Squaring a 2-digit Number Ending in 5

STEPS

Choose a 2-digit number ending in 5.
Multiply the first digit by the next consecutive number.
The product is the first two digit: XX__.
The last part of the answer is always 25: __25.

Example

If the number is 35, 3 x 4 = 12 (first digit times next number) : 12__
The last part of the answer is always 25: __25.
So 35 x 35 = 1225

Squaring a 2-digit number ending in 4

Steps
take a 2 digit number ending in 4.
square the 4; the last digit is 6: ___6 (keep carry, 1).
multiply the first digit by 8 and add the carry (1); the secondnumber will be the next to the last digit: __X6 (keep carry).
square the first digit and add the carry: XX__.

Example
if the number is 34, 4 x 4 =16 (keep carry, 1); the last digit is ___6.
8 x 3 = 24 (multiply the first digit by 8), 24 + 1 = 25 (add the carry): the next digit is 5: __56. (keep carry, 2).
square the first digit and add the carry, 2: 1156.
so 34 x 34 =1156.

Squaring a 2-digit ending in 3

Steps
take a 2-digit number ending in 3.
the last digit will be ___9.
multiply the first digit by 6: the second number will be the next to the last digit: __X9.
square the first digit and add the number carried from the previous step: XX__.

Example
if the number is 43, the last digit
is ___9.
6 x 4 = 24: __49.
4 x 4 = 16, 16 + 2 = 18 (add carry): 1849.
so 43 x 43 = 1849

Squaring a 2-digit number ending in 2

Steps
take a 2-digit number ending in 2.
the last digit will be ___4.
multiply the first digit by 4: the second number will be the next to the last digit: __X4.
square the first digit and add the number carried from the previous step: XX__.

Example
if the number is 52, the last digit is ___4.
4 x 5 = 20 : __04.
5 x 5 = 25, 25 + 2 = 27 (add carry): 2704.
so 52 x 52 = 2704.

SQUARING a 2-DIGIT NUMBER ENDING IN 1

STEPS

take a-2 digit number ending in 1.
substract 1 from the number.
square the difference.
add the difference twice to its square.
add 1.

EXAMPLE

If the number is 41, substract 1 : 41-1=40.
40 x 40 = 1600 ( square the difference ).
1600 + 40 + 40 =1680 ( add the difference twice to its square ).
1680 + 1 = 1681 ( add 1 ).
so 41 x 41 = 1681

Dividing a 2-digit number by 1 1/7

STEPS

1. Select a 2-digit number.
2. Multiply it by 7.
3. Divide the product by 8.

Example:

1. The number chosen to divide by 1 1/7 is 26.
2. Multiply by 7: 7 × 26 = 140 + 42 = 182
3. Divide by 8: 182/8 = 22 3/4
4. So 26 divided by 1 1/7 = 22 3/4.

See the pattern?

1. The number chosen to divide by 1 1/7 is 61.
2. Multiply by 7: 7 × 61 = 427
3. Divide by 8: 427/8 = 53 3/8
4. So 61 divided by 1 1/7 = 53 3/8

Dividing a 2-digit number by 1 1/8

STEPS

1. Select a 2-digit number.
2. Multiply by 8.
3. Divide the result by 9.

Example:

1. The 2-digit number chosen to divide by 1 1/8 is 42.
2. Multiply by 8: 8 × 42 = 320 + 16 = 336
3. Divide by 9: 336/9 = 37 1/3
4. So 42 divided by 1 1/8 = 37 1/3.

See the pattern?

1. The 2-digit number chosen to divide by 1 1/8 is 71.
2. Multiply by 8: 8 × 71 = 568
3. Divide by 9: 568/9 = 63 1/9
4. So 71 divided by 1 1/8 = 63 1/9.

Dividing a number by 0.125

steps

1. Select a number.
2. Multiply it by 8.

Example:

1. The number chosen to divide by 0.125 is 21.
2. Multiply it by 8: 8 × 21 = 168.
3. So 21 divided by 0.125 = 168.

See the pattern?

1. The 2-digit number chosen to divide by 1/8 is 68.
2. Multiply it by 8: 8 × 68 = 480 + 64 = 544.
3. So 68 divided by 0.125 = 544.

