Multiplying and Dividing Fractions Study Guide
Test Scheduled for Monday, October 27^{th}
This test will focus on multiplying and dividing fractions by whole numbers, other fractions, and mixed numbers. In order to solve these computation problems, you must also remember how to simplify fractions and convert improper fractions and mixed numbers.
Simplifying Fractions
How To Simplify Fractions:
Think of a number that can be divided evenly into both the numerator and the denominator in a fraction.
Divide both the numerator and denominator by that number.
Continue this process until the only common factor of the numerator and denominator is 1.
Example:
Simplify
Think of a number that can divide evenly into 10 and 15. 5 can!!
Divide both 10 and 15 by five. 10 ÷ 5 = 2 and 15 ÷ 5 = 3. So,
=
C onverting Mixed Numbers Improper Fractions
Vocabulary:
Improper Fraction- a fraction where the numerator is greater than the denominator.
Mixed Number- A combination of a whole number and a fraction.
Improper Fraction → Mixed Number
Divide the numerator by the denominator.
The quotient is your whole number.
The remainder is your numerator, the denominator stays the same.
Simplify if necessary.
Example:
Express as a mixed number.
2
4 11
- 8
3
Therefore, as a mixed number is 2 .
Remember: The quotient becomes the whole number in your mixed number, the remainder becomes the numerator, and the divisor becomes the denominator.
Mixed Number → Improper Fraction
Multiply the denominator by the whole number, and add the numerator to
that product, this is your new numerator.
The denominator stays the same.
Example:
+
X
2 =
Multiplying Fractions by Whole Numbers
To find the product of a fraction and whole number by using area model:
E x. 2 X
Shade of each of the first two columns.
A total of has been shaded.
Shade on the third column.
Simplify, if needed. (This problem does not require any simplifying)
To find the product of a fraction and whole number by using an equation:
Turn the whole number into a fraction by placing it over 1.
Multiply the numerator times the numerator and the denominator times the denominator.
Simplify the fraction (if necessary).
Ex. 6 X
X = , which equals 1 .
This mixed number can then be simplified to 1 by dividing the numerator and denominator by 2.
Multiplying Fractions by Fractions
To find the product of two fractions by using a model:
Ex. Find x . Write in simplest form.
Divide the rectangle into 4 rows. Then divide the rectangle into 3 columns.
Shade a section that is unit wide by unit long.
The section that is shaded represents x , or .
To find the product of two fractions by using an equation:
Multiply the numerator times the numerator and the denominator times the denominator.
Simplify the fraction (if necessary).
Ex. x = ÷2 = or
Simplify the fraction first (using cross products) and then multiply.
1
E
2
x. x
x =
Multiplying Mixed Numbers by Mixed Numbers
To find the product of two mixed numbers:
Change the mixed numbers into improper fractions.
Multiply and simplify the fractions as described above. Be sure to change all improper fractions back into mixed numbers!
Ex. 4 x 2
x =
can then be converted into the mixed number 11 .
Finally, 11 can be simplified to 11 by dividing the numerator and denominator by 2.
Dividing Whole Numbers by Fractions (The Model Method)
To find the quotient of a whole number and a fraction:
Represent the whole number (the dividend) in fraction form.
Split up the model into fractional parts based on the denominator.
Create “groups of” the divisor (the fraction’s numerator).
Example: 4 ÷ (4 is the dividend; is the divisor)
Explanation: 4 can be written as
Next, you must rewrite this fraction with the denominator of 4 to match your divisor. It
becomes .
You must then use the numerator of your dividend (16) to split up your model.
Lastly, use the numerator of your divisor (3) to circle groups in your model.
Dividing Whole Numbers by Fractions (The Computation Method)
To find the quotient of a whole number and a fraction:
Represent the whole number (the dividend) in fraction form.
Divide the numerators.
Divide the denominators.
Convert any improper fractions to mixed numbers and simplify (if needed).
Example: 2 ÷ (2 is the dividend; is the divisor)
Explanation: 2 can be written as
Next, you must rewrite this fraction with the denominator of 3 to match your divisor. It becomes .
You are now left with ÷ .
10 ÷ 4 = and 5 ÷ 5 = 1. Your answer is
____
1
Since our denominator is 1, the improper fraction on the numerator would be the same without it.
We can then convert the improper fraction to a mixed number to get 2 , which can be simplified to 2 .
Dividing Fractions by Fractions (The Model Method)
To find the quotient of a fraction and a fraction:
Model the dividend.
Divide the whole number into fractional parts.
Count the number of fractional parts made.
Convert any improper fractions to mixed numbers and simplify (if needed).
Example: Find ÷ . Write in simplest form.
Model the dividend, .
Divide the whole into thirds. ÷ means how many thirds are in .
Count the number of sections that are in .
Dividing Fractions by Fractions (The Computation Method)
To find the quotient of a fraction and a fraction:
Find a common denominator and convert the fractions.
Divide the numerators.
Divide the denominators.
Convert any improper fractions to mixed numbers and simplify (if needed).
Example: Find ÷ . Write in simplest form.
First you must find a common denominator between 4 and 3 and find the equivalent fractions. 12 is the lowest common denominator!
= and = . Therefore, your problem is now ÷ .
You must now divide the numerators and denominators.
9 ÷8 = and 12 ÷ 12 = 1
Our last step is to convert to a mixed number, which is 1 .
Dividing Mixed Numbers by Fractions or Other Mixed Numbers (The Computation Method)
To find the quotient of a mixed number and a mixed number/fraction:
Convert any mixed numbers to improper fractions.
Find a common denominator and convert the fractions.
Divide the numerators.
Divide the denominators.
Convert any improper fractions to mixed numbers and simplify (if needed).
Example: Find 4 ÷ 2 . Write in simplest form.
First you must convert the mixed numbers to improper fractions.
4 = and 2 = . Your problem is now ÷
Next, you must find a common denominator between 5 and 3 and find the equivalent fractions. 15 is the lowest common denominator!
= and = Therefore, your problem is now ÷ .
63 ÷35 = and 15 ÷ 15 = 1
Our last step is to convert to a mixed number. We get 1 , which can be simplified to 1 .
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