Math 60 - Basic problem solving - Hi

Hi, class. Today we're going to be doing some basic problem solving. And I'm going to be going through some basic problems that are going utilize some words that we discussed in the video called "Translating Words to Math.” So, if you need to go back and review that video to review those values, please feel free to do so. When we're dealing with these problems, what we're going to do is we're going to go through each word, one at a time and very slowly translate them from words into math. And we're going to have a couple different situations in this video that are typical things that you will come up with. And you're still just translating words into math, but you're dealing with certain situations that might deal with some percents [ % ] as this first one does, or situations where you have a constant increase to some value. So, we're going to take a look at this particular one and go through just reading it once, very slowly but kind of normal. And then going word by word translating it. And then going back through a third time to see if what we wrote down mathematically actually matches the words in the problem. And then we'll actually solve the problem. So, if we take a look at this first problem here, we have "What is 10% of 350?” So, we're going to start with this word "what.” And "what" we don't know what it is, so we're going to give it a variable. How about if we just call it "x"? So, we've translated this "what" into "x.” Now, as we read on, the next word is "is.” And if you'll remember from translating our words into math, that this word "is" here means "equals.” [ = ] So, now we'll go ahead and put an equals [ = ] down. Then the next thing we come to is this 10%. And any time you come to a percent [ % ], you want to put it down as a decimal, which is going to be 0.1. Then, the next thing that we come to in this is the word "of.” And if you'll remember, the word "of" when we look at it mathematically is "multiply.” So I'm going to put a dot down [ · ] for multiply. And then the final thing in the sentence is 350. So, I'm going to go ahead and write down our 350 to correspond with that. Now, let's go word for word and see if what we wrote matches. We have "what" which is x and we don't know what it is so that's what it should be. "Is" is equals. [ = ] 10% is 0.1 which is the equivalent as a decimal. "Of" means multiply [ · ] and then 350 is what we have here. Now we just need to solve. It does match. And to solve it all we need to do is multiply the .1 and the 350, [ 0.1 · 350 ] so x ends up being 35. [ x = 35 ] Now, that is a basic percent problem where we have, whoops! Just a second. Where we have problems where we're translating, and we're using our decimals. I'm going to erase this real quick, and then we'll go back to our situation. Now, this one had the "what" in the first spot that we have right here. We're going to take a look at a problem where maybe we don't have the "what" in that first spot. Notice here that we start out with a percent instead of "what.” So this one is "20% of what number is 8?” It's similar to the last problem, but because things are rearranged, it's going to translate slightly different. We had our variable by itself in the last problem. Let's see what happens in this one. We're going to deal with it the same way. 20% we're going to write down as a decimal. So this 20% is 0.2. If we continue on, "of" is multiply, so we will use a dot [ · ] for that to be multiplied. And then we get to "what number.” Well, we don't know what number, so let's let the "what number" be our variable, which we'll just use x. Continuing on, we have "is" which translates into "equals.” [ = ] And then we have the number 8. So, notice that it's a very similar problem to the last one, but things are slightly rearranged. Let's make sure what we have matches the sentence. So 20% is the 0.2. "Of" is multiply. [ · ] "What number" is x. "Is" is equals [ = ] and 8 is our value at the end. Now, to find the answer to this particular problem, all we need to do is solve this equation. And to solve it, we divide both sides by 2 tenths. [ 0.2 · x/ 0.2 = 8/ 0.2 ] These cancel. [ 0.2/ 0.2 ] And 8 divided by that ends up being 40. So, notice if you take 20% of 40, you will get 8, so that is our solution to this particular problem. Let's look at one more problem that is similar to that one however, things are rearranged into a third way of rearranging our percents. So, we started out with the "what" in the beginning spot. Then we went to the "what" in the middle. And now, rather than just having "what" we have "what percent” [ ? % ] Now for this particular problem, I'm going to give you a method that's very similar to what we just did, however you'll need to remember that at the end, you need a percent for your answer. And you're going to have a decimal that's going to have to be switched back into a percent. So, the "what" in this particular problem here, is going to be written down as an x The percent is going to tell us that we need to change that value that we get for x into a percent