Dividing a 2-digit number by 1 1/9

steps

1. Select a 2-digit number.
2. Multiply by 9.
3. Move the decimal point one place to the left.

Example:

1. The 2-digit number chosen to divide by 1 1/9 is 24.
2. Multiply by 9: 9 × 24 = 180 + 36 = 216
3. Move the decimal point one place to the left: 21.6
4. So 24 divided by 1 1/9 = 21.6.

See the pattern?

1. The 2-digit number chosen to divide by 1 1/9 is 91.
2. Multiply by 9: 9 × 91 = 819
3. Move the decimal point one place to the left: 81.9
4. So 91 divided by 1 1/9 = 81.9.

Multiplying two selected 3-digit numbers

steps

1. Select a 3-digit number with a middle digit of 0.
2. Choose a multiplier with the same first two digits, whose third digit sums to 10 with the third digit of the first 3-digit number.
3. The first digit(s) will be the square of the first digit:
   X _ _ _ _ or X X _ _ _ _.
4. The next digit will be the first digit of the numbers:
   _ X _ _ _ or _ _ X _ _ _.
5. The next digit is zero: _ _ 0 _ _ or _ _ _ 0 _ _.
6. The last two digits will be the product of the third digits:
   _ _ _ X X or _ _ _ _ X X.

Example:

1. If the first number is 407, choose 403 as the second number (same first digits, second digits add to 10).
2. 4 × 4 = 16 (square the first digit): 1 6 _ _ _ _.
3. The next digit will be the first digit of the numbers:
   _ _ 4 _ _ .
4. The next digit is zero: _ _ 0 _ _ .
5. 7 × 3 = 21 (the last two digits will be the product of the third digits: _ _ _ 2 1.
6. So 407 × 403 = 164021.

See the pattern?

1. If the first number is 201, choose 209 as the second number (same first digits, second digits add to 10).
2. 2 × 2 = 4 (square the first digit): 4 _ _ _ _.
3. The next digit will be the first digit of the numbers:
   _ 2 _ _ _ .
4. The next digit is zero: _ _ 0 _ _ .
5. 1 × 9 = 09 (the last two digits will be the product of the third digits: _ _ _ 0 9.
6. So 201 × 209 = 42009.

Multiplying two 2-digit numbers

steps

1. Both numbers should have the same second digit.
2. Choose first digits whose sum is 10.
3. Multiply the first digits and add one second: X X _ _.
4. Multiply the second digits together: _ _ X X.

Example:

1. If the first number is 67, choose 47 as the second number (same second digit, first digits add to 10).
2. Multiply the 1st digits, add one 2nd.
   6x4 = 24, 24+7 = 31. 3 1 _ _
3. Multiply the 2nd digits. 7x7 = 49 _ _ 4 9
4. So 67 × 47 = 3149.

See the pattern?

1. If the first number is 93, choose 13 as the second number (same second digit, first digits add to 10).
2. Multiply the 1st digits, add one 2nd. 9x1 = 9, 9+3 = 12.
   1 2 _ _
3. Multiply the 2nd digits. 3x3 = 9 _ _ 0 9
4. So 93 × 13 = 1209.

Multiplying two 2-digit numbers

steps

1. Select two 2-digit numbers with the same first digit.
2. Multiply their second digits (keep the carry). _ _ _ X
3. Multiply the sum of the second digits by the first digit,
   add the carry (keep the carry). _ _ X _
4. Multiply the first digits (add the carry). X X _ _

Example:

1. If the first number is 42, choose 45 as the second number (any 2-digit number with first digit 4).
2. Multiply the last digits: 2 × 5 = 10 (keep carry)
   _ _ _ 0
3. Multiply the sum of the 2nd digits by the first:
   5 + 2 = 7; 7 × 4 = 28; 28 + 1 = 29 (keep carry)
   _ _ 9 _
4. Multiply the first digits (add the carry)
   4 × 4 = 16; 16 + 2 = 18
   1 8 _ _
   
   
5. So 42 × 45 = 1890.

See the pattern?

