Math 300
BasicCollege Mathematics
Chapter 2
Multiplying and Dividing Fractions
2.1 Introduction of Fractions and Mixed Numbers
The following are some examples of fractions:
This way of writing number names is called fraction notation. The top number is called the numerator and the bottom number is called the denominator.
Example: Identify the numerator and the denominator.
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2.
2.1 Introduction of Fractions and Mixed Numbers (cont)
The number 1 in Fraction Notation
Any number divided by itself is equal to 1.
Example: Simplify.
The number 0 in Fraction Notation
Zero divided by any number is equal to 0.
Example: Simplify.
Excluding Division by 0
Any number divided by zero is “undefined” or “not defined”.
Example: Simplify.
Division by 1
Any number divided by 1 is equal to that number.
Example: Simplify.
2.1 Introduction of Fractions and Mixed Numbers (cont)
When you are given one picture and have to determine which part is shaded or indicated, you can write a fraction for the picture.
To write the correct fraction for the picture, you must understand the denominator of the fraction is the number of equal parts in which the picture is divided into and the numerator is the number of parts that are shaded on the picture.
Example: What part is shaded?
The picture is divided into 4 equal parts.
There is only 1 shaded area.
Therefore, the fraction is:
Writing Fraction from Real-Life Data
Example: Write each fraction.
1. In a family with 11 children, there are 4 boys and 7 girls. What fraction of the children is girls?
2.1 Introduction of Fractions and Mixed Numbers (cont)
Identifying Proper Fractions, Improper Fractions, and Mixed Numbers
A proper fraction is a fraction whose numerator is less than its denominator. Proper fractions are less than 1.
An improper fraction is a fraction whose numerator is greater than or equal to its denominator. Improper fractions are greater than or equal to 1.
A mixed number contains a whole number and a fraction. Mixed numbers are greater than 1.
Examples: Identify each number as a proper fraction, improper fraction, or mixed number.
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2.1 Introduction of Fractions and Mixed Numbers (cont)
There are some cases of when you will see more than one picture that is the same picture. This represents fractions that are greater than 1.
In that case, the write the correct fraction, you would identify how many equal parts is in each picture for your denominator and the numerator will consist of how many shaded areas there are altogether.
Example: What part is shaded?
In the picture above, there are 3 equal parts in each.
There are 7 shaded areas.
Therefore, the fraction is:
2.1 Introduction of Fractions and Mixed Numbers (cont)
Writing a Mixed Number as an Improper Fraction
To write a mixed number as an improper fraction, you would multiply the denominator times the whole number add the numerator and write it over the original denominator.
Example: Write each as an improper fraction.
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2.
Writing an Improper Fraction as a Mixed Number or a Whole Number
To write an improper fraction as a mixed number or a whole number, you would divide the denominator into the numerator.
You answer would be written as follows:
Whole number or just a whole number
Example: Write a mixed number or a whole number.
3. 5.
4. 6.
2.2 Factors and Prime Factorization
Factors
A number is a factor of another number if it divides evenly into it.
Example: Find all the factors of 15.
1, 3, 5, 15
Because all four numbers will divide evenly into 15.
Prime, Composite, & Neither
A prime number is a natural number that has exactly two different factors, 1 and itself.
Example: 2 is a prime number, because 1 2 = 2
4 is not a prime number, because
1 4 = 4, 2 2 = 4
A composite number is any natural number, other than 1, that is not prime.
Example: 6 is composite, because
1 6 = 6 and 3 2 = 6
The natural number 1 is neither prime nor composite.
2.2 Factors and Prime Factorization (cont)
Prime factorization is to express a number as a product of prime numbers only!!
There are 2 ways to do prime factorization.
1.) Prime factor using division
2.) Prime factor tree
Way # 1: Prime factoring using division
Example:Find the prime factorization of 72
First, make sure you know the list of primes
(at least the first 11 prime numbers)
Then write the number that you are prime factoring like this:
Put the first prime number that you are dividing by in front like this:
2
(Note: you may start dividing by any prime number that will divide evenly into that number.)
Continue dividing until you reach one at the top like this:
1
24
12
36
72
2
2
3
3
2
Once you reach one at the top, STOP DIVIDING!!!
Your prime factorization is the numbers that appear down the left side.
Write your answer like this: 72 = 2 2 2 3 3
2.2 Factors and Prime Factorization (cont)
Way #2: Prime factoring using the factor tree
Example:Find the prime factorization of 72
Begin with the number that has to be factor.
