Once a person is comfortable with multiplying fractions, dividing one fraction by another becomes fairly straightforward.
An alternative to division by any number (not just a fraction) is “multiplying by the reciprocal”. Dividing by two has the same effect as multiplying by one half. Multiplying by the reciprocal of a number will always produce the exact same result as dividing by the original number.
Using this approach, any division problem can be rewritten as a multiplication problem. This applies to fractions as well, which provides us with a useful approach for dividing one fraction by another: When you need to divide one fraction by another, transform the problem into a multiplication problem. Multiply the fraction in the numerator by the reciprocal of the fraction in the denominator. This is often referred to as “invert and multiply”:
Note that as many divisions as you wish from this problem can be rewritten as multiplication by the reciprocal, and the answer will remain the same:
To divide one fraction by another, invert the fraction in the denominator, then multiply.
Now that multiplication and division of fractions are familiar processes, what happens when a quantity is multiplied by the number 1? Nothing. Since one is the multiplicative “identity”, the original quantity will remain unchanged.
This is a very useful property when dealing with fractions. Consider the following:
Two expressions are equivalent if they can be converted into one another. They may look very different initially, but if they are equivalent and the rules of algebra are followed, one can be made to look exactly like the other.
Two quantities are equivalent if they represent the same number.
Both of the above definitions of equivalence work with fractions. Two fractions are equivalent if they represent the same decimal number. Two fractions are also equivalent if they simplify to the same fraction, which means they will also represent the same decimal number.
To create an equivalent fraction, multiply the original fraction by one… in the form of a fraction whose numerator and denominator are equal:
By using different versions of “one” (changing the number used in both the numerator and denominator), this process can create an infinite number of equivalent fractions:
All of the above fractions will simplify to a value of 4, so they are all equivalent fractions.
The concept of equivalent fractions is important to master at this stage, because it is what allows you to transform fractions with different denominators into fractions with the same denominator… something that is necessary before two fractions may be added or subtracted.
Earlier posts in this series:
Algebra Intro 1: Numbers and Variables
Algebra Intro 2: Addition
Algebra Intro 3: Subtraction
Algebra Intro 4: Negative Numbers, Zero, Absolute Values, and Opposites
Algebra Intro 5: Addition, Subtraction, and Terms
Algebra Intro 6: Multiplication
Algebra Intro 7: Properties of Multiplication
Algebra Intro 8: Division
Algebra Intro 9: Fractions, Reciprocals, and Properties of Division
Algebra Intro 10: Fractions and Multiplication
Posts that continue this series:
Algebra Intro 12: Adding and Subtracting Fractions