Mathematics
Grade 10
15 min
Divide fractions and mixed numbers
Divide fractions and mixed numbers
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1
Introduction & Learning Objectives
Learning Objectives
Convert mixed numbers to improper fractions and vice versa with 100% accuracy.
Find the reciprocal (multiplicative inverse) of fractions, whole numbers, and mixed numbers.
Divide any combination of proper fractions, improper fractions, and mixed numbers.
Simplify complex fractions involving multiple operations.
Solve multi-step word problems that require the division of fractions or mixed numbers.
Apply the principles of fraction division to solve for unknown variables in geometric formulas (e.g., finding the height of a triangle given its area and base).
Justify the 'invert and multiply' rule using the concept of multiplicative inverses.
How can you determine the precise number of smaller, 1/4-inch thick steel plates that can be sheared from...
2
Key Concepts & Vocabulary
TermDefinitionExample
Reciprocal (Multiplicative Inverse)The number by which another number can be multiplied to produce the value 1. To find the reciprocal of a fraction, you 'flip' the numerator and the denominator.The reciprocal of 3/5 is 5/3 because (3/5) * (5/3) = 15/15 = 1.
Mixed NumberA number consisting of a whole number and a proper fraction.4 2/3 (four and two-thirds)
Improper FractionA fraction in which the numerator is greater than or equal to the denominator. All mixed numbers must be converted to improper fractions before performing division.14/3 is the improper fraction equivalent of 4 2/3.
Complex FractionA fraction where the numerator, the denominator, or both contain a fraction. A complex fraction is another way of representing the division of fractions.(1/2) /...
3
Core Formulas
The Rule of Fraction Division
\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}
To divide one fraction by another, multiply the first fraction (the dividend) by the reciprocal of the second fraction (the divisor). This is often called the 'invert and multiply' or 'keep, change, flip' method.
Converting a Mixed Number to an Improper Fraction
A\frac{b}{c} = \frac{(A \times c) + b}{c}
To convert a mixed number to an improper fraction, multiply the whole number by the denominator, add the numerator to the result, and place this new value over the original denominator. This is a mandatory first step when dividing mixed numbers.
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Challenging
Which statement provides the mathematical justification for the 'invert and multiply' rule used in fraction division, such as in the problem a/b รท c/d?
A.Division is the inverse of multiplication, so the signs must be swapped.
B.Multiplying by the reciprocal (d/c) is equivalent to dividing by the original fraction (c/d) because it transforms the divisor into 1 while maintaining the expression's value.
C.The commutative property of division allows the terms to be rearranged.
D.Finding a common denominator and then dividing the numerators is too complex, so a shortcut was invented.
Easy
Which of the following correctly converts the mixed number 7 5/8 into an improper fraction?
A.35/8
B.61/8
C.56/5
D.12/8
Easy
What is the reciprocal, or multiplicative inverse, of the mixed number 3 1/4?
A.4/13
B.13/4
C.4/3
D.3/4
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