How to Square Numbers from 491 to 500 Fast: Easy Mental Math Trick

Can you square numbers from 491 to 500 without long multiplication? Yes. Once you recognize how close these numbers are to 500, the calculations become much faster and easier to check.

In this Maths Tricks guide, you will learn a fast method for finding squares from 491² through 500². The method uses the identity (500 − d)², where d is the amount below 500.

The Quick Pattern: 491² to 500²

Each number in this range can be written as 500 minus a small number. The first part of the answer moves upward from 241 to 250, while the final two digits follow a reverse square pattern:

491² = 241081   |   492² = 242064

493² = 243049   |   494² = 244036

495² = 245025   |   496² = 246016

497² = 247009   |   498² = 248004

499² = 249001   |   500² = 250000

The Main Shortcut: Use 500 as the Base

Write the number as 500 − d. Then use the difference-of-squares expansion:

(500 − d)² = 250000 − 1000d + d²

This formula is the same as expanding (a − b)² = a² − 2ab + b². Because 500² is 250,000, the arithmetic is easy to organize mentally.

Example 1: Find 491²

Since 491 is 9 less than 500, use d = 9:

491 = 500 − 9

491² = 250000 − (1000 × 9) + 9²

250000 − 9000 + 81 = 241081

Example 2: Find 497²

497 is 3 less than 500, so d = 3:

497² = 250000 − (1000 × 3) + 3²

250000 − 3000 + 9 = 247009

The Complete 491–500 Square Table

Number	Distance from 500	Square
491	500 − 9	241,081
492	500 − 8	242,064
493	500 − 7	243,049
494	500 − 6	244,036
495	500 − 5	245,025
496	500 − 4	246,016
497	500 − 3	247,009
498	500 − 2	248,004
499	500 − 1	249,001
500	500 − 0	250,000

A Fast Mental Pattern

When the distance from 500 is small, the formula is faster than ordinary multiplication. The answer can be found by starting with 250,000, subtracting 1,000 times the distance, and adding the distance squared.

500 − d → 250000 − 1000d + d²

The final two digits are especially easy to remember because they are the squares of 9, 8, 7, 6, 5, 4, 3, 2, 1, and 0: 81, 64, 49, 36, 25, 16, 09, 04, 01, 00.

Common Mistakes to Avoid

Remember that the distance from 500 must be squared. For 497, the distance is 3, so the final correction is 3² = 9—not 3. Also keep the zero in values such as 247,009 and 248,004.

Use commas only as reading separators. The mathematical values are 241081, 242064, 243049, and so on. Both formats represent the same numbers.

Practice Problems

Try these without a calculator:

1. 491²
2. 494²
3. 496²
4. 498²
5. 500²

Answer Key

1. 491² = 241,081
2. 494² = 244,036
3. 496² = 246,016
4. 498² = 248,004
5. 500² = 250,000

Final Takeaway

To square a number from 491 to 500, compare it with 500. If the number is 500 − d, use:

(500 − d)² = 250000 − 1000d + d²

This shortcut turns the 491–500 square table into a quick mental-math exercise and helps you spot the pattern behind the answers. Save this Maths Tricks guide for homework, test review, and fast calculations.

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