Squaring Numbers Ending in 5

Lesson overview

What you will learn

Every whole number ending in 5 has a square that follows a memorable two-part pattern. Remove the trailing 5, call the remaining digits n, multiply n by (n + 1), and append 25 to the result. Thirty-five squared: n = 3, 3 × 4 = 12, append 25 → 1,225.

The pattern works because algebraically (10n + 5)² expands to 100n(n + 1) + 25. The "append 25" step is literal: the last two digits of any square ending in 5 are always 25.

Large numbers obey the same rule. One hundred fifteen squared: n = 11, 11 × 12 = 132, append 25 → 13,225. Even 995 squared: n = 99, 99 × 100 = 9,900, append 25 → 990,025.

This trick is exact, not approximate. It is one of the few mental math shortcuts that produces perfect squares every time with no correction step.

Why it matters

Build the idea before the speed

The pattern gives instant exact squares and reveals useful algebraic structure.

In adult life

Where this skill shows up

Land area calculations often involve squares ending in 5: a square plot measuring 85 feet per side has area 85² = 7,225 square feet — computable in seconds during a site walk without a phone.

Financial modeling sometimes uses round half-percent increments; squaring numbers like 15, 25, or 35 appears when computing variance or standard deviation components mentally during quick reviews.

The method

Step by step

1. 1Remove the final 5 and call the remaining number n.
2. 2Multiply n by n + 1.
3. 3Append 25 to that product.

Worked examples

See the structure

Avoid these traps

Common mistakes

• Multiplying n by n instead of n by (n + 1) — computing 3 × 3 = 9 instead of 3 × 4 = 12 for 35².
• Appending 25 incorrectly when the n(n + 1) product already ends in digits that collide visually with 25.
• Applying the rule to numbers not ending in 5, where it fails completely.
• Forgetting that n includes all digits to the left of 5 — using only the last digit before 5 in multi-digit numbers.

Practice tips

• Memorize the first ten results (15² through 95²) as anchors; they appear frequently enough to warrant direct recall.
• Say "n times n-plus-one, then twenty-five" as a rhythm until automatic.
• Verify small cases with known facts: 25² = 625, so n = 2, 2 × 3 = 6, append 25 checks out.
• Use the pattern in reverse to estimate square roots: if 1,225 is a perfect square ending in 25, its root ends in 5.

Check your understanding

1. Why must the last two digits of any square of a number ending in 5 always be 25?
2. Compute 115² by the rule and explain what n represents in that problem.
3. Would this method work for 30²? Why or why not?

Your turn

Interactive practice

Answer one problem at a time. Each problem has a fresh 60-second timer, immediate correction, and a final accuracy score. Aim for understanding first; speed comes after accuracy.