factors other than 1.
The value of a fraction does not change if the numerator and the denominator are both multiplied (or divided) by
the same value.
a
b
a c
b c
a
b
c
c
For example,
4
5
4 2
5 2
8
10
Reducing a fraction to lowest terms
1. write the prime factorization of the numerator and denominator.
2. Divide the numerator and denominator by the greatest common factor (i.e. the product of all the factors
common to both).
Reduce to the lowest terms:
1. 15
75
2. 42
24
3
Multiplying fractions
The product of two or more fractions is the product of their numerators divided by the product of their
denominators.
a
b
c
d
a c
b d
Examples: Multiply the following fractions. Write your result in lowest terms.
1. 4
5
3
8
2. 5
3
11
3.
23
5
11
2
4
Recall in division, we divide the dividend by the divisor to get the quotient.
dividend
divisor
quotient
a
b
c
Two fractions are reciprocals of each other if their product is 1. The reciprocal of a is ba since a
b
b
b
a
1.
Dividing Fractions
To divide two fractions, multiply the dividend by the reciprocal of the divisor.
a
b
c
d
a
b
d
c
Examples: Divide the following fractions. Write your result in lowest terms.
1. 2
3
7
15
2. 6
7
3
5
Adding and Subtracting Fractions
The sum or difference of two fractions with common denominators is the sum or difference of the numerators
divided by the common denominator.
a
b
c
b
a
c
b
a
b
c
b
a
c
b
If the two fractions do not have a common denominator, we must find one before adding or subtracting them. That
is, we have to find equivalent fractions with the same denominators for each of the fractions we are adding or
subtracting. The denominator we generally choose to use is the lowest common denominator (or LCD).
Examples: Add or subtract as indicated. Write your result in lowest terms.
3
11
8
11
2. 3 3
8
11
8
3. 1
2
3
5
1.
6
4. 4
3
3
4
Using prime factorization to find the LCD:
5.
3
10
7
12
Using Fractions in Algebra:
1. Determine whether m
2 is a solution to m
3
1m
4
1
2
7
2. Translate the following statements to an expression or equation. Let the variable x represent the number.
a. Two-thirds of a number is 7 more than half that number.
b. Eight less than 3 of a number.
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3. The formula for converting the temperature in degrees Fahrenheit, F, to degrees Celsius, C, is
C
5 F
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32
If the temperature is 77 F, use the above formula to convert it to degrees Celsius.
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