2.6: Division of Rational Numbers
What if you had two numbers like
Watch This
CK12 Foundation: 0206S Dividing Rationals
Try This
For more practice dividing fractions, try the game at http://www.aaamath.com/div66ox2.htm or the one at http://www.mathplayground.com/fractions_div.html.
Guidance
An identity element is a number which, when combined with a mathematical operation on a number, leaves that number unchanged. For example, the identity element for addition and subtraction is zero, because adding or subtracting zero to a number doesn’t change the number. And zero is also what you get when you add together a number and its opposite, like 3 and 3.
Multiplicative Inverses
The inverse operation of addition is subtraction—when you add a number and then subtract that same number, you end up back where you started. Also, adding a number’s opposite is the same as subtracting it—for example,
Multiplication and division are also inverse operations to each other—when you multiply by a number and then divide by the same number, you end up back where you started. Multiplication and division also have an identity element: when you multiply or divide a number by one, the number doesn’t change.
Just as the opposite of a number is the number you can add to it to get zero, the reciprocal of a number is the number you can multiply it by to get one. And finally, just as adding a number’s opposite is the same as subtracting the number, multiplying by a number’s reciprocal is the same as dividing by the number.
The reciprocal of a number
To find the multiplicative inverse of a rational number, we simply invert the fraction—that is, flip it over. In other words:
The multiplicative inverse of
You’ll see why in the following exercise.
Example A
Find the multiplicative inverse of each of the following.
a)
b)
c)
d)
e)
Solution
a) When we invert the fraction
b) Similarly, the inverse of
c) To find the multiplicative inverse of
d) Don’t let the negative sign confuse you. The multiplicative inverse of a negative number is also negative! Just ignore the negative sign and flip the fraction as usual.
The multiplicative inverse of
e) The multiplicative inverse of
Look again at the last example. When we took the multiplicative inverse of
Divide Rational Numbers
Earlier, we mentioned that multiplying by a number’s reciprocal is the same as dividing by the number. That’s how we can divide rational numbers; to divide by a rational number, just multiply by that number’s reciprocal. In more formal terms:
Example B
Divide the following rational numbers, giving your answer in the simplest form.
a)
b)
c)
d)
Solution
a) Replace
b) Replace
c)
d)
Solve RealWorld Problems Using Division
Speed, Distance and Time
An object moving at a certain speed will cover a fixed distance in a set time. The quantities speed, distance and time are related through the equation
Example C
Anne runs a mile and a half in a quarter hour. What is her speed in miles per hour?
Solution
We already have the distance and time in the correct units (miles and hours), so we just need to write them as fractions and plug them into the equation.
Anne runs at 6 miles per hour.
Watch this video for help with the Examples above.
CK12 Foundation: Dividing Rationals
Vocabulary
The multiplicative inverse of a number is the number which produces 1 when multiplied by the original number. The multiplicative inverse of
To divide fractions, invert the divisor and multiply:
An object moving at a certain speed will cover a fixed distance in a set time. The quantities speed, distance and time are related through the equation
Guided Practice
Divide the following rational numbers, giving your answer in the simplest form.
a)
b)
Solution
a) Replace
b) Replace
Practice
For 15, find the multiplicative inverse of each of the following.
 100

28 
−1921  7

−z32xy2
For 610, divide the following rational numbers. Write your answer in the simplest form.

52÷14 
12÷79 
511÷67 
12÷12 
−x2÷57
Dividend
In a division problem, the dividend is the number or expression that is being divided.divisor
In a division problem, the divisor is the number or expression that is being divided into the dividend. For example: In the expression , 6 is the divisor and 152 is the dividend.identity element
An identity element is a value which, when combined with an operation on another number, leaves that other number unchanged. The identity element for addition is zero, the identity element for multiplication is one.Multiplicative Inverse
The multiplicative inverse of a number is the reciprocal of the number. The product of a real number and its multiplicative inverse will always be equal to 1 (which is the multiplicative identity for real numbers).Quotient
The quotient is the result after two amounts have been divided.Real Number
A real number is a number that can be plotted on a number line. Real numbers include all rational and irrational numbers.reciprocal
The reciprocal of a number is the number you can multiply it by to get one. The reciprocal of 2 is 1/2. It is also called the multiplicative inverse, or just inverse.Image Attributions
Description
Learning Objectives
Here you'll learn how to find the multiplicative inverse of a rational number. You'll also learn how to evaluate and simplify rational expressions involving division.