Section 1.4 Fractions LAWS & PROCESSES. Addition of Fractions DEFINITIONS & BASICS. 1. Common Denominator 2. Add numerators 3. Carry by denominator

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1 34 Fractions DEFINITIONS & BASICS 1) Numerator the top of a fraction 2) Denominator the bottom of the fraction 3) Simplify Fractions are simplified when the numerator and have no factors in common. 4) One any number over itself = 1. 5) Common Denominators Addition and subtraction require like things. In the case of fractions, like things means common s. 6) Prime Factorization Breaking a number into smaller and smaller factors until it cannot be broken down further. LAWS & PROCESSES Prime Factorization One of the ways to get the Least Common Denominator for adding and subtraction fractions that have large s is to crack them open and see what they are made of. Scientists get to use a scalpel or microscope. Math guys use prime factorization. Addition of Fractions 1. Common Denominator 2. Add numerators 3. Carry by 1. Observation 2. Multiply the s 3. Prime factorization EXAMPLE Add Step 1. The least common multiple of 4 and 2 is a 4, so we replace the with an equivalent fraction, which is. Step 2. Now that the s are the same, add the numerators.

2 35 Step 3. Carry the across. Changing from mixed numbers to improper fractions: Changing them back again: Subtraction of Fractions 1. Biggest on top 2. Common Denominator; Subtract numerators 3. Borrow by 4. Strongest wins 1. Observation 2. Multiply the s 3. Prime factorization EXAMPLE Do this: is bigger, so put it on top. - 3 The common is 9, so change the to a Subtract the numerators. Borrow by as needed.

3 36 Multiplication of Fractions Multiplication of Fractions 1. No common s 2. Multiply Numerators 3. Multiply Denominators EXAMPLES For multiplication don t worry about getting common s Multiply the numerators straight across Multiply the s straight across

4 37 Division of Fractions Division of Fractions 1. Improper Fractions 2. Keep it, change it, flip it. 3. Multiply. EXAMPLES Divide Turn the fractions into improper fractions Keep the first fraction the same Change the division sign to a multiplication sign Flip the second fraction s numerator and Multiply straight across the numerator and Divide Turn the fractions into improper fractions KEEP the first fraction the same Change the division sign to a multiplication sign Flip the second fraction s numerator and Multiply straight across the numerator and

5 38 Now that you have had a little time to multiply fractions together and simplify them, you may have noticed one of the slickest tricks that we can do with fractions, and that is that we can actually do the simplification before we multiply them. Take for example: Now, we can do this the normal way or we can try to notice if there is anything that we will be simplifying out later... and do that simplification before we multiply: Normal method: New and improved slick method: and now we try to simplify which probably took quite a while to get. So, and we try to see if any factors will cancel ahead of time What I was hoping to show is that the same answer was obtained and the same cancelling was done, but if you are able to see it before you multiply, then you will be able to simplify in a much simpler way. Here is another example: the 4 and the 8 can simplify before we multiply: This may seem like just a convenient way to make the problem go a bit quicker, but it does much more than that. It opens the door to a much larger world. Here is an example. If we travelled 180 miles on 12 gallons of gas, then we calculate the mileage by = 15 miles per gallon. Carrying that example just a bit further, what if gas were $3.2 per gallon? We can actually find how many miles we can drive for one dollar: = 4.7 miles per dollar. Another example: Carpet is on sale for 15 dollars per square yard. How much is that in dollars per square foot (9 ft 2 per yd 2 )? Now, knowing that we will be able to cancel anything on the top with anything that is the same on the bottom we write the multiplication so the yd 2 will cancel out, leaving us with dollars per ft 2 :

6 39 Then cancel the yd 2 : One more example: = dollars per square foot = $1.67 per square foot A rope costs $15 for 8 feet. How much does is cost per inch? We want to get rid of feet and get inches, so we write the multiplication: = = $.156 or 15.6 cents per inch Here are a few numbers that will help you with the conversions: 12 in = 1 foot 1 yd = 3 ft 16 oz = 1 pound 60 minutes = 1 hour 60 seconds = 1 minute 1 yd 2 = 9 ft watts = 1 kilowatt And also some exchange rates with the American dollar as they were sometime in 2010: 1 Mexican Peso = $ Euro = $ British Pound = $ Brazilian Real = $0.55