1. If the first number is 62, choose 67 as the second number
   (any 2-digit number with first digit 6).
2. Multiply the last digits: 2 × 7 = 14 (keep carry)
   _ _ _ 4
3. Multiply the sum of the 2nd digits by the first (add carry):
   2 + 7 = 9; 6 × 9 = 54; 54 + 1 = 55 (keep carry)
   _ _ 5 _
4. Multiply the first digits (add the carry)
   6 × 6 = 36; 36 + 5 = 41
   4 1 _ _
   
   
5. So 62 × 67 = 4154.

Multiplying two 2-digit numbers

steps

1. Both numbers should have the same first digit।
2. Choose second digits whose sum is 10.
3. Multiply the first digit by one number greater than itself; this number will be the first part of the answer:
   X X _ _.
4. Multiply the two second digits together; the product
   will be the last part of the answer: _ _ X X.

Note: If the two second digits are 1 and 9 (or, more generally, have a product that is less than ten), insert a 0 (zero) for the first X in step 4.

(Thanks to Michael Richardson, age 10, for this note.)

Example:

1. If the first number is 47, choose 43 as the second number (same first digit, second digits add to 10).
2. 4 × 5 = 20 (multiply the first digit by one number greater than itself): the first part of the answer is
   2 0 _ _.
3. 7 × 3 = 21 (multiply the two second digits together); the last part of the answer is _ _ 2 1.
4. So 47 × 43 = 2021.

See the pattern?

1. If the first number is 62, choose 68 as the second number (same first digit, second digits add to 10).
2. 6 × 7 = 42 (multiply the first digit by one greater), the first part of the answer is 4 2 _ _.
3. 2 × 8 = 16 (multiply the two second digits together); the last part of the answer is _ _ 1 6.
4. So 62 × 68 = 4216.

Squaring a 2-digit number beginning with 9

steps

1. Take a 2-digit number beginning with 9.
2. Subtract it from 100.
3. Subtract the difference from the original number:
   this is the first part of the answer.
4. Square the difference: this is the last part of the answer.

Example:

1. If the number is 96, subtract: 100 - 96 = 4, 96 - 4 = 92.
2. The first part of the answer is 92 _ _ .
3. Take the first difference (4) and square it: 4 × 4 = 16.
4. The last part of the answer is _ _ 16.
5. So 96 × 96 = 9216.

See the pattern?

1. For 98 × 98, subtract: 100 - 98 = 2, 98 - 2 = 96.
2. The first part of the answer is 96 _ _.
3. Take the first difference (2) and square it: 2 × 2 = 4.
4. The last part of the answer is _ _ 04.
5. So 98 × 98 = 9604.

Squaring a 2-digit number beginning with 5

1. Take a 2-digit number beginning with 5.
2. Square the first digit.
3. Add this number to the second number to find the first part of the answer.
4. Square the second digit: this is the last part of the answer.

Example:

1. If the number is 58, multiply 5 × 5 = 25 (square the first digit).
2. 25 + 8 = 33 (25 plus second digit).
3. The first part of the answer is 33 3 3 _ _
4. 8 × 8 = 64 (square second digit).
5. The last part of the answer is 64 _ _ 6 4
6. So 58 × 58 = 3364.

See the pattern?

1. For 53 × 53, multiply 5 × 5 = 25 (square the first digit).
2. 25 + 3 = 28 (25 plus second digit).
3. The first part of the answer is 28 2 8 _ _
4. 3 × 3 = 9 (square second digit).
5. The last part of the answer is 09 _ _ 0 9
6. So 53 × 53 = 2809.

Squaring a 2-digit number beginning with 1

the steps

1. Take a 2-digit number beginning with 1.
2. Square the second digit
   (keep the carry) _ _ X
3. Multiply the second digit by 2 and
   add the carry (keep the carry) _ X _
4. The first digit is one
   (plus the carry) X _ _

Example:

1. If the number is 16, square the second digit:
   6 × 6 = 36 _ _ 6
2. Multiply the second digit by 2 and
   add the carry: 2 × 6 + 3 = 15 _ 5 _
3. The first digit is one plus the carry:
   1 + 1 = 2 2 _ _
4. So 16 × 16 = 256.

See the pattern?

1. For 19 × 19, square the second digit:
   9 × 9 = 81 _ _ 1
2. Multiply the second digit by 2 and
   add the carry: 2 × 9 + 8 = 26 _ 6 _
3. The first digit is one plus the carry:
   1 + 2 = 3 3 _ _
4. So 19 × 19 = 361.