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### Course: 8th grade > Unit 2

Lesson 4: Equations word problems# Sum of integers challenge

The sum of three consecutive odd integers is 231. First let's identify the integers. If the smallest is x, what is the value of the next odd integer? Then we set up an equation, adding all three integers to get their total. We can solve for x, the smallest integer. Once we know the smallest, what is the largest integer? Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- I think there is a little mistake.

The statement says : the sum of three consecutive odd integers is 231. But Sal is using even integers which is x+2 and x+4 to solve for this question.(4 votes)- x, x+2, and x+4 are just any 3 consecutive numbers that differ by 2. They're neither even nor odd.

It's the fact that they add up to 231 that makes them odd - if they added up to 234 they would be even.(21 votes)

- hi i have a question.

find three consecutive odd numbers such that the sum of 3/5 of the first, 1/2 of the second and 3/8 of the third is 63.

i tried this and got so many different weird answers(7 votes)- Let us denote our variables as a, a+2, and a+4.

Based on your equation, we have 3/5a + 1/2(a+2) + 3/8(a+4) = 63

Simplifying, we get 3/5a + 1/2a + 1 + 3/8a + 3/2 = 63.

Then get a common denominator:

24/40a + 20/40a + 1 + 15/40a + 60/40 = 63

Then, adding, 59/40a + 1 + 3/2 = 63

Subtracting the integers, 59/40a = 121/2

Multiplying leaves us with 59a = 2420

Finally, dividing, a = ~41.02.

Therefore, there are no integers that will satisfy this equation, only repeating decimals.

Hope this helps(5 votes)

- what is the sum off all integers from 1 to 100(8 votes)
- I have a weird way of doing it. Not sure if it is correct? Pls tell me if it is not right. So for Sal's question there are 3 odd consecutive integers and the sum 231. so I divided 231 by 3 and got 77. then I go on and do 77+79+81=237 which is not equivalent to 231. I move a number back 75+77+79=321 and I get my answer(6 votes)
- Your method works. The answer you find would be the middle number. You are basically treating the starting number as the total to find an average. Dividing 3 finds the average result (which will be the middle number).

The potential issue is that is you were instructed to find the answer algebraically, then you aren't. Solving algebraically would require finding the pattern in the numbers and using a variable to create an algebraic equation, then solve that equation. It pays to learn / understand how to do this because there are other word problems that you must solve algebraically (not direct mathematical technique).(8 votes)

- Can someone please solve this equation that I found online. It's hard.

*The product of 3 whole numbers is 72. What is the maximum possible sum of these three whole numbers.*(5 votes)- In how many ways can you express 72 as the product of 3 natural numbers (unordered ... Then the possible cases are ... How do you find the sum of all natural numbers amongst first one thousand ... The number of ways the 3 powers of 2 is distributed over the three variables ... What is the maximum value of their product?(7 votes)

- Hi. I’m struggling with how the problem below is solved. I have the answer but not sure how it was worked out. I’ve only worked problems where the sum was given. I want to understand how they got the answer.

Problem: Two times the sum of three consecutive odd integers is the same as 23 more than 5 times the largest integer. Find the integers.(6 votes)- Although I did not calculate the equation, I think you can write the equation like this:

2(x+x+2+x+4)=5(x+4)

Hope it gives a sense of how to solve the problem!(2 votes)

- At2:10, I would have added 6 to both sides of the equation rather than subtracted 6 because that would get 3x + 12 on the left-hand side. Then, you can divide both sides by twelve to get x + 4, which is the largest integer.

Also, I may have set up the original equation as x+(x-2)+(x-4)=231, because then x= the largest integer and it will be easier to answer the question. Do you agree with my methods? Thanks!(5 votes)- Yes. Very clever but even easier is divide by three to get middle one.(2 votes)

- I thought that the number 1 is an odd number because in both videos you use 3 as the smallest odd iteger instead of 1. Why is that so?(4 votes)
- When Sal says that 𝑥 is the smallest odd integer, he means that it is the smallest of the three consecutive odd integers that add to 231.

Then to find the other 2 odd integers, he uses 𝑥 = 3 as an example.

If the smallest of three consecutive odd numbers is 3,

then the others are 5 and 7, which we can write as 3 + 2 and 3 + 4.

Thus, if 𝑥 is the smallest of three consecutive odd integers,

then the other two are 𝑥 + 2 and 𝑥 + 4.

– – –

By the way, 1 is only the smallest*positive*odd integer.

−1 is also an odd integer, as are −3, −5, −7 and so on, forever.

