Chapter 2 Fractions The Meaning of Fractions Fractions

Chapter 2. Fractions • The Meaning of Fractions • Fractions as Numbers • Equivalent Fractions • Comparing Fractions • Percent

The meaning of fractions • A and B are whole numbers, and B is not zero • If an object is divided in B equal parts, then 1/B of the object is the amount formed by 1 part, and A/B of the object is the amount formed by A parts • If A/B is a fraction, then A is called the numerator and B is called the denominator • Numerator tells you how many parts • Denominator tells you how many of what type of parts

Numerator: the number of parts Denominator: the type or name of parts Fractions are defined in relation to a whole, and this whole can be one object or a collection of objects

What does equal parts mean?

Remarks • If one is asked to divide a shape like the one we saw in the previous slide, into equal parts; one might not know whether the parts should be equal in size (area) and shape or only they need to be the same size. • When stating a problem, make sure it is explicit enough for the students to understand. • In realistic situations, it is seldom a problem to determine what constitutes equal parts. In most cases, equal parts will be defined by dollar value, length, area, volume, or number.

What are Proper and Improper Fractions? is called a proper fraction if A and B are whole numbers, with B not zero, and A < B is called an improper fraction if A and B are whole numbers, with B not zero, and A ≥ B

Two ways of writing an improper fraction

Fractions as Numbers on the Number Line

Representing Improper Fractions on The Number Line How do you write the fraction represented by the following figure? What assumption did you make? The fraction can be written as :

Example: Plot 11/8 on the number line. • How many parts in do I divide the segment between 0 and 1? • Is 11/8 greater or less than 1? • How many 1/8 parts should I “copy” on the number line? • What would be the mixed number form of the fraction?

Example: Plot 11/8 on the number line. • How many parts in do I divide the segment between 0 and 1? • Is 11/8 greater or less than 1? • How many 1/8 parts should I “copy” on the number line? • What would be the mixed number form of the fraction?

Example: Plot 11/8 on the number line. • How many parts in do I divide the segment between 0 and 1? • Is 11/8 greater or less than 1? • How many 1/8 parts should I “copy” on the number line? • What would be the mixed number form of the fraction?

Decimal Representation of Fractions • Any rational number can be represented as a fraction, for example 2=2/1; 0. 5 = ½; 1. 3333… = 4/3; etc. • To represent a fraction A/B as a decimal number, we divide the numerator A by the denominator B

Can two different fractions represent the same number?

Equivalent Fractions In fact, every fraction is equal to infinite number of fractions!

Equivalent Fractions In fact, every fraction is equal to infinite number of fractions!

Equivalent Fractions In fact, every fraction is equal to infinite number of fractions! Example Why ¾ is equivalent with 15/20?

Equivalent Fractions In fact, every fraction is equal to infinite number of fractions! Example. Why ¾ is equivalent with 15/20? X 5

In general, if N is any counting number, and A/B is a fraction of a whole (any object or collections of objects), then: Another way to explain the equivalence relationship above is by multiplying any fraction A/B with 1 in the form N/N, where N is any counting number.

To summarize:

Common Denominators When we work with fractions simultaneously, it is desirable to have common denominators (same denominators)

Example

Example. Write 3/8 and 5/6 with common denominators in three different ways. First Way Calculate the common denominator by multiplying the individual denominators: Then, create equivalent fractions for both given rational numbers:

Second Way Find the smallest number divisible by both 6 and 8. We will call this the lowest common denominator (multiple).

Second Way Find the smallest number divisible by both 6 and 8. We will call this the lowest common denominator (multiple). That number is 24 Then, create equivalent fractions for both given ones:

Third Way Find any number divisible by both 6 and 8 (no matter how loarge) It can be done by multiplying the lowest common denominator by any counting number. For example: Then, create equivalent fractions for both given ones

The Simplest Form of a Fraction A fraction of whole numbers A/B (where B is not zero) is said to be in the simplest form (or lower terms) if there is no whole number other than 1 that divides both A and B evenly. In general:

Example Put the fraction 18/24 in lowest terms (or simplest form):

Alternatively, since 18 and 24 are both even numbers, we can divide both of them by 2: But 9/12 is not in lowest terms since there is 3 that divides both:

The best way: Find the largest number that divides both 18 and 24 (also called the greatest common divisor (factor)). This number for 18 and 24 is 6 since:

Comparing Fractions There are four standard ways of comparing fractions: 1. Converting into decimals 2. Using common denominators 3. Comparing fractions that have common numerator 4. Cross-multiplication

1. Comparing Fractions by Converting to Decimals Recall that we can convert any fraction A/B into a decimal number by simply dividing A to B. Example:

2. Compare Fractions by Using Common Denominators If two fractions have the same denominator, then the one with the greater numerator is greater Example:

3. Comparing Fractions that have Common Numerators. Fractions with the same numerator and increasing denominators become smaller:

Examples

4. Comparing Fractions by Cross-Multiplying. Note: denominators are assumed to be positive, i. e. B>0 and D>0

Example > = > > = =

Percent The word percent, which is usually represented by the symbol %, means “out of a hundred”.

Percent The word percent, which is usually represented by the symbol %, means “out of a hundred”. Percent, Fractions, and Pictures

Solving Percent Problems Three quantities: whole amount, portion, percent There are three types of percent problems: I) Finding the portion when the percent and the whole amount are known II) Finding the percent when the whole amount and the portion are known III) Finding the whole amount when the percent and the portion are known

Solving Percent Problems Ways to solve the percent problems: • Using Algebra • Using Percent Tables • Making Equivalent Fractions

I) Finding the Portion When the Percent and the Whole Amount Are Known: Susie must pay 6% tax on her purchase of $44. 00. How much tax must Susie pay?

Using a percent table to solve the problem: Percent Amount 100% $44 1% 6% $44 100= $0. 44 6 x $0. 44= $2. 64

Using equivalent fractions:

II) Finding the Percent When the Whole Amount and the Portion Are known. Nellie leaves a $9 tip for a meal that costs $50. What percent was Nellie’s tip? Algebraic Solution:

Using equivalent fractions: X 2

Using a percent table to solve the problem: What percent is 9 of 50? Percent 100% 50 = 2% 2% x 9 = 18% Portion (Amount) $50 $1 $9

III) Finding the Whole Amount when the Percent P and the Portion Are Known. A store gave $ 15 000 to schools in the community. This $ 15 000 represents 3% of the store’s annual profit. What is the store’s annual profit? Algebraic Solution:

Using equivalent fractions: X 5000

Using a percent table to solve the problem: Percent 3% 1% 100% Portion (Amount) $ 15 000 3 = $ 5 000 $ 5000 x 100 = 500 000