[Music] hello and welcome to another algebra 1 lesson by emath instruction my name is kirk weiler and today we'll be doing unit 6 lesson 8 on percent review many of the exponential functions that we work with in the remainder of this unit are going to be based on percentages all right and so it's very important for you to feel comfortable with basic percent work just basic percent work so we're going to do a bunch of review on that in this lesson so let's jump right into it all right now the fundamental concept of percent or percentages there's a slight difference between those but i don't care all right you extensively studied percentages back in seventh grade that's the year that you kind of really dig into them all right the concept of a percent is one of the most used in everyday life alright a percent is simply a ratio out of a hundred that compares two quantities in a proportional relationship all right so it's kind of like saying you know like if i had like 20 kids in a class and three of them were left-handed right i could say the ratio of left-handed kids to all the kids was three to twenty but if i wanted it in terms of a percent right then i wouldn't want it as a ratio out of twenty i'd want it as a ratio out of a hundred so maybe i'd multiply both numbers by 5 right 20 times 5 is 100 and i'd say well 3 out of 20 is the same as 15 out of 100 so 15 of the kids are left-handed all right that's the kind of idea right a percentage is really just a fraction out of a hundred fifty-six percent is fifty-six out of a hundred twenty-two percent is twenty-two out of a hundred three percent is three out of a hundred it is so so very important that you understand that idea that being said let's jump right into it in problem number one jonathan is getting a raise from 12.50 cents per hour to 14.75 cents per hour his supervisor michaela got a raise from 22 dollars per hour to 25 dollars and 30 cents per hour letter a how much in terms of dollars per hour is jonathan's salary going up all right well you shouldn't need any help from me on that by the way make sure to have your calculators handy today we are certainly going to be using them a lot to do some messy decimal division and you may even want it on this problem to just do the subtraction pause the video now and figure out just in terms of dollars how much did jonathan's salary go up all right well it's simple right this is just subtraction right i can say well you know 14.75 minus 1250 is going to give me two dollars and 25 cents that jonathan's salary went up now that doesn't seem like much of a race two dollars and 25 cents right but we want to compare that two dollars and 25 cents to where he started in the next problem let's take a look at letter b by what percent is jonathan's salary increasing all right so again this is really a ratio i want to compare this to the original amount whenever we do a percent problem we're always kind of comparing to the original amount now the ratio of how much he got to what he originally started with is 225 this is kind of like my left-handers to everyone compared to 1250 right that's my actual ratio of how much the raise was compared to the total but what is that in terms of a decimal for this i definitely need my calculator and it's simple enough right i'm going to have 225 divided by 1250 simple enough and that gives me 0.18 0.18 all right now keep in mind 0.18 is the same as eighteen one hundredths and eighteen one hundredths is by definition eighteen percent right so in other words i'm comparing i'm just doing a ratio but i need that ratio out of a hundred and quite frankly decimals are a great way to get there because .18 is out of 100 that's 18 out of 100 so jonathan got an 18 raise let's take a look at letter c who received the larger increase in salary in terms of dollars per hour so who got the larger salary increase jonathan or michaela why don't you take a moment you already know what jonathan got in terms of a raise why don't you go ahead and figure out what michaela got in terms of arrays all right well we just got to figure out michaela's raise michaela's raise was 25 30 minus 22 dollars she got a raise of 3 dollars and 30 cents so michaela got the larger raise right i mean jonathan got 225 michaela got 330 there's no question michaela got the larger raise but let's take a look at letter d who received the larger percent increase in salary so we know right we know at this point that jonathan got an 18 raise okay what we don't know yet is what percent raised michaela got why don't you figure that out and then tell me who got the larger percent raise all right well we're going to do the same thing for michaela that we did for jonathan we're going to take the raise that she got which was three dollars and 30 cents we're going to divide it by her original salary which was 22 dollars i'm gonna figure out what that is in terms of a decimal 330 divided by 22 that gives me 0.15 0.8 which is the same as 15 one hundredths which is the same as 15 so even though michaela got the overall larger raise jonathan got the larger raise in terms of percent he got an 18 raise michaela got a 15 so jonathan got the larger percent raise now you might say well you can't buy bread with a percent and that's true right there's no question that