at the end. And then we have "of" which is multiply [ · ] And we're multiplying by 30. Then we come to "is" which is equals [ = ] And what it equals is 6. So let's look at what we have here. "What" is the x. We're going to switch that x to a percent at the end. "Of" is your multiply. [ · ] 30, whoops. "Is" is equals, [ = ] and 6 is our number on the end. Now, let's solve this. [ x · 30 = 6 ] To solve it we divide both sides by 30. [ x · 30 /30 = 6/30 ] And x ends up being 6 divided by 30 [ x = 6/30 ] which you can do in your calculator. You get 0.2. Which, remember we need a percent, so that's the same thing as 20%. [ 0.2 = 20% ] So, 20% is what you're looking for, because you need that as a percent. So that's another type of a problem that deals with your percents and has things written in slightly different manner. Let's go ahead and erase this and take a look at one other type of problem. Now, what we're going to do now is take a look at translating some words directly into an equation, and then solving the equation. So, our situation here is that: "three times the sum of 5 and a number is 48" and we're to find the number. So, when you have "3 times" that means we're going to multiply 3 by something. We now need to see what we're going to multiply it by. And it's going to be "the sum" Well, remember "sum" is the answer after you add? So you need to find out what you're going to add together, because that entire thing is what you're going to be multiplying the 3 by. And we're going to be "adding 5 and a number" Now, we don't know what the number is so we can use any variable for it. How about we use "n" this time? So, we're going to multiply 3 by our sum of 5 and a number we're going to call "n" this time. [ 3( 5 + n ) ] Then we get "is" here, which is equals [ 3( 5 + n ) = ]. And what it equals is the 48 [ ... = 48 ] that we have on the end [ 3( 5 + n ) = 48 ] When you hit the end of a period for a sentence, that's normally the end of an equation. So, let's look at what we have here. "3 times" is the 3. And we need to see what we're multiplying by. We're multiplying by the sum of 5 and a number. So that's the 5 plus n [ ( 5 + n ) ] in parentheses, because we're multiplying by the entire sum. "Is" is equals, and it equals the 48. [ = 48 ] Now we need to find the number, so we're just going to go ahead and solve this. First you need to distribute your 3 to get 15 plus 3n equals 48. [ 15 + 3n = 48 ] Now, we need to move this 15 here, because it's being added to our variable and all our variables are on one side. The opposite of a positive 15 is a negative 15. So let's subtract 15 from both sides. [ 15 - 15 + 3n = 48 -15 ] That will make these cancel. [ 15 - 15 = 0 ] And we get 3n equals 33. [ 3n = 33 ] Now, to move the 3, since it's being multiplied by the n, we're going to divide both sides by 3. [ 3n/3 = 33/3 ] And the number ends up being 11. Now let's see if that number is correct. If you add, take a look at your thing and you add 5 and the number, well, our number was 11. [ 3( 5 + 11) = 48 ] So adding 5 to that gives us 16. [ 3( 16 ) = 48 ] Then if we multiply 3 by it, we get 48, [ 48 = 48 ] and that's what we were supposed to come up with. So, we know that 11 is the number that we're looking for. Now, that was one where we were just directly translating numbers into an equation. What if we have a situation that is more real life and we are translating it into an equation? So, we're going to do this particular problem here where you're dealing with average mortgage payments over time. Now, this says that the average mortgage payment in 2004 was $960 per month. Payments have increased by an average of $11.40 per year, and this is a hypothetical situation. If the trend continues, How many years after 2004 will mortgage payments average $1,074 per month? And, we also want to know in what year this occurs. So, the first thing we're going to do on this, because there's a lot of information is figure out what is the unknown that we don't know. And we don't know how many years after 2004 this thing here will occur. So, why don't we name a variable and, since we're looking for years, I'm going to use 'y' It doesn't matter what you use. I'm going to say, "Let y equal the number of years after 2004.” Now, since it's the number of years after 2004, when you're in 2004, your year would be zero. If you're in 2005, your year would be one, 2006 your year would be 2, and so on. Now, let's think about what happens. Every year your payment is increasing by an average of $11.40 per year. So, your first year, you're going to have $960 for a payment. Then the next year, you're going to have $1,140 more, and then the next year another $1,100, Ugh, $11.40 more So, each year, y, you're going to add another $11.40. So, if you don't know what y is, what you could do is just take the $11.40 and multiply it by the number of years, then add it to what your payment was in 2004, and you'll know your payment. So, this here [ $960 ] is your starting amount, plus how much it changes each time, [... + $11.40 ] times each period [ · y ] which is