More equation practice
Integer sums Adding sums of consecutive integers
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- Welcome to the presentation on finding sums of integers.
- You're probably wondering why are we doing this within
- the context of averages.
- Well, if you think about it, all an average is is you take
- a sum of a bunch of numbers and then you divide by the
- number of numbers you have.
- What we're going to do here is do a couple of algebra problems
- that involve just the sum parts first, and actually they can
- carry over into average problems as well.
- Let's get started with a problem.
- Let's say I told you that I had the sum of five consecutive
- integers is equal to 200.
- What is the smallest -- I apologize for my handwriting
- -- what is the smallest of the five integers?
- Well there's a couple of ways to do this, but I guess the
- most straightforward way is just to do it algebraically,
- I would say.
- So let's say that x is the smallest of the integers,
- right, so x is actually what we're going to want to figure out.
- Well if x is the smallest, what are the other four going to be?
- We have a total of five.
- Well, they're consecutive.
- Consecutive just means that they follow each other,
- like 5, 6, 7, 8, 9, 10.
- All of those are consecutive integers, right?
- And if you remember, integers are just whole numbers, so it
- can't be a fraction or a decimal.
- So if x is the smallest, so then the next integer is
- going to be x plus 1.
- And the one after that's going to be x plus 2.
- And the one after that's going to be x plus 3.
- And the one after that's going to be x plus 4, right?
- It might seem confusing I'm writing all of these x's.
- But if you think about it, if x was 5, then this would be 6,
- this would be 7, this would be 8, and this would be 9.
- And that's all I'm writing here, right?
- So these would be, assuming that x is the smallest of the
- integers, the five integers would be x, x+1, x+2, x+3, and x+4.
- And we know that the sum of these five consecutive integers is 200.
- What is the sum of these five, I guess we could say, numbers or expressions?
- Well let's see, we have five x's -- 1, 2, 3, 4, 5.
- So x plus x plus x plus x plus x is equal to just 5x.
- Or you could just say 5 times x.
- And then that's plus 1 plus 2 is 3, 3 plus 3
- is 6, 6 plus 4 is 10.
- So the sum of these five integers is going to be 5x plus
- 10, and all I did is add up the x's and added up the constants.
- And we know that that is going to equal 200.
- Now this is just a level two linear equation.
- We can just solve for x.
- So we get 5x is equal to 190 -- I just subtracted 10 from both sides, right?
- And then x is equal to -- let me divide 5 into 190.
- 5 goes into 19 three times, 3 times 5 is 15.
- 9 minus 5 is 4, bring down the 0.
- 5 goes into 40, eight times.
- So x is equal to 38.
- Pretty straightforward problem, don't you think?
- Now what if I were to ask you what is the average of the five consecutive numbers?
- Well now, there's two ways of doing this.
- Now that we already know that x is 38, we know that the other
- numbers are going to be -- well this is 38, 39, 40, 41, 42.
- Well we could just average these four numbers.
- You could just say 38 plus 39 plus 40 plus 41 plus 42.
- And well we already know what those -- I don't
- even have to do the math.
- You already know that they average up, they sum up to 200
- and then we divide the sum by 5, because there are 5 numbers.
- So the average is 40.
- There are a couple ways you could think about that.
- One, you see 40's just a middle number so that makes sense.
- And the only time we can really say it's the middle number
- is when the numbers are distributed evenly around 40.
- If we had a number that was much smaller than 40 or
- something, you couldn't just necessarily pick the middle number.
- But in this case these are consecutive and makes sense.
- Another way we could have done this problem, if you were, say,
- taking the SAT and they were to ask you the sum of five
- numbers is 200, what's the average of the numbers?
- Well you say, well, all I have to do is divide that 200
- by 5 and I'll get 40.
- Let's do another problem and I'll make it a
- little bit harder.
- Let's say the sum of seven odd numbers, and let me make up a
- good -- I hope this one works, I'm going to try to do it in
- my head -- is 217, what is the largest number?
- I shouldn't say number -- seven odd integers.
- Actually it becomes a much harder problem if it was just
- seven odd -- well actually, the only thing that could be odd
- are integers anyway, so you could almost assume it.
- But the sum of seven odd integers is 217.
- What is the largest of the integers?
- As you can tell I'm doing this on the fly.
- Actually my wife just diagnosed me with, she thinks I
- have benign vertigo.
- I got very dizzy this morning when I went to work, so you
- have to forgive me for that as well.
- That's impairing me even more.
- So let's do this problem.
- Let's say that x is the largest.
- Then what would the number right below x be?
- Would it be x minus 1?
- Well, if x is an odd number, x minus 1 would
- be an even number.
- So in order to get the number right below it, we have to
- do x minus 2 to get another odd number.
- My apologies -- it should say the sum of seven
- consecutive odd.
- I don't know if you assumed that.
- I'm trying my best today to confuse you.
- The sum of seven consecutive odd integers is 217.
- What is the largest of the integers?
- So if x is the largest, then to next smallest one would be x
- minus 2, right, because it's consecutive odd numbers,
- not just consecutive.
- So consecutive odd numbers are like 1, 3, 5, 7 -- you're
- skipping the evens, right?
- So that's why you're going up or down by two, depending
- how you view it.
- So the next one down would be x minus 2, then we'll have x
- minus 4, x minus 6, x minus 8, x minus 10, x minus 12.
- I think that's it.
- One, two, three, four, five, six, seven, right.
- Those are seven numbers.
- They're separated by two.
- X is the largest of them, right?
- We can assume that they're odd because apparently
- the problem will work out so that they're odd.
- So what is the sum of these seven numbers?
- Well the seven x's just add up to 7x.
- And then let's see, 2 and 4 is 6, 6 and 6 is 12, 12 and 8 is
- 20, 20 and 10 is 30, 30 and 12 is 32.
- So 7x minus 32 is equal to 217.
- We just solved for x.
- 7x is equal to -- let's see, if we add 32 to both sides
- of this equation we get 249.
- Let's see, 7 goes into 249 -- is that right?
- Right.
- So 7 goes into 249 -- did I do this addition properly?
- I want to make sure.
- 2 plus 4 is 6, 6 plus 6 is 12, 12 plus 8 is 20, 20 plus 10
- is 30, 30 plus 12 is 42.
- Oh, here you go.
- See, my mathematical spider sense could tell that something
- was fishy about this.
- So that's 7x minus 42.
- So if we add 42 to both sides it's 7x is equal to 259.
- See how brave I am, I do this thing in real time.
- 259.
- So 7 goes into 259 -- let's see, 7 goes into 25 three
- times, 3 times 7 is 21, 49 -- it goes into it 37 times.
- So we get x is equal to 37 and we're done.
- So just to review because I think had a lot of errors in
- this problem when I presented it.
- The question was the sum of seven consecutive odd integers is 217.
- What is the largest of the integers?
- I said x is the largest, and then if x is the largest, the
- next smaller one will x minus 2.
- Because we're not saying just consecutive integers,
- we're saying consecutive odd integers, right?
- So if x is 37, which is what we solved for, then x minus 2 is
- 35, this is 33, this is 31, this is 29, this is
- 27, this is 25.
- And then we just added up all the x's and I'll add up all
- the constants and said, well they add up to 217.
- And then we just solved for x.
- I think you're now ready to try some of these problems.
- Have fun.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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