Download Multiply/Divide Fractions

Math 0300
Multiplication and Division of Fractions
Multiplication of Fractions
The product of two fractions is the product of the numerators over the product of the
denominators.
a c ac
× =
b d bd
Example:
Multiply:
Where b ≠ 0 and d ≠ 0
2 1
×
5 3
2 1 2 ×1 2
× =
=
5 3 5 × 3 15
(Multiply the numerators and the denominators)
If a is a natural number, then 1 is called the reciprocal or the multiplicative
a
inverse.
The product of a number and its multiplicative inverse is 1.
Example:
1
1
×8 = 8× = 1
8
8
The product:
Of an odd number of negative fractions is negative
Of an even number of negative fractions is positive
Student Learning Assistance Center - San Antonio College
1
Math 0300
Example 1:
− 3 − 2 − 10
×
×
8
5
21
⇒ There are an odd number of negative fractions, so the product will be negative.
⇒ Use the Order of Operations Agreement. Multiply the first two fractions. The
product is positive.
−3 −2
×
8
5
⇒ The product of the first two fractions and the third fraction is negative.
 3 × 2 × 10 
−

 8 × 5 × 21 
⇒ Multiply the numerators and the denominators.
 60 
−

 840 
⇒ Write the product in the simplest form.
First write the prime factorization of each number
3× 2 × 2 × 5 

−

 2 × 2 × 2 × 5× 3× 7 
Then strikethrough the common factors


3× 2× 2×5
−

 2× 2 × 2 × 5 × 3 ×7 
Write what you have left.
−
1
14
Numerator = 1, because it is a factor of all values.
Denominator = 2 × 7
Student Learning Assistance Center - San Antonio College
2
Math 0300
Example 2:
3×
Multiply:
5
8
⇒ Write the whole number 3 as a fraction
3×
3
1
5 3 5
= ×
8 1 8
⇒ Multiply the fractions.
There are no common factors in the numerator and denominator.
3 5
= ×
1 8
⇒ Write the improper fraction as a mixed number
=
Example 3:
Is
⇒ Replace “ x ” with
15
7
=1
8
8
−2
−1
3
a solution of the equation x =
?
3
4
2
−2
and then simplify
3
3  −2  −1
×  =
4  3  2
 3 2  −1
− ×  =
 4 3 2
 3 × 2  −1
−
=
 2× 2×3  2
⇒ The result is
−1 −1
=
2
2
YES,
−2
is a solution of the equation.
3
Student Learning Assistance Center - San Antonio College
3
Math 0300
Division of Fractions
♦ The reciprocal of a fraction is the fraction with the numerator and denominator
interchanged.
♦ Inverting the fractions is the process of interchanging the numerator and the
denominator of a fraction.
The reciprocal of
a
b
The reciprocal of
Example:
is
b
a
4
3
is
4
3
⇒ To find the reciprocal of a whole number rewrite the whole number as a fraction with a
denominator of 1. Then invert the fraction.
6=
So, the reciprocal of 6 is
6
1
1
.
6
Division of Fractions
To divide two fractions, multiply by the reciprocal of the divisor.
a c a d
÷ = ×
b d b c
Where b ≠ 0, c ≠ 0, and d ≠ 0.
SIGN RULES FOR DIVIDING POSITVE AND NEGATIVE FRACTIONS
(Same as dividing integers)
The quotient of two numbers with the same sign is positive.
The quotient of two numbers with opposite signs is negative.
Student Learning Assistance Center - San Antonio College
4
Math 0300
Example 1:
− 7 − 14
÷
10
15
Simplify
⇒ The signs are the same. The quotient is positive.
=
7 14
÷
10 15
⇒ Rewrite the division as multiplication by the reciprocal.
=
7 × 15
10 × 14
⇒ Multiply and simplify the fractions.
=
7 × 3× 5
3
=
2×5× 2× 7 4
To divide a fraction and a whole number, first write the whole number as a fraction with a
denominator of 1.
Example 2:
Find the quotient of
2
and 4.
3
⇒ Write the whole number 4 as the fraction
=
4
.
1
2 4
÷
3 1
⇒ Rewrite the division as multiplication of the reciprocal.
=
2 1
×
3 4
=
2 ×1
3× 4
=
2 ×1
1
=
3× 2 × 2 6
⇒ Multiply the fraction.
Student Learning Assistance Center - San Antonio College
5
Math 0300
When a number in a quotient is a mixed number, first write the mixed number as an
improper fraction. Then divide the fractions.
Example 3:
1
Divide 2 ÷ 1 .
4
⇒ Write the mixed number 1
=
1
as an improper fraction
4
5
 .
4
2 5
÷
3 4
⇒ Rewrite the division as multiplication by the reciprocal.
=
2 4
×
3 5
=
2× 4 8
=
3 × 5 15
⇒ Multiply the fractions.
Student Learning Assistance Center - San Antonio College
6