going to be a year in this case. Now, what we really want to know is when is this going to be $1,074. so we want to know when is this value [ equation ] = $1,074. Now that's the equation that we believe we're going to be solving. Let's read through and see how we got that. First thing we knew is, it was asking for how many years after 2004. So we let our variable be the number of years after 2004, because that was the unknown. Then we had to see how you get your payment for those years after. You start with this value from 2004 [ $960 ] and then you add on $11.40 for each year. So that's the $960 plus $11.40 for each year. [ $960 + $11.40y ] And we want to know when is that going to equal your $1,074. So now all we have to do is solve this [ $960 + $11.40y = $1074 ] First thing we need to do is move our 960 by subtracting 960 from both sides, since we're solving for our y [ 960 - 960 + 11.40y = 1074 - 960 ] These cancel [ 960 - 960 = 0 ] and we get $11.40 times our y is going to be $114. [ 11.40y = 114 ] Now, we just need to move this [ $11.40] which is being multiplied by our y, so we'll divide by $11.40. [ 11.40y/11.40 = 114/11.40 ] And when you divide those two values, you end up with y being 10 years. [ y = 10 years ] So now we know the answer to the first question, which is how many years. But then it says, "In what year will this occur?” Well, 10 years after 2004 is going to be the year 2014. So, that is when this will happen based on this situation. So that's another type of situation that you may have to deal with when you're dealing with problem solving. Let's take a look at another one. This is a typical one that you will deal with when you are shopping. Oftentimes you will have a ticket on an item that you're purchasing that will just have the sales price. And maybe on the top of the rack it says what the percentage off was supposed to be, and that the prices are marked. Well, you might want to know if the price marked is correct or not. So, this situation becomes very valuable for you. And let's say that after a 20% reduction, a shirt was marked $40. And you want to know what the original price is, just to see if things were marked correctly. For this, you have to kind of think of how the sale price of $40 was obtained. So, let's go ahead. We don't know the original price. Let's let P be the original price, since "price" starts with P. The way this $40 was obtained was, you took the original price and you subtracted from it 20% of that original price, which means 20% times that original price. And 20% as a decimal would be .2 So 20% of that price subtracted from the original price gave you the $40. So, all we need to do is solve this and we'll know what the original price was to find out if that $40 was correct or not. If the original price didn't match, then you knew that it was marked incorrectly. So, this P right here is really a 1P. So if you take one minus 0.2, [ 1 - 0.2 ] you get 8 tenths times P. [ 0.8P ] Now if we go ahead and divide both sides by .8, [ 0.8P/0.8 = 40/0.8 ] we find out that that original price should have been $50. If it wasn't $50, and it was actually more than $50, you're getting a good deal. Somebody mismarked it in your favor. If it's less than $50, then you know that somebody didn't mark it correctly, and you should have them correct it before you actually pay. Now, let's take a look at one that's similar to this kind of situation, but is another one that people typically do incorrectly. And this one's dealing with salary. So, we're going to take a look at this particular situation where we're going to say that you have gotten a raise, and the raise made it so that your new income was $33,000 a year. And that was after you got this wonderful 10% raise over last year's salary. And we want to know what was the last year's salary? Well, we need to think about how they should have obtained this $33,000. And we don't know last year's salary, let's say. So, let's let, since salary starts with S, let's let S equal last year's salary. The $33,000 should have been obtained by taking last year's salary, and then adding on to it 10% which is 0.1 as a decimal times last year's salary. That's how we should have obtained this $33,000 for the new salary. So if we solve this, [ S + 0.15 = $33,000 ] we'll know what last year's salary should have been, to figure out if they calculated the new salary correctly. So, remember this S here has an invisible 1 here. So if you add 1 and one tenth, [ 1 + 0.1 ] you get 1 and one tenth, 1.1S equals 33,000. [ 1.1S = 33,000 ] Now, just divide both sides by 1.1 [ 1.1S/1.1 = 33,000/1.1 ] and you would find out that last year's salary should have been $30,000. If it wasn't, then you know that your new salary was calculated incorrectly. Those are some basic situations that you will be dealing with with your basic problem solving sections in your book, so you're ready now to try some just basic problems. When you're done with those, come back, and I'll show you a nice way to deal with percent problems that might deal with more than one account that's getting interest.