Math A100 IUSB Sec. 2.6 Detailed Notes Calculating percentages There are several methods that you can use when you need to calculate percentages: 1) The “literal” method – use the formula “part” = “whole” * “%” Another version of this formula is “is” = “of” * ”%” 2) Trial & Error – in this method, you have three choices: a. Multiply b. Divide c. Divide the other way NOTE: In the Trial & Error method, only one of the 3 options will give you a number that makes sense. So, you will find the answer, but may have to do all three choices to find it. 3) Proportions – the basic form for this proportion is: is = % or of 100 part = % whole 100 then cross-multiply to solve. NOTE: when using this method, you do NOT have to convert the % to a decimal, just write it without the % sign above the 100. “Percent” means “per 100” or “out of 100” so 25% means 25 out of 100 = 25/100. This is the only one of these methods where you do not have to convert the percent to a decimal. 4) The “Circle” method – draw a circle, then drawing a horizontal straight line thru it going thru the center (like a diameter), then draw a vertical line in the bottom half going from the center to the outside edge (like a radius). Put your “is” (a.k.a. “part”) number on top. Put your “of” (a.k.a. “whole”) number and percent in the bottom two sections (it doesn’t matter which goes where). Put the “x” in the spot for the missing numbers and fill in the other numbers. Cover up the “x”: if the two numbers left are next to each other  multiply If the two numbers left are one above the other  divide No matter which method you use, when you finish with your calculations, ask yourself “does my answer make sense?” Example 1: What is 30% of 150? The Literal Method: the % is 30%; the “of” number is 150 (because it is next to the “of”; x is the “is” “is” = “of” * ”%” x = (30%)(150) x = (0.30)(150) x = 45 45 makes sense because 30% is less than 50% (i.e. less than half). 50% of 150 is 75. 45 is less than 75. Trial & Error Method: a. Multiply: (30%)(150) = (0.30)(150) = 45  makes sense b. Divide: 150/30% = 150/0.30 = 500  500 is 30% of 150?? does NOT make sense c. Divide: 30%/150 = 0.30/150 = 0.002 0.002 is 30% of 150?? does NOT make sense Proportions Method: the % is 30%; the “of” number is 150 (because it is next to the “of”; x is the “is” is = %  x = of 100 150 30  cross multiply & solve: x = 45 100 Circle Method: Draw your circle. the % is 30%; the “of” number is 150 (because it is next to the “of”; x is the “is”, so: x =goes on top 30% and 150 go in the bottom Cover up the “x”  two numbers left are next to each other  multiply x = 45 Example 2: 84 is what percent of 250? The Literal Method: the % is ” x”; the “of” number is 250; the “is” number is 84 “is” = “of” * ”%” 84 = (x)(250) 84/250 = x 0.336 = x  change to a %: x = 33.6% 33.6% makes sense because it is less than 50% and 84 is less than half of 250. Trial & Error Method: a. Multiply: (84)(250) = 21,000  84 is 21,000% of 250?? does NOT make sense b. Divide: 250/84 = 2.97 297%  84 is 297% of 250?? does NOT make sense c. Divide: 84/250 = 0.336  33.6%  makes sense Proportions Method: the % is ” x”; the “of” number is 250; the “is” number is 84 is = %  of 100 84 = x  cross multiply & solve: x = 33.6  33.6% 250 100 Circle Method: Draw your circle. the % is ” x”; the “of” number is 250; the “is” number is 84, so: 84 =goes on top x and 250 go in the bottom Cover up the “x”  two numbers left are one on top of the other  divide x = 84/150 = 0.336  change to a percent = 33.6% Example 3: 36 is 15% of what ? The Literal Method: the % is 15%; the “of” number is “x”; the “is” number is 36 “is” = “of” * ”%” 36 = (15%)(x) 36 = 0.15 x 240 = x 240 makes sense because 15% is a small amount and 36 is small compared with 240. Trial & Error Method: a. Multiply: (15%)(36) = .15*36 = 5.4  36 is 15% of 5.4?? does NOT make sense b. Divide: 36/15% = 36/0.15 = 240  36 is 15% of 240?? makes sense c. Divide: 15%/36 = 0.15/36 = 0.00417  36 is 15% of 0.00417??  