72
Choose two factors of that number. (Two factors that when you multiply them together, you get the number in which your factoring.)
72
8 9
For each number that is not prime, factor it by choosing 2 more numbers that are factors of that number.
For each number that is prime, bring down to the next row.
72
8 9
2 4 3 3
2 2 2 3 3
Your end result should be ALL PRIME NUMBERS!!!
The prime factorization of 72 is 2 2 2 3 3
2.2 Factors and Prime Factorization (cont)
Rules for Divisibility by 2, 3, 4 or 5
If a number is even (the last digit is 0, 2, 4, 6, 8), then it will divide by 2!!
Example: 4786 is divisible by 2, because the last digit is 6.
If you add up the digits in a number and the result divides evenly by 3, then the number will divide evenly by 3!!
Example: 201 is divisible by 3, because 2 + 0 + 1 = 3
If the last two digits of a number is divisible by 4, then the number is will divide evenly by 4.
Example: 1712 is divisible by 4, because the last two digits (12) is divisible by 4.
If a number ends in 0 or 5, then it will divide by 5.
Example: 790 is divisible by 5, because the number ends in 0.
2.2 Factors and Prime Factorization (cont)
Rules for Divisibility by 6, 9, or 10
A number is divisible by 6 if it is both an even number and divisible by 3.
Example: 810 is divisible by 6, because it is an even #
and
8 + 1 + 0 = 9, well 9 is divisible by 3.
A number is divisible by 9 if the sum of their digits is divisible by 9.
Example: 29, 223 is divisible by 9, because
2 + 9 + 2 + 2 + 3 = 18.
A number is divisible by 10 if the number had a zero in the ones place.
Example: 300 is divisible by 10, because there is a zero in the ones place.
2.3 Simplest Form of a Fraction
To simplify a fraction means to write a fraction in simplest or lowest terms. A fraction is considered in simplest or lowest terms when the numerator and denominator have no common factors other than 1.
Example: Simplify.
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2.3 Simplest Form of a Fraction (cont)
A Test for Equality
A test for equality is how we compare fractions.
The signs that you would use is either equal to or not equal to.
You would use cross products or you may find common denominators to determine the equality.
Example: Determine whether each pair of fractions is equivalent.
1. and
2. and
Solving Problems by Writing Fractions in Simplest Form
Example: Write each fraction in simplest form.
3. Two thousand baseball caps were sold one year at the U.S. Open Golf Tournament. What fractional part of this total does200 caps represent?
2.4 Multiplying Fractions
Multiplying a Fraction and a Whole Number
To multiply a fraction and a whole number, you would multiply the numerator (top number) by the whole number and keep the denominator (bottom number) the same.
Example: Multiply.
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Multiplying a Fraction by a Fraction
To multiply a fraction by a fraction, you would multiply the numerator times numerator and denominator times denominator.
Example: Multiply.
2.
2.4 Multiplying Fractions (cont)
Multiplying and Simplifying
This section is where you would put both multiplying and simplifying together in one problem.
There are 2 ways you can approach multiplying and simplifying:
- You may multiply the fractions first, then simplify your answer
Or
- You may simplify first by using cancelling correctly, then multiply your answer.
Example: Multiply and simplify.
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2.
2.4 Multiplying Fractions (cont)
Multiplying Fractions and Mixed Numbers
To multiply with mixed numbers, first write the mixed number as a fraction and then multiply.
Example: Multiply.
1.
Solving Problems by Multiplying Fractions
To solve real-life problems that involve multiplying fractions, use the steps for problem-solving steps from Chapter 1. A new key word that implies multiplication is used…that key word is “of.”
Example: Solve. Write answer in simplest form.
3. A recipe calls for of a cup of flour. How much flour should be used if only of the recipe is being made?
2.5 Dividing Fractions
To find a reciprocal of a fraction, interchange the numerator and the denominator. (In other words, flip the fraction over.)
To find a reciprocal of a whole number, first place the whole number over 1 to make it a fraction then flip the fraction over.
Zero has no reciprocal.
Example: Find the reciprocal.
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2. 9
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2.5 Dividing Fractions (cont)
Division of Fractions
Steps for dividing fractions:
- Keep the first fraction the same.
- Change the division symbol to a multiplication symbol .
- Take the reciprocal of the second fraction.
