# How to Tame Them How to Tame Them

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How to Tame Them How to Tame Them
Fractions How to Tame Them Fractions Difficulty in handling fractions is probably the most common students have in understanding and applying mathematics. This module is intended to help in solving the cognition problem associated with fractions. By calling things by their correct names and carefully defining each part, following up with examples, it is hoped that clarity will be brought to the topic. With focus and a little effort the student can master the concept and usage of fractions – they will indeed be tamed. 6/25/2013 Fractions

Basic Parts Numerator and Denominator
How to Tame Them Basic Parts Numerator and Denominator For real numbers A and B with B ≠ we can write a fraction as Denominator shows the denomination to be counted Halves, thirds, fourths, fifths, tenths, etc. Numerator A B 1 A B = Denominator Numerator and Denominator We begin by naming the parts of a fraction and what each represents. By custom, fractions are written in a form that represents a ratio that compares the standard unit to be counted with how many units there are. The horizontal line is the demarcation between how many units there are above the line (the numerator) and the size of the units below the line (the denominator). The things we’re going to count are named in the denominator (thirds, fourths, dimes, dollars, pesos, quarts, pints, etc.). It should be clear that when we say we are counting thirds, for example, there is a 3 in the denominator, representing the fraction 1/3, the 1/B shown in the illustration. Fractions Fractions 2 5/25/2013 6/25/2013 6/25/2013 Fractions

Basic Parts Numerator and Denominator 1 A A • =
How to Tame Them Basic Parts Numerator and Denominator Numerator enumerates, or counts, how many items of a given denomination Three halves, two thirds, four thirds, six tenths, etc. A B Numerator Denominator A B 1 = Fractions Fractions 3 5/25/2013 6/25/2013 6/25/2013 Fractions

Basic Parts Numerator and Denominator Fraction written as a ratio
How to Tame Them Basic Parts Numerator and Denominator Fraction written as a ratio Three halves is ratio three-to-two or A fraction is a rational number A B Numerator Denominator A B 1 = 3 2 1 2 3 = Fractions Fractions 4 6/25/2013 5/25/2013 6/25/2013 Fractions

Multiplying Fractions
How to Tame Them Multiplying Fractions Simplest of All Fraction Operations General Rule: Multiply numerators together and multiply denominators together Form: A B C D = = 3 8 1 2 4 Example: –1 4 3 2 = –3 8 = 3 –8 = 3 8 Example: Fractions Fractions 5 5/25/2013 6/25/2013 6/25/2013 Fractions

How to Tame Them Adding Fractions Same Denominator Adding fractions requires a single denomination or denominator Example: 3 pints + 4 pints = 7 pints 4 gallons + 5 gallons = 9 gallons Fractions Fractions 6 6/25/2013 5/25/2013 6/25/2013 Fractions

How to Tame Them Adding Fractions Same Denominator Adding fractions requires a single denomination or denominator 2 3 4 + = 2 + 4 3 Example: = 6 3 = 2 NOTE: The sum is reduced to lowest terms Fractions Fractions 7 5/25/2013 6/25/2013 6/25/2013 Fractions

How to Tame Them Adding Fractions Common Denominator Adding fractions requires conversion to a single denomination, i.e. a common denominator Since the sum can have only one denomination Example: Add 3 pints to 4 gallons Fractions Fractions 8 6/25/2013 5/25/2013 6/25/2013 Fractions

Adding Fractions Common Denominator Example: •
How to Tame Them Adding Fractions Common Denominator Example: Add 3 pints to 4 gallons 4 gallons = 8 pints 1 gallon 4 gallons = 32 pints Thus 3 pints + 4 gallons = 3 pints + 32 pints = 35 pints Question: What if we converted all to gallons ? Fractions Fractions 9 6/25/2013 5/25/2013 6/25/2013 Fractions

( ) ( ) Adding Fractions Common Denominator Example: •
How to Tame Them Adding Fractions Common Denominator Example: Add 3 pints to 4 gallons 3 pints 1 gallon 8 pints = 3 pints gallons 3 8 = Thus gallons 3 8 + 4 ( ) = + 4 gallons gallons 3 8 gallons = ( ) 3 8 + 4 gallons 35 8 = Fractions Fractions 10 5/25/2013 6/25/2013 6/25/2013 Fractions

