Multiplication Tricks for Faster Calculations

1. The Power of Base Multiplication

One of the most effective ways to multiply numbers quickly is by using base multiplication. This method is particularly useful when multiplying numbers close to a base like 10, 100, or 1000.

Example: Multiplying Numbers Near 100

Let’s multiply 98 and 97.

1. Find the difference from the base (100):
   • 98 is 2 less than 100.
   • 97 is 3 less than 100.
2. Subtract diagonally:
   • 98 – 3 = 95, or 97 – 2 = 95. This gives the first part of the answer.
3. Multiply the differences:
   • 2 × 3 = 06. This gives the second part of the answer.
4. Combine the results:
   • The final answer is 9506.

This method works because:

98×97=(100−2)×(100−3)=1002−(2+3)×100+(2×3)=9506.98×97=(100−2)×(100−3)=1002−(2+3)×100+(2×3)=9506.

2. The Distributive Property (Breaking Numbers Apart)

The distributive property allows you to break down one of the numbers into smaller, more manageable parts. This is especially helpful for multiplying larger numbers.

Example: Multiply 12 × 14

1. Break 14 into 10 and 4
2. Multiply 12 by 10: 12×10=120
3. Multiply 12 by 4: 12×4=48
4. Add the results: 120+48=168

This method can be extended to larger numbers, such as 123 × 45, by breaking 45 into 40 and 5.

3. The Lattice Method

The lattice method is a visual technique that simplifies multiplication, especially for multi-digit numbers. It involves creating a grid and filling it with partial products.

Example:

Multiply 12 × 13

1. Draw a 2×2 grid (since both numbers have 2 digits).
2. Write 13 on top and 12 on the side.
3. Multiply each digit and place the result in the corresponding cell:
   • 1 × 1 = 1
   • 2 × 3 = 6
   • 1 × 3 = 3
   • 1 × 2 = 2
4. Add diagonally to get the final result: 156.

Multiply 23 × 45

1. Draw a 2×2 grid (since both numbers have 2 digits).
2. Write 23 on top and 45 on the side.
3. Multiply each digit and place the result in the corresponding cell:
   • 2 × 4 = 08
   • 2 × 5 = 10
   • 3 × 4 = 12
   • 3 × 5 = 15
4. Add diagonally to get the final result: 1035.

4. Multiplying by 5, 10, and 25

Multiplying by these numbers can be simplified using shortcuts:

• Multiply by 5: Divide the number by 2 and multiply by 10.
  • Example: $24\times 5=(24/2)\times 10=120$.
• Multiply by 10: Simply add a zero at the end.
  • Example: $24\times 10=240$.
• Multiply by 25: Divide the number by 4 and multiply by 100.
  • Example: $24\times 25=(24/4)\times 100=600$.

5. The 11 Rule

Multiplying any two-digit number by 11 is incredibly simple:

1. Add the two digits of the number.
2. Place the sum between the original digits.

Example: Multiply 35 × 11

1. Add 3 and 5: 3+5=8.
2. Place 8 between 3 and 5: 385.

If the sum of the digits is greater than 9, carry over the extra digit:

• Example: 57×11:
  • 5+7=12.
  • Place 2 between 5 and 7 and carry over 1: 627.

6. The 9 Rule

Multiplying by 9 can be done using the following trick:

1. Multiply the number by 10.
2. Subtract the original number from the result.

Example: Multiply 14 × 9

1. 14×10=140.
2. 140−14=126.

7. Squaring Numbers Ending in 5

Squaring a number that ends with 5 is straightforward:

1. Multiply the first digit(s) by the next higher number.
2. Append 25 to the result.

Example: Square 45

1. Multiply 4 by 5: $4\times 5=20$.
2. Append 25: 2025.

8. The Butterfly Method for Fractions

When multiplying fractions, you can simplify the process by cross-canceling common factors before multiplying.

Example: Multiply 34×8943​×98​

1. Cancel 3 and 9: 14×8341​×38​.
2. Cancel 4 and 8: 11×2311​×32​.
3. Multiply: 2332​.

9. The Russian Peasant Method

This ancient method involves doubling and halving numbers:

1. Write the two numbers side by side.
2. Halve the first number and double the second number repeatedly.
3. Add the doubles of second numbers where the halves of the first number are odd.

Example: Multiply 14 × 12

1. Halve 14: 7, 3, 1 (ignore remainders).
2. Double 12: 24, 48, 96.
3. Add the doubled numbers corresponding to odd halves: 24+48+96=168.

10. Mental Math Practice

Regular practice is key to mastering these tricks. Start with smaller numbers and gradually move to larger ones. Use flashcards, apps, or timed drills to improve speed and accuracy.