Math A-100 IUSB Sec. 1.5 Notes – Week 1 Multiplying fractions – no need to change them, because you are finding a “part” of another “part”. You have two options: a) You can “multiply across the top” (i.e. multiply the two top numbers) and “multiply across the bottom” (i.e. multiply the two top numbers), then reduce the fraction. Example: 3/8 x 6/11 = ? 3 x 6 = 18 and 8 x 11 = 88, giving you: 18/88 which reduces to 9/44 b) You can “cross cancel” (a.k.a. “cancelling”) the fractions – compare (not multiply) the top of one with the bottom of the other fraction to see if they reduce, to make the numbers smaller before multiplying. Example: 3/8 x 6/11 = ? Nothing goes into both 3 & 11, but 2 goes into both 8 & 6, so take 2 out of each: 3/4 x 3/11 = 9/44 which does not reduce Multiplying whole numbers & fractions – make the whole number into a fraction by putting “1” in the bottom (because 5/1 = 5, it does not change the number at all). Now you have two fractions, so you can multiply them as above. Example: Becomes 3/8 x 7 = ? 3/8 x 7/1 21/8 NOTE: MyMathLab’s directions often ask for “simplified” fractions. An improper fraction like 21/8 is a simplified fraction, so the answer is correct (i.e. it is okay to use improper fractions). If you give MyMathLab a mixed number or a decimal answer when it asks for a “simplified fraction” then it will mark your answer wrong for not following the directions. Dividing fractions – instead of dividing, we flip the second fraction upside down and multiply the fractions instead. The new “flipped” fraction is called the “reciprocal”. Example 1: 3/8 divided by 6/11 = ? Flip 6/11 to become 11/6 (the reciprocal) and multiply instead 3/8 divided by 6/11 becomes 3/8 x 11/6 = 33/48 which reduces to 11/16 Example 2: 3/8 divided by 7 = ? Make 7 into 7/1, and flip it to become 1/7 (the reciprocal) and multiply instead 3/8 divided by 7 becomes 3/8 x 1/7 = 3/56 which does not reduce NOTE: Why does this work? Think of it this way: 1 dozen divided by 2 = 6 ½ of a dozen = ½ x 1 dozen = 6 2 = 2/1, so dividing by 2/1 is the same as multiplying by 1/2 Adding or subtracting fractions – to add fractions, we are adding parts together to see how many total we have (or have left). In order to do that, all the parts must be the same size. i.e. the bottom numbers MUST be the same. So, we need to change each fraction into a form with the same bottom number. Example 1: 3/8 + 7/10 = ? the bottoms are not the same, so the “parts” are not all the same size. Both 8 and 10 can be changed into size 40 (or 80 if you like): 8 x 5 = 40 and 10 x 4 = 40 So make 3/8 into (3 x 5)/(8 x 5) = 15/40 and make 7/10 into (7 x 4)/(10 x 4) = 28/40 Now we have 15 pieces that are size 40 and 28 pieces that are size 40, giving us a total of 15 + 28 = 43 pieces that are size 40, which can be written as 43/40 Example 2: 1/4 + 7/10 = ? the bottoms are not the same, so the “parts” are not all the same size. Both 4 and 10 can be changed into size 20 (or 40 if you like): 4 x 5 = 20 and 10 x 2 = 20 So make 1/4 into (1 x 5)/(4 x 5) = 5/20 and make 7/10 into (7 x 2)/(10 x 2) = 14/20 Now we have 5 pieces that are size 20 and 14 pieces that are size 20, giving us a total of 5 + 14 = 19 pieces that are size 20, which can be written as 19/20 NOTE: you can subtract fractions the same way, except you will subtract instead of adding once the parts have the “same size” Questions with multiple things going on, a.k.a. Order of Operations First off, there is no such thing as the Order of Operations, because there is no 1 set order that works in every question. Each question must be taken on a case-by-case basis. Forget all about PEMDAS, because it is a lie. Here are some things to think about and why we handle them earlier than other things: Exponents – they are simply short-hand, numbers with exponent are NOT REAL numbers, so we have to find out what the REAL number is Absolute values - |__| is not a real number either, so we need to find out what the real number is before we can go on Square roots - __ is also not a real number, so we need to find out what the real number is before we can go on Parentheses – they are like a “bag” containing a single value. We have to find out what that single value is BEFORE we can go on Multiplying – this represents “groups of ____”. We need to find out how many total pieces you have Dividing – this represents breaking something up into smaller groups, so we need to find out how many are in each smaller group before we go on Adding – we need to know how many we have before we can add more to it Subtracting - we need to know how many we have before we can take some away from it. Example 1: simplify: 7 – 4(5 + (6 – 32)2) Steps: 32 is not a real number, so find out what that real number is and replace it 32 = 9 replacing it gives us 7 – 4(5 + (6 – 9)2) (6 – 9) is a package, so find out what single value is it and replace 6 – 9 = -3 replacing gives us 7 – 4(5 + (-3)2) (-3)2 is not a real number, replace it (-3)2 = (-3)(-3) = 9 replacing it gives us 7 – 4(5 + 9) (5 + 9) is a package, so replace it 5 + 9 = 14 replacing it gives us 7 – 4(14) you now have 4 groups of 14, find the total and replace it 4(14) = 56 replacing it gives us 7 – 56 -49 Example 2: simplify: (-13 – 5)/6 - 2(-4 + 9) Both (-13 – 5) and (-4 + 9) are packages, so replace them -13 – 5 = -18 and -4 + 9 = 5 replacing gives us (-18)/6 – 2(5) -18/6 is breaking it up into groups, so replace it -18/6 = -3 replacing gives us -3 – 2(5) 2(5) mean 2 groups of 5 so replace it 2(5) = 10 -3 – 10 -13 replacing gives us