Find Three Consecutive Numbers Whose Sum Is 108

You're staring at a homework problem, or maybe a puzzle in a newspaper, and it reads: Find three consecutive numbers whose sum is 108.*

Your brain might freeze for a second. "Consecutive." "Sum." "108." It feels like a trick. That said, it’s not. It’s just algebra wearing a trench coat Most people skip this — try not to..

Let’s solve it. Then let’s talk about why this specific type of problem shows up everywhere, the three different ways to crack it open, and the traps that catch almost everyone the first time.

What Is a "Consecutive Number" Problem Anyway?

At its core, this is a translation exercise. You’re taking a sentence written in English and turning it into math symbols.

Consecutive numbers are just integers that follow each other in order, without gaps. 4, 5, 6. -1, 0, 1.100, 101, 102. The difference between any two neighbors is always exactly one Which is the point..

The problem asks for three* of them. And it tells you the sum — the result of adding them together — is 108.

That’s the whole universe of the problem. Step size of one. Three numbers. Total equals 108.

The Variable Setup: Two Ways to Start

Most textbooks teach you to call the first number x.

• First number: x
• Second number: x + 1
• Third number: x + 2

That works. It’s standard. But it’s not the only way, and sometimes it makes the arithmetic slightly messier than it needs to be.

The cleaner approach? Let the middle number be x.

• First number: x - 1
• Second number: x
• Third number: x + 1

Watch what happens when you add those three: (x - 1) + x + (x + 1). The -1 and +1 cancel out instantly. You’re left with 3x.

That cancellation is the secret weapon. It turns a three-term addition problem into a one-step division problem.

Why This Specific Problem Matters

You might wonder: Who cares about three numbers adding to 108?*

Fair question. The numbers themselves don't matter. The structure matters And it works..

This problem type — "find consecutive integers given a sum/difference/product" — is the gateway drug to algebraic modeling. It teaches you to:

1. Worth adding: identify the unknown. Practically speaking, 2. Express relationships relative* to that unknown. But 3. On the flip side, build an equation from a word sentence. 4. Solve and, crucially, interpret the answer back into the original context.

It shows up on the SAT, the ACT, the GRE, civil service exams, and coding interview whiteboards (often disguised as "find a subarray of length k with sum S").

If you can do this one fluidly, you can model rent increases, depreciation schedules, or physics displacement problems. The math is identical; only the nouns change.

How to Solve It: Three Distinct Methods

Here’s the meat. I’ll walk through the standard algebraic way, the "middle number" shortcut, and a purely arithmetic approach that requires zero variables.

Method 1: The Textbook Algebra (First Number = x)

Set it up:

• Number 1: x
• Number 2: x + 1
• Number 3: x + 2

Equation: x + (x + 1) + (x + 2) = 108

Combine like terms: 3x + 3 = 108

Subtract 3 from both sides: 3x = 105

Divide by 3: x = 35

So the first number is 35. The second is 36. The third is 37 Less friction, more output..

Check: 35 + 36 + 37 = 108. Done.

Method 2: The Middle Number Shortcut (Middle Number = x)

This is faster. Less writing, fewer sign errors.

• Number 1: x - 1
• Number 2: x
• Number 3: x + 1

Equation: (x - 1) + x + (x + 1) = 108

Simplify the left side. The -1 and +1 vanish. 3x = 108

Divide by 3: x = 36

x is the middle number. So the numbers are 35, 36, 37.

Same answer. Plus, half the steps. Because of that, this is why experienced math people always pick the middle variable for odd-count consecutive integer problems. For an even count (like "four consecutive numbers"), there is no single middle integer, so you’d pick the two middle ones or revert to the "first number" method.

Method 3: Pure Arithmetic (No Variables At All)

This is the "mental math" way. Use it to check your work or to solve it in a conversation without pen and paper.

Logic: The sum of three consecutive numbers is always three times the middle number.

Why? Plus, the deficits and surpluses cancel. Because the first number is one less* than the middle, and the third is one more*. The average is the middle number Worth knowing..

So: Sum = 3 × Middle Number.

108 = 3 × Middle Number.

Middle Number = 108 ÷ 3 = 36.

The other two are 35 and 37.

That’s it. But ten seconds. But no x, no distribution, no combining like terms. This works because consecutive integers form an arithmetic sequence with a common difference of 1. In any arithmetic sequence with an odd number of terms, the median equals the mean Turns out it matters..

Common Mistakes: Where the Wheels Fall Off

I’ve graded hundreds of these. Now, the errors are almost never about division. They’re about setup and reading comprehension.

1. Answering "x = 35" and Stopping

The problem asks: "Find three consecutive numbers..."
The student writes: x = 35.*
Incomplete. You found the first* number. You haven't listed the three numbers. Always write the final answer as a set or a list: 35, 36, 37.

2. The "Consecutive Even/Odd" Trap

This problem says "consecutive numbers." That implies consecutive integers (difference of 1).

If the problem said "three consecutive even numbers whose sum is 108," the setup changes completely.

• First: x
• Second: x + 2
• Third: x + 4
• Equation: 3x + 6 = 108 → 3x = 102 → x = 34.
• Numbers

This is the bit that actually matters in practice.

are 34, 36, 38.

Notice the difference: with even or odd numbers, you jump by 2s, not 1s. If you use x + 1 for even numbers, you will end up with a decimal, which is a massive red flag. If you are solving for integers and you get a fraction, stop immediately and re-read the prompt.

3. Misinterpreting "Consecutive" vs. "Consecutive Even/Odd"

This is the most common reading error.

• Consecutive numbers: 1, 2, 3, 4... (Difference = 1)
• Consecutive even numbers: 2, 4, 6, 8... (Difference = 2)
• Consecutive odd numbers: 1, 3, 5, 7... (Difference = 2)

Always check if the problem specifies "even" or "odd." If it doesn't, assume the difference is 1 But it adds up..

Summary Table for Quick Reference

To make this easier for future problems, keep this mental framework handy:

Conclusion

Solving for consecutive integers is less about complex algebra and more about pattern recognition. While Method 1 (the standard algebraic approach) is the safest way to ensure you don't make a logic error, Method 2 (the middle number) and Method 3 (the arithmetic average) are your best friends for speed and verification The details matter here..

Not the most exciting part, but easily the most useful.

Mastering these three approaches gives you a "triple-check" system: you can solve it one way, verify it with another, and use the third to ensure your logic is sound. Once you stop seeing these as "math problems" and start seeing them as "patterns of numbers," you'll find that the answers reveal themselves much faster.