which of the following is the quantity you would label as “what you got”? This is a topic that many people are looking for. bluevelvetrestaurant.com is a channel providing useful information about learning, life, digital marketing and online courses …. it will help you have an overview and solid multi-faceted knowledge . Today, bluevelvetrestaurant.com would like to introduce to you M46: 2.4 & 2.6 Introduction to Problem Solving and Percent & Mixture Problems. Following along are instructions in the video below:
We have another little double up of sections on this. Video this is two four four and two. Six uh so.
24 is introduction to. Problem solving and 26. Is and mixture problems as well as some other problem.
Solving now the way our book works. Anytime. You see the phrase problem.
Solving you can kind of clue in this means. Word. Problems.
This is a sneaky publisher trick to not put the phrase word problems because most students tend to go when they see word problems. But thats what problem solving means and hopefully we can break things down and make them not as terrifying or as worrying as some of you might be feeling. And so im combining these two sections because they do go hand in hand.
Really well together. So we have two objectives to do the first is to again from the last video. Write.
Algebraic expressions from phrases and the second is to apply the steps of problem. Solving which we will go over after we see a couple examples or one example. So our first example.
We did this already were just reiterating to remind you so write the following as algebraic expressions. Then simplify so we have the sum of three consecutive integers remember we need to name our integers. We need to name them they tell us the first one needs to be x.
So if we have x the next consecutive integer. The one that comes right after x is one more than x. So x.
Plus. One and the one that comes after x. Plus.
One is x plus. Two. Then we can make the sum x.
Plus. X. Plus.
One plus x. Plus. Two.
And that simplifies. One. Two three xs total.
And one plus two is three three x plus. Three in part b. The perimeter.
Remember perimeter is just a geometric way of saying sum. So you add everything up. So.
The perimeter is going to be x. Plus. 5x.
Plus. 6x minus 3. Because those are our three side lengths.
Again. We collect like terms. We have 1 plus.
5. Plus. 6.
For a total of 12 xs and the minus 3. Is all alone so that simplifies to 12x minus 3. And i know that we did these in the last video and were just reminding and reiterating how to translate from one to the other this is the same practice problem so if you havent done it yet try it if you did do it.
See if you can do it again. And check your work for what you got last time. Were going to go on to our first word problem here so the three most populous metropolitan regions in the us.
Are the cities of new york la and chicago in that order the new york area has 61. Million. More than the double more than the double the population that shouldnt be there more than double the population chicago.
So were going to stop right here for a second. Were going to pause. And were going back to this so the three most populous regions.
So we have three cities new york chicago and la and we know they tell us that in this order is how big they are so this is the number one this is number two and this is number three. So while these arent variables in the traditional sense. They are acting as labels as names so we know what were talking about so now we can come back to where we pause and reread the.
Sentence the new york area has 61. Million more than double the population of. Chicago okay so new york is 61.
More remember more means. Plus. Then double the population of the chicago area double is two times.
So two times started to draw to write an h with a c two times. Chicago the population of the la area is. 01 million less than twice the population of the chicago area so la 01.
Less less opposite of more means subtraction so we were subtracting the number that they told us because its this much less so minus this number twice the population is a number another way to say double this is two times. The chicago area now it says write the sum of the populations of the three areas as an algebraic expression. Okay.
Thats what were doing up here and were going to let x be the population of the chicago area so were going to write this as. X which. Means we can translate this to 61 plus 2x and we could translate this to 2x minus.
01. And now were doing the sum so were adding everything up were doing new. York.
Plus chicago plus la so this is. 61. Plus 2x plus.
X plus 2x minus. 01. And then we simplify so we collect like terms we have 2 plus.
1.
Plus 2 is 5 total xs and then 61 minus. 01. Is going to be a positive.
6. So this is our algebraic expression for the sum of those populations. So this is really just another version of this that we did up here and in the last video.
But masked in a big block of text and so youll notice i like to underline and circle things and i tell myself where ive stopped break it up i recommend you do this stuff. It helps you break down. Whats being told and look at things individually sentence by sentence and make sure youre naming things and labeling things so that you know what youre talking about the best thing to do is always start with labeling.
What you need to talk about now. I didnt label. Any variables at first.
I just gave myself. Some some descriptions. Ny.
Ch and la and thats always a good start. If you want to jump to labeling your variables first because you can thats great too so if we flip. The paper here.
Theres a practice problem. Very very similar to the one that we just did about athletes in the us from the olympics. So you should try that one on your own.
But what i want to talk about now is the general strategy for problem. Solving and this is everything that we just did in the last example minus. Actually solving for something so number one is understand understand the problem.
Which means a couple things because a lot of times. You can say understand the problem. And you go great easy for you to say youre the teacher of course you just say understand it well.