So, in fact, there is no smallest odd integer.(4 votes)

- So I have a question, how do you solve this problem:

Find the sum of all the integers between 10 and 899 that are divisible by 4?

I'm not sure how exactly to start off this problem, but I understand what is being asked.(2 votes)- First, we need to find the lowest integer (first term) and the highest integer (last term) that need to be added. Because the integers being added are divisible by 4, the lowest is 12 and the highest is 896. So the first term is 12 and the last term is 896.

A neat property of any series whose terms are equally spaced (i.e. any arithmetic series) is that the average (mean) of all the terms equals the average (mean) of the first and last terms alone! So the average (mean) of all the terms is just (12+896)/2=454.

So now we need to find the number of terms. The terms are the multiples of 4 from 12=4*3 to 896=4*224, inclusive. Therefore, there are 224-3+1=222 terms.

Finally, the sum of the terms is the average (mean) of the terms times the number of terms: 454*222=100,788.(7 votes)

- Why does he use even numbers to solve this question. When it says, the sum of three consecutive odd intergers? I have been stuck on that for a while.(4 votes)
- If you use "2n+1 + 2n+3 + 2n+5 = 231" you get the correct answer, and all these numbers are actually odd, regardless of what n is.

"x, x+2, and x+4" could represent either consecutive odd, or even, integers (depending on what? on whether x is even or odd).

Sal's way works because the sum constrains the numbers to come out right.(3 votes)

## Video transcript

We're told that the sum
of three consecutive odd integers is 231. What is the largest integer? So let's think about
this a little bit. Let's say that x is the
smallest integer. x is equal to the smallest
odd of these three. It's not the smallest odd
integer of all integers, it's is the smallest odd of
these three, the smallest odd integer. So what's the next
one going to be? Well, if I have one odd integer,
what's going to be the next odd integer? Let's think about this. If x was 3, what's the
next odd integer? It's 5. And then what's the next
one after that? It's 7. And the next one after that? It's 9. So every time we add 2. So if the smallest one is x, the
next smallest odd integer is x plus 2, is equal to the
next smallest odd integer-- I'll write integer here--
odd integer. And then what would
be the next one? Well, we're going to add
2 to this one, right? So it's going to be x plus 4. Think about it. If the smallest is 3, then you
have x plus 2, which is 5. And then you have x plus
4, which is 7. So this will be the largest of
the consecutive odd integer in this group. And they tell us that the sum
of these consecutive odd integers is 231. What is the largest integer? So if I take x, x plus 2 and x
plus 4, and I sum them, they should be equal to 231. So let's do that. So we have x plus x plus 2, plus
x plus 4, and this needs to be equal to 231. And when they ask us what's the
largest one, we're going to have to tell them what
x plus 4 is equal to. So let's just solve
this equation. So let's add our x terms. We
have one x, two x, three x's, so we get 3x plus-- and then
what are our constants? We have a 2 and we have a 4. So 3x plus 6 is equal to 231. Now, let's get rid
of the 6 from the left-hand side of the equation. The best way to do that it is to
subtract 6 from both sides. So let's subtract 6
from both sides. The left-hand side, we're
just left with the 3x. The 6's cancel out. The right-hand side,
231 minus 6 is 225. We have 3x is equal to 225. To isolate the x, let's just
divide both sides by 3. The left-hand side, the 3's
cancel out-- that was the whole point behind dividing by
3-- we get just an x is being equal to-- and 225
divided by 3. Let me do it over here. So 3 goes into 225. It goes into 22 7 times. 7 times 3 is 21. 22 minus 21 is 1. Bring down the 5. 3 goes into 15 5 times. 5 times 3 is 15. Subtract, no remainder. So 225 divided by 3 is 75. So the smallest, the smallest
of the odd integers is 75. So this one is 75. What's x plus 2 going
to be equal to? Well, that's going
to be 2 more, 77. And what's x plus 4 going
to be equal to? Well, that's the largest
of them. x is 75 plus 4 is
going to be 79. And notice, we have three
odd integers. They're consecutive. They're the, you know, they're
the odd integers that come directly after each other. And let's verify that when we
add them up, we get 231. So if we get 75 plus-- let me
just write it like this-- 75 plus 77, plus 79, want
to add them all up. 5 plus 7 is 12. 12 plus 9 is 21. Carry the 2. 2 plus 7 is 9. 9 plus 7 is 16. 16 plus 7 is 23. So there you have it. The three consecutive odd
integers, when you add them up, you got 231. They're consecutive and odd. They ask, what's the largest? The largest is 79.