michaela's buying power went up more than jonathan's buying power but if they continued getting raises jonathan continued getting an 18 raise michaela continued to get a 15 raise eventually jonathan's salary would become higher than michaela's salary simply because the percent raise he's getting is higher than michaela's percent all right let's keep going ratios out of a hundred now you can almost always solve percent problems by setting up proportional equations that involve 100. we didn't really do that in the last problem but we're going to take a look about that a little bit in exercise two and three typically you use cross multiplication to solve these types of proportions so let's take a look at exercise number two gabe is buying a pair of jeans at a local store that are priced at forty five dollars he knows that the county he lives in has an eight percent sales tax added on to the list price if gabe has a gift card for fifty dollars will it cover the cost of the jeans and the tax show work to justify your answer all right so this is classic right one of the places that we see percent all the time in the real world is taxes money that the government charges you when you buy something okay now what we're have to do is we have to find out what 8 of 45 is right we need to find eight percent of 45 dollars and what that really means right is we want to say well look x is to 45 as 8 is to 100 x is to 45 as 8 is to 100 now a lot of teachers will say all right do 45 times 8 do 100 times x cross multiply divide and all that no not in this situation right i want to take you all the way back to the first day that we solved linear equations what's happening to x well we're dividing x by 45. how do i undo that and get x all by itself well i am just going to multiply both sides by 45 right it would be actually just a little bit even silly to cross multiply right i just need to multiply both sides by 45 in order to get to undo the division by 45. now x right is gonna be eight one hundredths times forty five definitely just wanna bring that up that's simple enough maybe minimum whoop that's not what i want come back come back to me there it is all right i want eight one hundredths times forty-five what is that that's eighteen fifths that doesn't help me ah three point six all right 3.6 since we're talking about money i'm going to say it's 3.60 now keep in mind that is certainly not what gabe is paying for these genes if it was that'd be awesome right that's actually just the tax right this is just the tax that's the eight percent of our 45. so what i now need to do is take my 45 dollars add to it my three dollars and sixty cents get forty eight dollars and sixty cents and that is what gabe is paying and so the fifty dollar gift card is enough right that was ultimately what we were trying to answer will he have enough to pay for both the jeans and the tax on the jeans given that he has a 50 gift card and yeah it's enough because well you know 50 is more than 48.6 simple enough all right let's take another look at an tax problem or not a tax problem but a percent problem let's see number three population of deer and a forest preserve is predicted to decline right that's another word for decrease by 5 this year if the current population is 560 what population is predicted for next year what percent of the deer will remain all right well given what we just did i'd like you to play around with this problem it's very similar to the last one except at the end of the day you know after you find out what 5 of 560 is you don't want to add that to the 560 right you want to subtract it from the 560 because that's what's going on right it's declining pause the video now and see if you can do that all right so let's go through it now the first thing that we need to do is we need to figure out what 5 of 560 is again we can do that just like we did the last problem i can say x divided by 560 is the same as 5 divided by 100. now you can cross multiply that's fine but again i really advise you to just think about it in terms of solving those equations right i divide x by 560 to undo it i multiply by 560 again i need to have my calculator out right now because i do not want to try to do that in my head okay i've got five one hundredths times 560 gives me 28. wonderful 28. now again that's just how much the population has declined so my new population is going to be 560 minus 28 which is equal to 532. now there is one other part of this problem right it asks what percent of the deer will remain well to do that right one way i can do it is i can do 532 those are the number of deer remaining divided by 560 which is the number of deer i started with so percent remaining will be 532 divided by 560. now that's a ratio out of 560 if i want a ratio out of a hundred what i'm going to do is i'm going to bring this up here 532 divided by 560. that's 19 out of 20. thank you so much calculator that's 0.95 which is 95 one hundredths which is 95 percent now to put it mildly that answer shouldn't surprise us at all if we've taken a population and we've decreased it by 5 what remains is 95 right should just make all the sense in the world if i had actually decreased the population by let's say 20 percent what would have remained would have been 80 percent of what