does NOT make sense Proportions Method: the % is 15%; the “of” number is “x”; the “is” number is 36 is = %  of 100 36 = 15  cross multiply & solve: x = 240 x 100 Circle Method: Draw your circle. the % is 15%; the “of” number is “x”; the “is” number is 36, so: 36 =goes on top x and 15% go in the bottom Cover up the “x”  two numbers left are one on top of the other  divide x = 36/15% = 36/0.15 = 240 Special Cases Percentages larger than 100% These are worked just like the others, except note that percentages more than 100% become decimal numbers larger than 1. Extending Example 1 from above: What is 230% of 150? The Literal Method: the % is 230%; the “of” number is 150 (because it is next to the “of”; x is the “is” “is” = “of” * ”%” x = (230%)(150) x = (2.30)(150) x = 345 345 makes sense because 230% is more than twice as big, and 345 is more than twice as a big as 150. Trial & Error Method: a. Multiply: (230%)(150) = (2.30)(150) = 345  makes sense b. Divide: 150/230% = 150/2.30 = 65.217  65.2 is 230% of 150?? does NOT make sense c. Divide: 230%/150 = 2.30/150 = 0.015 0.015 is 30% of 150?? does NOT make sense Proportions Method: the % is 230%; the “of” number is 150 (because it is next to the “of”; x is the “is” is = %  x = of 100 150 230  cross multiply & solve: x = 345 100 NOTE: since the % is bigger than 100%, the both fractions will have a top number bigger than bottom number. Circle Method: Draw your circle. the % is 230%; the “of” number is 150 (because it is next to the “of”; x is the “is”, so: x =goes on top 230% and 150 go in the bottom Cover up the “x”  two numbers left are next to each other  multiply x = 345 Word Questions Unfortunately, “word questions rarely identify the “is” and “of” numbers some clearly. Most often “is” represent “equals” or “equal to” and “of” represents multiplying, if either word is even found in the question! One strategy is to decide if the percent is smaller than 100% or bigger than 100%: the percent is smaller than 100%  the “is” number is smaller than the “of” number a.k.a. , both fractions in the proportion method have smaller numbers on top the percent is BIGGER than 100%  the “is” number is BIGGER than the “of” number a.k.a. , both fractions in the proportion method have BIGGER numbers on top Example 3: If you paid $169.52 in income taxes on a gross monthly earnings of $1,304, what is your income tax rate? “Income tax rate” means Income tax %. Your income tax % BETTER be less than 100%, so $169.52 is the “is” number and $1304 is the “of” number Proportions Method is = %  of 100 169.52 = x  cross multiply & solve: x = .13 = 13%, makes sense 1304 100 Example 4: The doctor says you need to consume fewer calories. You should be consuming only 2,000 calories, but you are eating 2,600 calories. The amount of calories you are eating is what percentage of what you should be consuming. You are eating more than your doctor recommends. i.e. you are eating MORE THAN 100% of what you should, so the % is BIGGER than 100%  The fraction would be: is = %  of 100 Percent Change 2600 = x  cross multiply & solve: x = 1.3 = 130%, makes sense 2000 100 When a number changes, you can look at it two ways: 1) How much it actually went up or down, which is called the Total Change (or Absolute Change) 2) How much it changes compared to its original amount, i.e. what percent of the original amount did it go up or down by, which is called the Percent Change (or Relative Change) To calculate the “Total Change”, simply find the difference between the ending value and the starting value. If the number went up, it is called “an increase” and is represented by a positive number when you are actually doing the calculation. In real-life, we say “it went up by _____” or “an increase of _____” If the number went down, it is called “a decrease” and is represented by a negative number when you are actually doing the calculation. In real-life, we say “it went down by _____” or “a decrease of _____” NOTE: often students ask me which order to subtract the numbers. You can either: a) Memorize “new – old” or “ending – beginning” (or something similar), or b) Look at what the number did and use that to decide. a. If the number got bigger, then Total change is Bigger # - smaller # b. If the number got smaller, then the change is negative, so it is smaller # - Bigger # Example 1: your hourly wage went from $9.50 per hour to $12 per hour. What is the total change? The number got bigger, so the change is positive, so 12 – 9.5 = 2.5  the total change