The Square-Ending-in-5 Trick
Squaring numbers that end in 5 is remarkably simple. Take the digit(s) before the 5, multiply by the next consecutive integer, and append “25” to the result. For 35²: take 3, multiply by 4 (3 × 4 = 12), append 25 → 1,225. This works for any number ending in 5, from 15² to 995².
Why It Works
A number ending in 5 can be written as (10n + 5). Squaring: (10n + 5)² = 100n² + 100n + 25 = 100n(n + 1) + 25. The n(n + 1) part gives the leading digits, and 25 is always the ending. This algebraic proof confirms the trick works universally.
Practice Examples
15²: 1 × 2 = 2 → 225. 25²: 2 × 3 = 6 → 625. 45²: 4 × 5 = 20 → 2,025. 75²: 7 × 8 = 56 → 5,625. Notice how quickly you can compute these once the pattern is internalized. This trick is a favorite in mental math competitions.
Applications
This technique is useful for area calculations (a 35-foot square room = 1,225 sq ft), probability (squaring percentages), and any situation involving perfect squares. Combined with the multiply-by-11 trick and cross multiplication, you can handle most two-digit arithmetic mentally.
Practice Examples
Example 1: 35² = ?
- Take the tens digit: 3
- Multiply by next number: 3 × 4 = 12
- Append 25: 1,225
Answer: 1,225
Example 2: 65² = ?
- Tens digit: 6
- 6 × 7 = 42
- Append 25: 4,225
Answer: 4,225
Example 3: 85² = ?
- 8 × 9 = 72
- Append 25: 7,225
Answer: 7,225