we started with so some of you probably could have answered that right away now others of you i hope right now are saying man i don't want to figure out what five percent of 560 is by setting up this equation and cross multiplying or even doing this i know a faster way of doing it and that's what we want to review right now let's take a look although you can find a certain percentage of a whole by using ratios which is what we've been doing it's critically important that you don't okay in other words it's critically important that you do it by just multiplying by a decimal so let's review how that works it's certainly something you've found before let's take a look at problem number four consider the following problem let's say that i want to live a 15 tip on a meal that costs 35 letter a find the tip by setting up a ratio involving 100 wonderful okay so what i'd like you to do is work this problem out letter a just like what we were doing right x divided by 35 equals 15 divided by 100 etc etc and figure out what tip we would leave on a 35 check if we were paying 15 percent in tip pause the video now all right well again you know it's simple enough it's all right i have x over 35 is equal to 15 over 100 i'm going to multiply both sides by 35. and by 35 over here cancel cancel right x is going to be 15 over 100 times 35 bring me out my calculator let's see so i've got 15 divided by 100 times 35 it's going to give me that nice fraction again and that's going to give me a tip of 5.25 all right now again that that's not a bad way to do it i'm not saying that but we need to be able to do it faster than that i need to you to be able to say if i say figure out what 15 of 35 is i need you to be able to do it in a single calculation right away so let's take a look at letter b find the tip by multiplying 35 dollars by a decimal less than one why is this the same in a well actually let's use a to like look at it right now right how did we actually calculate 15 of 35 well we did 15 divided by 100 times 35. okay and i really want you to understand this this really goes to really a deep understanding about what a percent is a percent is simply a fraction out of a hundred so when i say that the tip is fifteen percent what i'm really saying is it is fifteen one hundredths of the original right i am multiplying the original by 15 one hundredths now what is fifteen one hundredths as a decimal well fifteen one hundredths as a decimal is point one five right so i can find out what 15 that is literally 15 15 and 0.15 are the same thing right because fifteen percent is fifteen one hundredths and fifteen one hundredths is the same as point one five and so i need you to be able to figure out what fifteen percent of thirty five is not by doing that cross multiplying or anything else i need you to just be multiplying 0.15 times 35 and there's that 525 right that is how we are going to be finding percents each and every time from now on all right so let's summarize that right here right finding percentages by multiplying to find p percent of a total of t simply find the product of p divided by a hundred times t where often p divided by a hundred is expressed as a decimal right i mean you could you could write it as a fraction it just gets bulky right and really remember that dividing by a hundred is just going to take the decimal point in your percent and move it two spaces to the left so let's get some work on this this is just critical exercise number five find each of the following write down the product you use to find your answers all right so look if i want to find 20 of 85 what i'm really doing is i'm multiplying by 20 one hundredths right so just just right away i'm going to write this down just a couple times right but literally what i'm doing to find 20 percent of 85 is i'm multiplying i'm multiplying 85 by 21 hundredths but 21 hundredths is the same as 0.20 so i'm just doing that and i want to be able to do that right away right i just want to all right i've got you know 0.2 time 0.20 which is the same as 0.2 times 85 gives me 17 right 20 percent of 85 is 17. simple enough all right now you should be able to do letter b pretty easily why don't you pause the video and go ahead and do that now okay well let me just um view full screen move this over here so i want 12 of 200. now at this point i want to think about like well what is 12 or what is 12 divided by 100 well 12 divided by 100 i just move that decimal point two places to the left and 12 of 200 will be 0.12 0.12 times 200. without a calculator i think i can do that that's going to be 24. 