is a $2.50 increase Example 2: You weight went from 140 lbs. down to 110 lbs. What is the total change? The number got smaller, so the change is negative, so 110 – 140 = -30 lbs.  the total change is a 30 lb. decrease NOTE: if you are given units (i.e. labels like $, inches, lbs., etc.) they need to be included in your answer AND you need to tell me the direction of the change (i.e. was it a decrease or increase) . In this example, a “$2.50 per hour increase” is the correct answer; 2.5 is NOT the correct answer Percent change is found by comparing the amount theta it went up or down to where it started. Your textbook will give you equations like: (new – old)/old or (ending – initial)/initial. You have 2 ways to think of it: Option 1 (the textbook method): Percent Change = The change itself => as a percent The starting point Option 2 (the Math 123 method): Percent Change = The ending value The starting point - 1 => as a percent You can use either method, although MML HW follows the Option 1 one more than Option 2. Extending Example 1: your hourly wage went from $9.50 per hour to $11 per hour. What is the percent change? Option 1: The total change is a $2.50 increase The “starting point” is $9 Percent Change = The change itself = 2.50 = 0.2631 ==> as a percent = 26.3% increase The starting point 9 Option 2: Percent Change = The ending value - 1 = 12 - 1 = 1.2631 - 1 = 0.2631 => 26.3% increase The starting point 9.5 Extending Example 2: You weight went from 140 lbs. down to 110 lbs. What is the percent change? Option 1: The total change is a 30 lb. decrease The “starting point” is 140 lbs. Percent Change = The change itself = -30 = - 0.2143 ==> as a percent = 21.4% decrease The starting point 140 Option 2: Percent Change = The ending value - 1 = 110 - 1 = 0.7857 - 1 = -0.2143 => 21.4% decrease The starting point 140 NOTE: The units (i.e. label) for Percent change is ALWAYS %. And again make sure you state whether it is an increase or decrease (because no one says “I had a -21.4% change in my weight.”) Using % change to make calculations You can use the Percent Change to calculate: a) a new (or ending” number) if you know the starting point and the percent change b) the starting point, if you know the ending value and the percent change. You have two ways to do this: 1) the “traditional” method (this is the method you are used to: a. use the percent change to calculate the total change b. add or subtract the total change to/from the starting point 2) Use the “multiplier” method – change the percent change into a new percent that represent the ending value and multiply it. a. Start at 100% - if the percent change is an increase (i.e. positive, then add the percent change to 100% to get the new percent b. Start at 100% - if the percent change is a decrease (i.e. negative), then subtract the percent change from 100% to get the new percent Example 1, version B: your hourly wage started at $9.50 per hour and went up 30%. What is your new hourly wage? Method 1 (Traditional - from above): 30% * 9.50 = $2.85  an increase of $2.85 $9.50 + $2.85 = $12.35  your new hourly wage is $12.35 Method 2 (Multiplier - from above): 30% increase  means you have more than you started with, so 100% + 30% = 130%  as a decimal = 1.3 = the “multiplier” 1.3 x $9.50 = $12.35  your new hourly wage is $12.35 Example 2, version B: You weight was 140 lbs. down when you started your diet. You had a 15% loss. What is you new weight? Method 1 (Traditional - from above): 15% * 140 lbs. = 21 lb.  a decrease of 21 lbs. 140 lbs. - 21 lbs. = 119 lbs.  your new weight is 119 lbs. Method 2 (Multiplier - from above): 15% loss  means you have less than you started with, so 100% - 15% = 85%  as a decimal = 0.85 = the “multiplier” 0.85 x 40 lbs. = 119 lbs  your new weight is 119 lbs. This works well if you have your starting point and are trying to find your ending value. But what about the other way around: what is you have your ending value and need to find your starting point> Example 1, version C: your current hourly wage is $12 per hour. It is up 30% from last year. What was last year’s hourly wage? Method 1 (Traditional - from above): 30% * 12 = $3.60  an increase of $3.60 $12 - $3.60 = $8.40  last year’s hourly wage is $8.40 Method 2 (Multiplier - from above): 30% increase  means you have more than you started with, so 100% + 30% = 130%  as a decimal = 1.3 = the “multiplier” 1.3 x (last year’s hourly wage) = $12 divide both sides by 1.3 (undo the multip.) 