Squaring a number ending in 5, without a calculator

The rule

To square a number ending in 5:

1. Take the left part (everything except the final 5), call it $n$.
2. Compute $n \times (n + 1)$.
3. Stick 25 at the end.

Two examples

$35^2$:

• Left part: $n = 3$
• $n \times (n+1) = 3 \times 4 = 12$
• Stick 25: 1225

$75^2$:

• $n = 7$
• $7 \times 8 = 56$
• Result: 5625

Works on bigger numbers too

$115^2$:

• $n = 11$
• $11 \times 12 = 132$
• Result: 13225 ✓

Why?

A number ending in 5 writes as $10n + 5$. Its square:

$(10n + 5)^2 = 100n^2 + 100n + 25 = 100 \cdot n(n+1) + 25$

The $100 \cdot n(n+1)$ term pushes $n(n+1)$ into hundreds, and the $+25$ tacks on at the end. Mathematical magic!

?Your turn

What is 45²?

Check

?Your turn

Without a calculator, what is the square of 95?

A.9025B.9525C.8125D.9125

Check