12 of 200 is 24. now letter c 6 of 550 that can throw people off because some people will say well that's 0.6 times 550 but keep in mind right that six one hundredths which is what six percent is is not .6 but it's point zero six right six one hundredths is point zero six so six percent of 550 would be point zero six times 550 and that one i don't think i so much want to do in my head so let's bring it over here um let me uh let me just clear this thing out real quick so we clear up the screen a little bit 0.06 times 550 gives me 33. okay easy enough what i'd like you to do is pause the video now and see if you can work through letter d e and f i know letter d and letter f might look a little bit worse but i bet you can still handle them pause the video now and see what you get all right 4.5 of 120. now again right if i absolutely need to i really want to think about 4.5 divided by 100 i move that decimal place 2 whoops two places to the left and that's .045 right so i want to multiply by 4.5 out of 100 or 0.045 so that's going to be .045 times 120 you don't want to multiply by 0.45 that would actually be 45 percent but i can multiply by .045 times 120 and i'll get 5.4 okay 36 percent of 96 that's a little bit more straightforward of one 36 one hundredths is .36 so 0.36 times 96 might not be the prettiest answer but it certainly isn't a hard one to set up 0.36 times 96 gives us 34.56 and finally two and a quarter percent of 350 all right now very often you will run into percents that also have fractions mixed into them oh the tax rate is eight and one eighth percent oh my mortgage is four and three quarters of a percent right that's where you have to feel comfortable turning common fractions okay into decimals in other words you need to be comfortable with the idea that two and a quarter is the same as 2.25 so if i then move that decimal place two units to the left it'll be .0225 times 350. so .0225 times 350 and that's 7.875 okay so we have to be ready especially at the high school level for things like that right we have to feel comfortable at this point that when i see something like two and a quarter that's the same as 2.25 i mean if you saw two and five-sevenths i wouldn't expect you to know what that is but then again percents typically don't come like that right you don't find a mortgage that has two and five-sevenths percent interest rate on it you know it's quarters eighths halves right tenths things like that all right just to make sure you understand this take a look at number six which of the following calculations would find 8.5 percent of 250. all right take a look and tell me which one's correct well again 8.5 percent right 8.5 8.5 per 100 is going to be .085 and so we're gonna multiply 250 by .085 it is really a percent honest to goodness is just a way of writing a fraction whose denominator is a hundred without having to put the 100 there that's all it is in fact that's why it's per cent right cent means 100 right that's what it means a century has 100 years in it right cent means a hundred that's why there's a hundred pennies 100 cents in a dollar so 8.5 percent means 8.5 per 100 which is the equivalent of 0.085 believe it or not there used to be something called a per meal right and that was a per 1000 but it didn't like catch on the way a percent per 100 did although it's still kind of used if you look at batting averages in baseball you know when you see somebody that's hitting 350 that's actually a per meal that is 350 per 1000. anyway whatever don't want to confuse you more than i already have to all right exercise number seven let's take a look at one last one the price of gasoline this week is 3.60 per gallon it is projected to increase by 15 next week letter a how much will it increase and letter b what will be its price next week all right we're going to work in the next lesson exclusively with things that are increasing by a certain percent or decreasing by a certain percent i'd like you to try to work through this one well the first thing i want to figure out is how much will it increase they are literally telling me it's going to increase by 15 of this amount right so my increase itself my increase itself will be equal to 0.15 times 3.60 all right i don't know what that is so let's bring it out i've got .15 times 360. that's going to give me an increase of 0.54 or 54 cents right but that's just the increase right if i actually want to know what my price is next week then i'm going to take my 3.60 per gallon and add 54 cents to it that's easy enough so it's going to be 360 plus 0.54 and i'm pretty sure that's going to be four dollars and 14 cents all right easy enough right if i'm told where i start and what percent that i either increase or decrease it's easy enough to just figure out using that decimal multiplication and then add it to the original or subtract it from the original to get my final amount all right let's wrap this up so today we did a lot of different review right most importantly just the idea of a percent being a ratio out of 100 but the real takeaway you want to come out of here today with is that when i'm finding a particular percent of a number i want to do that by multiplying by the decimal version of that percent so if i'm finding 25 percent of a number i want to multiply by 0.25 if i'm finding 13 percent of a number 0.13 2 of a number 0.02 not 0.2 that would be 20 of a number all right so you do want to be careful and you can always figure out what you're going to multiply by by taking your percent and dividing it by 100 on your calculator of course all that does is effectively move the decimal place two units to the left in your number because that is the decimal equivalent of the percent all right we're going to get a ton of work with percents as we move on in this unit for now i just want to thank you for joining me for another algebra one lesson by emath instruction my name is kirk wyler and until i see you again keep thinking and keep solving problems