( ) Adding Fractions Common Denominator Example:
How to Tame Them Adding Fractions Common Denominator Example: Technique: Multiply each fraction by 1 in “clever disguise” (i.e., b/b) to produce the same denominator (15 in this case) = + 2 3 ( ) 5 4 2 3 4 5 + = 12 15 + 10 = 22 15 Fractions Fractions 11 6/25/2013 5/25/2013 6/25/2013 Fractions

Adding Fractions Find Common Denominator Example: + = + = + + = = +
How to Tame Them Adding Fractions Find Common Denominator Example: 7 15 + 12 = + 7 3 4 5 = + 7 3 4 5 Changes form but not value + = 7 3 4 5 = 28 60 + 35 Question: = 63 60 Is 12 • 15 = 180 also a common denominator ? = 21 20 Fractions Fractions 12 6/25/2013 5/25/2013 6/25/2013 Fractions

Subtracting Fractions
How to Tame Them Subtracting Fractions Find Common Denominator Example: 7 15 12 = 7 3 4 + 5 –7 = 7 3 4 5 + –7 = + 7 3 4 5 –7 Changes form NOT value = –28 60 + 35 = 7 60 Question: Is 60 the least common denominator? Fractions Fractions 13 5/25/2013 6/25/2013 6/25/2013 Fractions

( ) Dividing Fractions Simplifying A Complex Fraction
How to Tame Them Dividing Fractions Simplifying A Complex Fraction Multiply numerator and denominator by a fraction that reduces the denominator to 1 I In general, for non-zero C = A B C D ( ) A B C D = A B C D = 1 A B D C = A B D C Simplifying Complex Fractions Be sure that the numerator, A/B, and denominator, C/D, are properly identified. We employ the already described technique of multiplying the overall complex fraction by 1 in some “clever” disguise, in this case This is done to change the form of the complex fraction, without changing its value. Of course, the reason for choosing D/C as numerator and denominator of our “1 in disguise” is that when the multiplication is carried out the denominator of the complex fraction is reduced to 1. The complex fraction is thus converted to a simple product of two fractions. Factoring this product yields the original numerator and the inverted form of the original denominator. Change form, NOT value = A B D C General Rule: Invert denominator, multiply by the numerator Fractions Fractions 14 5/25/2013 6/25/2013 6/25/2013 Fractions

Dividing Fractions Example: Simplify = = = … using the General Rule OR
How to Tame Them Dividing Fractions Example: Simplify 5 7 3 4 = 5 7 3 4 = 35 21 20 = 21 20 … using the General Rule OR 5 7 3 4 Dividing Fractions In the first example we follow the algorithm for deriving the invert-and-multiply rule. Hence we multiply the complex fraction by the value 1 disguised as another complex fraction. This change the form of the given fraction without changing its value, and at the same time converts the complex fraction into a simple product of fractions. This product is exactly the result of applying the invert-and-multiply rule. In the second example we simply apply the invert-and-multiply rule directly. = 3 4 7 5 = 21 20 Fractions Fractions 15 6/25/2013 5/25/2013 6/25/2013 Fractions

Exercises Exercise 1: Simplify = = = = = 4 15 3x 5 3x 5 15 4
How to Tame Them Exercises Exercise 1: Simplify 4 15 3x 5 = 3x 5 15 4 = (3x)(3)(5) (5)(4) = (5)(3)(3x) (5)(4) Exercises In Exercise 1 we apply the invert-and-multiply rule directly to produce the equivalent product of fractions, which we then simplify. In Exercise 2 we first simplify the numerator and denominator of the complex fraction to simple fractions. Then we apply the invert-and-multiply rule as before. = 5 4 3 3x ( ) = 9x 4 Fractions Fractions 16 5/25/2013 6/25/2013 6/25/2013 Fractions

Exercises Exercise 2 Simplify : = + = = = 1 7 x – + 1 7 – 7x x x + 1 x
How to Tame Them Exercises Exercise 2 Simplify : 1 7 x + = 1 7 7x x + = x + 1 x 7 7x 1 Exercises In Exercise 1 we apply the invert-and-multiply rule directly to produce the equivalent product of fractions, which we then simplify. In Exercise 2 we first simplify the numerator and denominator of the complex fraction to simple fractions. Then we apply the invert-and-multiply rule as before. = x x + 1 7x 7 1 = 7(x + 1) x(7x 1) Fractions Fractions 17 5/25/2013 6/25/2013 6/25/2013 Fractions

Think about it How to Tame Them Fractions Fractions 18 6/25/2013
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