1. The Rule of 9
If you need to multiply a number by 9 quickly, try this trick: subtract 1 from the number, and then the remaining digits must add up to 9. For example, to multiply 7 by 9, subtract 1 to get 6, and the remaining digits (6 + 3) equal 9, so 7 x 9 = 63.
2. The Magic of Multiplying by 11
Multiplying by 11 reveals two elegant shortcuts that demonstrate the beauty of number patterns. The first method: add the two digits and place their sum between the original digits. For example, with 23 × 11, add 2+3=5, then place 5 between 2 and 3, giving us 253. When the sum exceeds 9, carry over: for 85 × 11, 8+5=13, so place 3 between 8 and 5 and add 1 to the first digit, resulting in 935. The second method uses the pattern that multiplying by 11 is the same as multiplying by 10 and adding the original number (n × 11 = n × 10 + n). So 23 × 11 = 23 × 10 + 23 = 230 + 23 = 253. Both methods elegantly arrive at the same result, showcasing different ways to understand this mathematical pattern.
3. Squaring Numbers Ending in 5
For any number ending in 5, the square will end in 25. Multiply the first digit(s) by itself plus one, then append 25. For example, to square 75, multiply 7 by (7+1) to get 56, then append 25, giving 5625.
4. Doubling and Halving
If you need to multiply two numbers, one even and one odd, you can simplify the calculation by doubling one number and halving the other. For example, 12 x 35 can be transformed into 24 x 17.5 or 6 x 70.
5. Fast Division by 5
To divide any number by 5, multiply by 2 and then move the decimal point one place to the left. For example, 245 ÷ 5 becomes 245 x 2 = 490, then moving the decimal one place to the left gives 49.0.
6. The Power of 15
To quickly multiply a number by 15, you can multiply the number by 10 and then add half of that product to it. For example, to multiply 24 by 15: 24 x 10 = 240, and 240 ÷ 2 = 120. Add 240 and 120 to get 360.
7. The 10% Trick
To quickly find 10% of any number, simply move the decimal point one place to the left. For example, 10% of 475 is 47.5. This trick also works for other percentages; to find 5%, halve the 10% value (for 475, it would be 23.75).
8. Multiplying by 5
To multiply a number by 5 quickly, multiply the number by 10 and then halve it. For example, 8 x 5: 8 x 10 = 80, and half of 80 is 40.
9. Adding Large Numbers
When adding large numbers, round them to the nearest multiple of 10, add them, and then adjust the result. For instance, 467 + 239 can be rounded to 470 + 240 = 710, then subtract the rounding difference (4) to get 706.
10. Subtracting from 1000
To subtract a number from 1000, subtract each digit from 9, except the last digit, which is subtracted from 10. For example, 1000 - 648: 9 - 6 = 3, 9 - 4 = 5, 10 - 8 = 2, resulting in 352.
11. Multiplying by 4
To multiply a number by 4, simply double the number twice. For example, 7 x 4: double 7 to get 14, and double 14 to get 28.
12. Dividing by 4
To divide a number by 4, halve the number twice. For example, 32 ÷ 4: half of 32 is 16, and half of 16 is 8.
13. Multiplying by 9 Using Fingers
To multiply a single-digit number by 9, use your fingers. For example, to multiply 7 by 9, lower your 7th finger. You'll have 6 fingers to the left and 3 fingers to the right, giving you 63.
14. The Rule of 72
To estimate how long it will take for an investment to double with compound interest, divide 72 by the annual interest rate. For example, if the interest rate is 6%, 72 ÷ 6 = 12 years.
15. Multiplying by 12
To multiply a number by 12, multiply by 10 and then add twice the original number. For example, 15 x 12: 15 x 10 = 150, and 150 + (15 x 2) = 180.
16. Division by 9
To divide a number by 9, sum the digits of the number repeatedly until you get a single digit. If this sum is a multiple of 9, then the number is divisible by 9. For example, to divide 108 by 9, sum the digits (1+0+8=9), which is a multiple of 9, so 108 is divisible by 9. Therefore, 108 ÷ 9 = 12.
17. Quickly Finding Squares
To find the square of a number ending in 1, add the number to the base and then add the square of the base. For example, 41²: 40 + 41 = 81, and 40² = 1600, 1600 + 81 = 1681.
18. Multiplying Numbers Close to 100
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Find how far each number is from 100:
- Subtract each number from 100 to get the "gaps."
- For example, for 96 × 97: 100 - 96 = 4, and 100 - 97 = 3.
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Subtract diagonally to find the base:
- Subtract one number's gap from the other number (96 - 3 = 93).
- Multiply this base (93) by 100 to get the first part of the answer (9300).
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Multiply the gaps to get the second part:
- Multiply the two gaps: 4 × 3 = 12.
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Add the two parts together:
- Combine 9300 + 12 to get the final answer: 9312.
Final Answer: 96 × 97 = 9312