When i say understand i mean read and reread the problem youll notice in the last one. I read it i got to a point. Where there was a lot of information being thrown at me so i stopped and i reevaluated what they had already told me.
And i reread and then i started making decisions so in this process you need to choose variables choose variables to represent unknown quantities and i like to draw a diagram if necessary right in the last example. Were talking about populations. Were not going to draw a picture of populations.
But sometimes its helpful to draw a picture for certain problems so once youve done that youve read you feel like you understand the gist of whats being asked of you remember you dont need to understand everything all at once you just need to understand whats being asked of you what variables have you labeled. What do they represent now you can move on to the translating part so translate to an algebraic expression. We did that a couple times.
Once youve translated using your variables. The next thing to do is solve. So this is the step that we didnt do we didnt solve anything so solve the equation.
The problem etc. Whatever theyre asking you for this is that step so in the last example they were asking us for the sum. They werent asking us to solve the sum they just wanted the sum thats the solving the problem.
Weve answered the question that they asked of us once youve solved. Its always important to interpret your results because word problems are always real world situations. Which means they have real world context and that context needs to come through in your answers.
Sometimes thats a sentence sometimes its a short paragraph more often than not the most common version of this is units. So you can have a sentence. A paragraph or units for a measurement is the most common interpretation.
If youre asked for the distance between here and some other city you give it to them you give the answer in miles theyll understand i asked for distance. They told me this number miles. I know exactly what the distance is so this is the most common form of interpretation that well use as units.
And you have to have it if you dont have your units. You dont have a complete answer. So now were going to revisit a problem that we saw in the last section.
So in the last section. We just wrote out and translated expressions now were going to actually solve were going to find the numbers. So lets walk through the steps first so find three numbers such that the second number is three more than two okay.
So im going to pause right here find three numbers. So just like we talked about in the last video that means were talking about three things i need to label them. Im going to call them x y and z.
Because im a traditionalist you can call them like we talked about any letters that you want and im going to say x is my first y is my second and z is my third number and i know i need to do that because they started talking about second number first number third number. So this is for me. This is so i know.
Which of my variables corresponds to what theyre talking about okay. So now we can continue on such that the second number y is is our as we said english way of saying equals. Without saying.
The word equals three more so three plus then twice the first number two times. My first number is x okay. The third number so we could pause there the third number for us is or for me is z is four times.
The first number four times x. Okay. The sum of three numbers of the three numbers.
Is 164. Sum of course. Meaning.
Plus. X. Plus.
Y. Plus. Z.
Equals. 164. So.
This. Is where we left off in the last figures. Okay.
This is translated completely now were going to solve now were going to find the three numbers. So what were going to do the other thing. We said x equals.
X. Because they didnt tell us anything about the first number. So it equals itself.
Were going to take these quantities and substitute them in to the equation. That has everything so were going to have x plus. 3.
Plus. 2x. Plus.
4x equals 164.
Now were going to pause here from actually solving it so i can talk a little bit about what i just did whenever you substitute. An expression in for a variable put it in parentheses. Now in this example.
It wasnt a big deal because nothings going to change. But imagine that this was a minus sign right here we talked about in the last one of the last videos. The minus sign needs to get distributed to both well.
If you forget to put parentheses. That might just look like 3 plus 2x. But in reality youre subtracting and adding the whole thing so every time you substitute something just write parentheses.
It takes less than half a second to do and it will save you a lot of headache. When you need to be distributing things so when you have all addition you just continue on like you never had parentheses. So im gonna collect like terms i have one plus two plus 4 is 7x and then plus 3 equals 164.
Im going to subtract 3 from both sides using my addition property you can think of it as adding negative. 3. If it helps you this cancels.
We have 7x equals 161 were going to divide by 7 so on the side here. 7. Goes into 16 twice with two left over bring down the one thats three so x equals.
23. Now were not done here. Because it wants us to find all three numbers.
So we need to continue on and find y and z. Now so the way that we do that is we take our 23 and we substitute it back in to our two expressions here so y equals. 3.
Plus. 2. Times.
23. This is 3 plus. 46.
So this is 49 and z. Equals 4 times. 23.
Which equals. 92. Now i have a shortcut for how i did that so fast.
If youre curious about it so 4 times. 25. Would be 100.
So this is just a little scratch work over here. How i do this in my head 4 times 25. Would equal 100.
This is 23. So this is 2 times. 4 less so its a hundred minus two times four.
Which is ninety two two times four is eight. So thats just a little trick for multiplying numbers a little bit faster. If youre not going to your calculator every time.
But so. Now we have our answers. So now.
X. Equals. 23.
Y. Equals. 49.
And z. Equals. 92.
