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Sum of integers challenge

Sal solves the following problem: The sum of three consecutive odd integers is 231. What is the largest integer? Created by Sal Khan and Monterey Institute for Technology and Education.

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  • mr pink red style avatar for user lily zhu
    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.
    (1 vote)
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  • blobby green style avatar for user jadescar214
    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
    (4 votes)
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    • duskpin ultimate style avatar for user Neal
      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
      (1 vote)
  • blobby green style avatar for user saba
    what is the sum off all integers from 1 to 100
    (5 votes)
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  • aqualine ultimate style avatar for user Thegeniustwins
    two consecutive odd integers sum to -8 what are they? guys can I have an answer plz
    (3 votes)
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    • stelly blue style avatar for user Kim Seidel
      Odd integers increase by 2. So you need to use X=1st odd integer and X+2=2nd odd integer
      "Sum" means you add them to get -8
      X + X + 2 = -8
      Now, solve for X to find the 1st number.
      Then, add 2 to find the 2nd number.
      Hope this helps.
      (5 votes)
  • aqualine tree style avatar for user Semira Fernando
    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.*
    (3 votes)
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    • aqualine tree style avatar for user NeverGiveUp
      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?
      (4 votes)
  • duskpin seed style avatar for user Laura Hollis
    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.
    (4 votes)
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  • blobby green style avatar for user 22mercb
    how'd you get +6? in the equation?
    (2 votes)
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  • area 52 purple style avatar for user Aadya
    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
    (2 votes)
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    • stelly blue style avatar for user Kim Seidel
      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).
      (3 votes)
  • starky sapling style avatar for user Swetha S. #20
    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.
    (1 vote)
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    • primosaur seed style avatar for user Ian Pulizzotto
      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.
      (4 votes)
  • primosaur seedling style avatar for user Claire Farma
    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?
    (1 vote)
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    • cacteye blue style avatar for user Jerry Nilsson
      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.
      (2 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.