- Multiply and simplify.
Example: Divide and simplify.
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2.5 Dividing Fractions (cont)
To divide with a mixed number, first write the mixed number as a fraction and then follow the steps for dividing a fraction.
Example: Divide.
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2.5 Dividing Fractions (cont)
To solve real-life problems that involve dividing fractions, we continue to use our problem-solving steps from Chapter 1.
Example: Write each answer in simplest form.
1. If there are grams of fat in 4 ounces of lean hamburger meat, how many grams of fat are in an ounce?
2. A heart attack patient in rehabilitation walked on a treadmill miles over 4 days. How many miles is this per day?
Math 300 Chapter 2
Glossary
Fraction – number used to express equal parts of a whole.
Numerator - Top number in a fraction, it is the number of parts of the whole being considered.
Denominator – Bottom number in a fraction, it is the number of equal parts in the whole.
Proper fraction - a fraction with the numerator less than the denominator.
Improper fraction – a fraction with the numerator greater than or equal to the denominator.
Mixed Number - A number and a fraction together.
Natural number – Natural numbers are whole numbers starting with 1.
Prime number – a natural number that has exactly two different factors, 1 and the number itself.
Composite number – any natural number other than 1 that is not a prime number.
Prime factorization – factorization of a number such that all the factors are prime number.
Equivalent fractions – two fractions that represent the same portion of a whole. They are equal.
Simplest form of a fraction - a fraction is in its simplest form when the numerator and denominator have no common factors other than 1.
Reciprocal - Two numbers are reciprocals of each other when their product is 1.
Cross product – used to determine if two fractions are equivalent. The cross product of a/b and c/d is (a)(d) and (b)(c). If those products are equal, then the fractions are equal.
Properties
Multiplication of fractions:
a/b * c/d = a * c / b * d
Division of fractions:
a/b divided by c/d is a/b * d/c
Math 300 Chapter 2
Glossary (cont)
Hints
The mixed number 2 ¼ represents 2 and ¼
Convert a mixed number to an improper fraction
- Multiply the denominator of the fraction by the whole number
- Add the numerator of the fraction to the product from step 1 above
- Write the sum from 3 above as the numerator of the improper fraction with the denominator from the original fraction.
Writing an improper fraction as a mixed number
- Divide the denominator into the numerator
- The whole number part of the mixed number is the quotient. The fractional part of the mixed number is the remainder from step 1 above over the original denominator.
When converting an improper fraction to a mixed number, if the remainder is 0, then the answer is just a natural number without the fraction.
Prime factorization
- Factor tree
- Division
Multiplication is commutative so 2x2x3 = 2x3x2 = 3x2x2 = 22x3
Divisibility tests – a whole number is divisible by:
- 2 if the last digit is even (0, 23, 4, 6, 8)
- 3 if the sum of the digits is divisible by 3. For example, 255 = 2+5+5=12 and 12 is divisible by 3
- 5 if the last digit is a 0 or a 5
Math 300 Chapter 2
Glossary (cont)
Hints (continued)
- 4 if the last two digits are divisible by 4. For example, 324 is divisible by 4 since 24 is divisible by 4
- 6 if it is divisible by 2 and 3
- 9 if the sum of the digits is divisible by 9
Factor Tree
36
123
43
22
2x2x3x3=22 x 32
Any nonzero number when divided by itself is 1. 12/12 = 1
When putting a fraction into simplest form, it is helpful to write the prime factorizations of the numerator and denominator and then removing any common factors.
When simplifying a fraction, remember if all of the prime factors can be removed there is still 1 left.
To determine if two fractions are equal, just find the cross product.
It is not always necessary to do a complete prime factorization to simply a fraction. For example, if both the numerator and denominator end in 0, you can divide both by 10 (not just 2 and then 5).
Multiplication of mixed number is done by converting the mixed numbers into improper fractions and multiplying.
The reciprocal of a fraction if done by interchanging the numerator and denominator. The product of the reciprocals is 1.
Math 300 Chapter 2
Glossary (cont)
Hints (continued)
To divide two fractions, simply multiply the first fraction time the reciprocal of the second fraction.
Dividing fractions and mixed number is done by converting all mixed number s to improper fractions and then divide by multiplying the first fraction by the reciprocal of the second.
When dividing fractions, do not remove common factors until after the multiplication is complete.
Math 300 M-G Chapter 2; Rev: Jan 2009Page 1 of 25