And we can check our work. Before. We submit our work by just plugging it back into this so.
If we want to check. It over here 23 plus. 49.
Plus 92. Does that equal 164 and youll find that yes. It does when you check it so this is taking what we started with last time in the last video and then actually going through how to solve it now so you flip the page.
Theres a practice problem here again very very similar to what we just did so you can walk through the steps. I also have this nice table. Which is kind of a breakdown of how to interpret certain things for percentages and fractions and also english phrases so for example 10 percent the short.
Cutter the meaning that you might see this written as is 01. Or 1 10. Of a number or move.
The decimal point one place to the left 10 of 60 is 6 or. 6. Is an example of it same thing with 200 percent.
This means 2 or double a number 100 means one or all of that number 25 percent is a quarter or one fourth of a number. So theres a little bit of breakdown. So you can better uh.
Interpret and translate your problems so. Oh. It looks like oh.
Its on the back okay nevermind. So were going to try another example here percentage. Problem now so suppose a computer store just announced an eight percent decrease in the price of a particular computer model maybe they got the new model in for this year uh if this computer sells for two thousand one hundred sixty two dollars must be an apple after the decrease find the original price.
So what were looking at is an eight percent decrease. So decrease means that the starting price is higher than 2162 because it tells us the computer sells for 2162 after the decrease. So we want theres two ways we can do this the easiest way is if i had a hundred percent take away 8 percent for that decrease.
I have 92 percent of whatever the value was is now 2162. So now were just going to call this x. So now our job is to take this and make it something we can actually calculate so.
92. Percent as a decimal. Is.
092 of is our way to say multiply in english so x. 092. X.
Is is the equal sign.
2162 now we have an equation and we can solve divide by. 092 divided by 092. And again this would be a calculated problem.
I wouldnt expect you to know this off the top of your head and when i punch that into my. Calculator i get x. Equals.
2350. Now. We are not done.
This is a word problem. So. The last step is to interpret so what is this thing well this is dollars.
So that means the starting price was two thousand three hundred fifty dollars have to interpret so i have one last example to go through in this section. Theres a practice problem thats another percentage problem for you to try out on your own. So here.
We have a mixture problem. So xavier pike. A pharmacist needs.
70 liters of a 50 alcohol. Solution. He has 30 percent alcohol.
Solution and an 80 alcohol. Solution. Available.
How many liters of each would he need to mix to obtain 70 liters of a 50 solution. So. This is a little bit different so we want to think about is how much we need so were going to make a little table okay.
This is the amount needed this is our percentage of alcohol. And well leave this one blank for a second so. We know if we have our percentages here.
30. 80. Are what we have fifty percent is what we need we know the fifty percent.
We need is seventy. So now we dont know how much of the thirty or eighty percent. We need so thats an unknown variable so im going to call this x.
Now a lot of people might jump to okay. Well. Lets call this y.
Then but that gives us two unknown variables. And we dont have a good way of solving an equation for two variables. Yet thats something well be looking at later.
But we do know if we have x liters of our 70 liters as the 30 percent. Then this needs to be whatever 70 minus x. Would be in liters.
So now what were going to do is multiply. Here to here as a decimal. So this is point three x.
Here to here. This is point eight times the quantity seven minus. X and here to here is going to be 05.
Times 70. And the reason we do that is because we are mixing. Were mixing r2 solutions that we have so.
That means were going to have. 03. X added to.
08. Times. 70 minus.
X and we know that that has to equal. 05. Times 70 so now were going to distribute.
The 08 so we have 03. X. Plus 56 minus.
08. X. Equals.
35. One half of 70 collect like terms point three minus point. Eight.
Is minus point five x. Plus. Fifty six should equal thirty five.
Were going to subtract 56 from both sides using our addition property of equality and we have negative. 05 x should equal a negative 21 divide by negative. 05 divide by negative.
05 negative over a negative makes a. Positive. 21 divided by 05.
Doubles up 21 to make 42 and this is liters now we need to interpret this x. Coming back here represented 30 percent. So this is 42 liters of the 30 percent solution and that means 70 minus.
42. Which is 28 well quick math here. 70.
Minus. 42. Equals.
28. So 28 liters of the 80 solution. And you can do a little little check a little reality check for yourself.
If we have in 30 and an 80 and we need to get a 50. It makes sense that we should have more 30 than 80. Because the more of this had we have the closer to 80 will be so since we need to be lower.
We should have more of the less percentage solution and this would be our final answer here. Okay. So i know that this one was a little long 25 minutes.
So ill try and make up some time in the next one to make it up to you. But this is the end of 2 4. And 2 6.
So two sections for the price of an extra five minutes .
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