Squaring Two-Digit Numbers Ending in Five: A Simple Math Trick

Squaring two-digit numbers ending in five can seem daunting, but there's a simple trick that makes it a breeze! This method allows you to calculate squares mentally, saving you time and effort.

The Trick

Here's how it works:

1. Identify the tens digit: Take the two-digit number ending in five and identify the tens digit. For example, in the number 35, the tens digit is 3.
2. Multiply the tens digit by itself plus one: Multiply the tens digit by the number that comes after it. In our example, 3 * (3 + 1) = 3 * 4 = 12.
3. Add 25: Take the result from step 2 and add 25 to it. In our example, 12 + 25 = 37.
4. Add the two digits together: The result from step 3 is the square of the original number. So, 35² = 37.

Example

Let's try another example: 75²

1. Tens digit: 7
2. 7 * (7 + 1) = 7 * 8 = 56
3. 56 + 25 = 81
4. Therefore, 75² = 81.

Why Does This Work?

This trick works because of the algebraic expansion of (10a + 5)², where 'a' represents the tens digit. Expanding this expression gives us:

(10a + 5)² = 100a² + 100a + 25 = 100a(a + 1) + 25

This equation clearly shows the steps we follow in the trick. The first part, 100a(a + 1), represents multiplying the tens digit by itself plus one and then multiplying by 100. The second part, + 25, represents adding 25 to the result.

Practice

Now that you understand the trick, let's practice with a few more examples:

1. 45²
2. 95²
3. 25²

Try solving these problems using the trick. You'll be surprised at how quickly and easily you can square two-digit numbers ending in five!

Conclusion

This simple math trick for squaring two-digit numbers ending in five is a valuable tool for anyone who wants to improve their mental math skills. It's a fun and engaging way to learn about the properties of squares and how to manipulate numbers effectively.