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Sums of consecutive integers

When solving a math problems involving consecutive integers, we can use algebra to set up an equation to solve for the smallest of the four integers. Created by Sal Khan.

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Video transcript

The sum of four consecutive odd integers is 136. What are the four integers? So before even attempting to tackle it, let's think about what it means to be a consecutive odd integer, what four consecutive odd integers could be referring to. So we could start, let's say, we had 3. The next odd integer-- so we want consecutive odd integers. So we want the next odd integer-- would be 5. Then the one after that would be 7, and the one after that would be 9. These are consecutive odd integers. Another example-- we could start at 11. Then the next odd integer is 13. The next one is 15. The next one is 17. The example of non-consecutive odd integers, if someone went from 3 straight to 7, these are not consecutive. The next odd integer after 3 is 5, not 7. So these are examples of consecutive odd integers. These are not consecutive odd integers. So with that out of the way, let's actually try to tackle this question. And I encourage you to pause this video right now and try to tackle this on your own before I take a go at it. Well as you can imagine, a little bit of algebra might be handy here. So let's let x be equal to the smallest of the four. So if x is the smallest of the four consecutive odd integers, how can we express the other three in terms of x? We'll just use these examples right over here. If that is x, then how would we express this in terms of x? Well, this would be x plus 2. The next odd integer is going to be the previous odd integer plus 2. You're essentially skipping the next number, which is going to be even. So if you added 1, you'd just get to an even number, so you have to add 2 to get to the next odd one. Well if you add 2 again, if you add 2 to x plus 2, well now you get to x plus 4. You add 2 to that. You get to x plus 6. And you see that again right over here. If this was x, then this is x plus 2. This is x plus 4, and this is x plus 6. So in general, if x is the smallest of the integers, we can define the other three as x plus 2, x plus 4, and x plus 6. And let's take the sum of them and set that equal to 136 and then solve for x. So the smallest one is x. Now the next smallest one is going to be x plus 2. Now the one after that's going to be x plus 4, and then the one after that is going to be x plus 6. So this is the sum of the four consecutive odd integers, and they tell us that that is going to be equal to 136. This is equal to 136. And now we can just literally solve for x. We have this one unknown here. So let's add these x terms together. So we have one x, two x's, three x's, four x's. So we can rewrite those literally as 4x. And then we have 2 plus 4, which is 6, plus another 6 is 12. 4x plus 12 is equal to 136. So to solve for x, a good starting point would be to just to isolate the x terms on one side of the equation or try to get rid of this 12. Well to get rid of that 12, we'd want to subtract 12 from the left-hand side. But we can't just do it from the left-hand side. Then this equality wouldn't hold anymore. If these two things were equal before subtracting the 12, well then if we want to keep them equal, if we want the left and the right to stay equal, we've got to subtract 12 from both sides. So subtracting 12 from both sides gives us, well on the left-hand side, we're just left with 4x. And on the right-hand side, we are left with 136 minus 12 is 124. Did I do that right? Yeah, 124. So what is x? Well, we just divide both sides by 4 to solve for x. And we get-- do that in the same, original color-- x is equal to 124 divided by 4. So let's see. 100 divided by 4 is 25. 24 divided by 4 is 6. 25 plus 6 is 31. And if you don't feel like doing that in your head, you could also, of course, do traditional long division. Goes into 124-- 4 doesn't go into 1. 4 goes into 12 three times. 3 times 4 is 12. You subtract, bring down the next 4. 4 goes into 4 one time. You get no remainder. So x is equal to 31. So x is the smallest of the four integers. So this right over here, x is 31. x plus 2 is going to be 33. x plus 4 is going to be 35. And x plus 6 is going to be 37. So our four consecutive odd integers are 31, 33, 35, and 37.