1.3 x (last year’s hourly wage)/1.3 = $12/1.3 = $9.23  last year’s wage was $9.23 WAIT!!! WHY are they NOT the same?!?!?!?!?!?! Let’s check each to see which is correct: Method 1 (Traditional - from above): 30% x $8.40 = $2.52 $8.40 + $2.52= $10.92 NOT $12 Method 2 (Multiplier - from above): 30% increase  means you have more than you started with, so 100% + 30% = 130%  as a decimal = 1.3 = the “multiplier” $9.23 x 1.3 = $12 IS $12 NOTE: The “Traditional Way is ONLY good when you have the starting point and are looking for the new ending value. It DOES NOT WORK when you have the ending value and are looking for the starting point, because you are multiplying by the wrong number (you multiply by the ending value ($12 in this example), instead of the starting point (which you were looking for in the first place). Let’s try that again to be sure Example 2, version C: At the end of your diet, you weight was 110 lbs. You lost 15% during your diet. What was your weight when you started the diet? Method 1 (Traditional - from above): 15% * 110 lbs. = 16.5 lb.  a decrease of 16.5 lbs. Adding it back: 110 lbs. + 16.5 lbs. = 127.5 lbs.  your weight was 127.5 lbs. Let’s check each to see if it is correct: (Recall: At the end of your diet, you weight was 110 lbs.) 127.5 lbs.. x 15% = 19.125 lbs. 127.5 lbs. - 19.125 lbs. = 108.375 lbs. NOT  110 lbs. This method did not work correctly Now try: Method 2 (Multiplier - from above): 15% loss  means you have less than you started with, so 100% - 15% = 85%  as a decimal = 0.85 = the “multiplier” 0.85 x (your starting weight) = 110 lbs. divide both sides by 0.85 0.85 x (your starting weight)/.85 = 110 lbs./.85 = 129.4 lbs  your old weight was 129.4 lbs. Let’s check each to see if it is correct: (Recall: At the end of your diet, you weight was 110 lbs.) 15% loss  means you have less than you started with, so 100% - 15% = 85% 0.85 x 129.4 lbs. = 110 lbs.  as a decimal = 0.85 = the “multiplier” CORRECT NOTE: The Multiplier Method works no matter whether you are looking for the ending value OR the starting point! Here is another way to set up the Multiplier Method by using proportions: “new percent” = ending value 100% starting point Extending Example 1, version B: your hourly wage started at $9.50 per hour and went up 30%. What is your new hourly wage? 30% increase  means you have more than you started with, so 100% + 30% = 130%  “new percent” Starting point = $9.50 As a proportion “new percent” = ending value  100% starting point Cross multiplying gives you: 130% = 100% x__  130 9.50 100 (130)(9.5) = (100)(x)  1235 = 100x x = $12.35 which is the same as above = x 9.5 Example 1, version C: your current hourly wage is $12 per hour. It is up 30% from last year. What was last year’s hourly wage? 30% increase  means you have more than you started with, so 100% + 30% = 130%  “new percent” Ending value = $12 As a proportion “new percent” = ending value  100% starting point Cross multiplying gives you: 130% = 100% 12__  130 = x 100 12 x (130)(x) = (100)(12)  130x = 1200 x = $9.23 which is the same as above Example 2, version B: You weight was 140 lbs. down when you started your diet. You had a 15% loss. What is you new weight? 15% loss  means you have less than you started with, so 100% - 15% = 85% = new percent Starting point = 140 lbs As a proportion “new percent” = ending value  100% starting point Cross multiplying gives you: 85% = 100% x__  85 140 100 = x 140 (85)(140) = (100)(x)  11,900 = 100x x = 119 lbs. which is the same as above Example 2, version C: At the end of your diet, you weight was 110 lbs. You lost 15% during your diet. What was your weight when you started the diet? 15% loss  means you have less than you started with, so 100% - 15% = 85% = new percent Ending value = 110 lbs. As a proportion “new percent” = ending value  100% starting point Cross multiplying gives you: 85% = 100% 110__  85 = x 100 110 x (85)(x) = (100)(110)  85x = 11,000 x = 129.4 lbs. which is the same as above % change Multipliers Several of your HW questions will ask you to convert from either percent change to the multiplier or from the multiplier to the percent change. Remember that: a) it all starts with 100% and goes up or down depending on the percent change b) The multiplier is simply the “new percent” written as a decimal, so you can put it in your calculator. Example 3: What multiplier goes with a percent increase of 6%? 