The sum of 3 consecutive integers is 138. what is the middle number?

I like to toy around with these types of questions with a simple tactic. First, I divide my sum by how many consecutive numbers there are (in this case, that’d be 3). This gives me a general area of where my 3 numbers are.

138 / 3 = 46

Since I got a whole number, 46 is one of my three numbers, but this doesn’t conclude it as our first, middle, or last number; however, we can toy around with the general area to determine it.

There is a quick way to finding it, too. In order to determine that, look at the final digit in the total sum (for 138, it’s 8). The ones’ places of our numbers must add up to 8 or a variation of it (18, 28, and so on). Since we already have one number, 46, we can determine the other two easily. Either the ones’ digits will be 4 and 5, 5 and 7, or 7 and 8. (Hopefully you’re catching my drift here. Feel free to ask for clarification if you’re confused by this tactic.)

Now, we must add these three integers up and determine which of them sum up to 18/28/38.

4 + 5 + 6 = 15, so 44, 45, and 46 are not applicable integers. This also concludes that 46 is potentially a middle or first number.

5 + 6 + 7 = 18, so 45, 46, and 47 are potentially applicable integers, and 46 is the middle number in this case; however, we shouldn’t conclude this as our answer yet as we haven’t attempted our remaining potential solution.

6 + 7 + 8 = 21, so 46, 47, and 48 are NOT applicable numbers; therefore, our solution is 45, 46, and 47. (45 + 46 + 47 = 138.)

Our middle number is 46.

I hope this helps!

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