6% increase  means you have more than you started with, so 100% + 6% = 106%  as a decimal = 1.06 = the “multiplier” Example 4: What multiplier goes with a percent decrease of 4%? 4% decrease  means you have less than you started with, so 100% - 4% = 96%  as a decimal = 0.96 = the “multiplier” Example 5: What multiplier goes with a percent increase of 102%? 102% increase  means you have more than you started with, so 100% + 102% = 202%  as a decimal = 2.02 = the “multiplier” Example 6: What multiplier goes with a percent decrease of 75%? 75% decrease  means you have less than you started with, so 100% - 75% = 25%  as a decimal = 0.25 = the “multiplier” Now let’s go back the other way, from multiplier to percent change: Example 8: The multiplier is 1.46. What is the percent change? Work it backwards: multiplier = 1.46 which as a percent is 146% 146% mean you have more than you started, so take away the 100% you started with: 146% - 100% = 46%  46% increase Example 9: The multiplier is 0.46. What is the percent change? Work it backwards: multiplier = 0.46 which as a percent is 46% 46% mean you have LESS than you started, so take away the 100% you started with: 46% - 100% = - 54%  54% decrease Example 10: The multiplier is 1.007. What is the percent change? Work it backwards: multiplier = 1.007 which as a percent is 100.7% 100.7% mean you have more than you started, so take away the 100% you started with: 100.7% - 100% = 0.7%  0.7% increase Example 11: The multiplier is 0.09. What is the percent change? Work it backwards: multiplier = 0.09 which as a percent is 9% 9% mean you have LESS than you started, so take away the 100% you started with: 9% - 100% = - 91%  91% decrease NOTE: the “new percent” is NEVER EQUAL to the percent change!!!! Special Cases Sales tax & total cost – Sales tax is an amount they add to your purchase, so a 7% sale tax is a 7% increase, so use the examples above to do the calculations. NOTE: you can use both the traditional method OR the Multiplier method to do these. The MML HW will ask you to do them both ways. 2-part %: commission & agent’s cut - these questions are especially confusing. You can do them two options: Option 1: Break them up into two parts: a. Do the first % given, then b. Use the answer from part a) to do part b) Option 2: Since you have a starting point and are taking a percent (i.e. a part) of it twice, you can multiply both percents by the starting point at the same time. Example 12: You want to sell your car for $17,000 through a dealership. The dealership gets 8% of the sale price for selling your car. The agent who actually makes the sale gets a 30% commission from the dealership’s cut . How much $$ does the agent get? Option 1: Break them up into two parts: a. $17,000 x 8% = $17,000 x 0.08 = $ 1360 b. $1360 x 30% = $ 408  the agent gets $ 408 Option 2: multiply both percents by the starting point at the same time $ 17,000 x 8% x 30% = $ 17,000 x .08 x .3 = $ 408  the agent gets $ 408 Videos that may help: Percent Problems (3 examples) http://www.khanacademy.org/math/algebra/solving-linear-equations-andinequalities/percent_word_problems/v/percent-problems Convert Between Percents, Decimals, and Fractions http://www.youtube.com/watch?v=z1p9o6ymteI Solving Percent problems using the percent equation http://www.youtube.com/watch?v=LkTYkHbUiU4 Percent with Sales Tax or Tipping at a Restaurant http://www.youtube.com/watch?v=7cb1lsfU5Rw http://www.youtube.com/watch?v=yFaa2CMx9rk&feature=youtu.be Income Tax – How the Basic Tax Rate Schedule Works http://www.khanacademy.org/science/core-finance/taxes-topic/taxes/v/basics-of-us-incometax-rate-schedule Income Tax (Checkpoint 6) http://media.pearsoncmg.com/cmg/pmmg_mml_shared/flash_video_player/player.html?/ph /esm/esm_blitzer_bztm5e_11/video/bztm5e_8_1_cp6 Percent Increase and Decrease http://www.youtube.com/watch?v=vB0Zo8Kxgtk