11s
11 X 48 = _______
Step 1: Write the one’s digit down.
11 X 48 = ______8
Step 2: Add the ten’s place to the whole number.
4 + 48 = 52
Answer:
11 X 48 = 528
12s Double Double Add
12 X 48 = _____
Step 1: Double the one’s, carry if necessary.
8 X 2 = 16
12 X 48 = _____6
Carry the one
Step 2: Double the ten’s place and add it to the whole number.
4X2=8
8 + 48 = 56
Plus the one we carried
56 + 1 = 57
Answer:
12 X 48 = 576
General Multiplication
43 X 7 = ______
Decompose larger number into two parts – 40 and 3
40 X 7 = 280
3 X 7 = 21
Add both answers together
280 + 21 = 301
Extension: They may add a zero to one of the numbers. Follow the same
procedure and add an extra zero to the answer.
1)
2)
3)
4)
5)
43 X 70
40 X 7 = 280
3 X 7 = 21
280 + 21 = 301
43 X 70 = 3,010
Combinations
5 X 7 X 6 = ________
First multiply the two numbers that have a product ending in zero.
5 X 6 = 30
Then multiply the product by the remaining number.
30 X 7 = 210
Remember, you just multiply 3 X 7 = 21 and then add the zero. 30 X 7 = 21
25s Divide by 4
25 X 36
Step 1: Divide number by 4
36 ÷ 4 = 9
25 X 36 = _9___
Step 2:
If there is no remainder, add two zeros
25 X 36 = 900
If the remainder is one, add 25
25 X 37 = 925
If the remainder is two, add 50
25 X 38 = 950
If the remainder is three, add 75
25 X 39 = 975
50s Divide by 2
50 X 36
Step 1: Divide number by 2
36 ÷ 2 = 18
50 X 36 = 18____
Step 2:
If there is no remainder, add 2 zeros.
50 X 36 = 1,800
If the remainder is 1, add 50.
50 X 36 = 1,850
Reverses
973 – 379
Step 1: Subtract hundreds, minus one more.
9–3=6
Minus one more…
6–1=5
973 – 379 = 5_____
Step 2: Write a 9
973 – 379 = 59____
Step 3: The last digit and the first digit should add up to 9.
Since the first digit is 5 and 5 + 4 = 9, the last digit is 4.
973 – 379 = 594
75s Divide by 4 times 3
75 X 36
Step 1: Divide the number by 4
36 ÷ 4 = 9
Step 2: Multiply the answer by 3
9 X 3 = 27
Add 2 zeros
75 X 36 = 2,700
Multiply Numbers Ending in 5
65 X 85 =
Step 1: Multiply first two digits
6 X 8 = 48
Step 2: Average first two numbers (if after adding, answer is odd, subtract one) and add this to
the previous number.
6 + 8 = 14
14 ÷ 2 = 7
7 + 48 = 55
65 X 85 = 55___
Step 3: If when you added the first two numbers, the answer was even, add 25, if the answer
was odd, add 75
Since 6 + 8 = 14
Since 14 is even, we add 25
65 X 85 = 5,525
Example 2:
45 X 95
Step 1: 4 X 9 = 36
Step 2: 4 + 9 = 13 ÷ 2 = 6 remainder 1, ignore the remainder
6 + 36 = 42
Since 4 + 9 is odd, we add 75
45 X 95 = 4,275
With this method, it is easy to mix up the steps or leave one out resulting in an incorrect solution. Work
through many example problems slowly and notice common mistakes you are making.
Difference of Squares
If you know the squares of a number, say 25 X 25 = 625, you can find the product
of two numbers equally spaced from 25.
Example 1:
24 X 26
Since 24 and 26 are both 1 away from 25, we take 1 X 1 = 1 and subtract from
625.
Since 25 X 25 = 625
625 – 1 = 624
24 X 26 = 624
Example 2:
23 X 27
Since 23 and 27 are both 2 away from 25, we take 2 X 2 = 4 and subtract from
625.
625 – 4 = 621, therefore
23 X 27 = 621
Example 3:
16 X 24
Since 16 and 24 are both 4 away from 20, we take 4 X 4 = 16.
Since 20 X 20 = 400, we subtract 16 from 400.
400 – 16 = 384, therefore
16 X 24 = 384
Double-Half Method
14 X 15
Sometimes it’s easier to multiply half of one number times double of the other.
7 X 30 = 14 X 15
Since 7 X 30 = 210
Then 14 X 15 = 210
Example 2:
22 X 45
11 X 90 = 990, therefore
22 X 45 = 990
Example 3:
16 X 35
8 X 70 = 560, therefore
16 X 15 = 560
*Difference of Squares and Double-Half Method Problems look very similar, the
key is learning to distinguish quickly between them and applying the correct
technique*
Remainders:
Remainders dividing by 3
Find the remainder of 2572 ÷ 3
Add each digit 2 + 5 + 7 + 2 = 16
Keep adding until you get a single digit
1+6=7
Divide the single digit by 3
7 ÷ 3 = 2 remainder of 1
So 2572 also has a remainder of 1
Remainders dividing by 4
Find the remainder of 1857 ÷ 4
Use only the last two digits when dividing by 4
57 ÷ 4 = 14 remainder of 1
So 1857 ÷ 4 also has a remainder of 1
Remainders dividing by 5
Multiples of 5 land on 0 or 5 so use only the last digit
If last digit is zero or five, then remainder is 0
Examples: 1765, 15770, 287295
If last digit is less than 5
Examples: 3784, 28903, 20982
Then last digit is the remainder
If the last digit is greater, then subtract 5 and you have the remainder
Example: 19878 ÷ 5
8 – 5 = 3, so the remainder is 3
Remainder dividing by 9
Since 9 = 3 X 3, this trick is similar to the 3s
Find the remainder of 2572 ÷ 9
Add each digit 2 + 5 + 7 + 2 = 16
Keep adding until you get a single digit
1+6=7
The difference between 9s and 3s is that the last digit is the remainder
So 2572 ÷ 9 has a remainder of 7
* If the single digit turns out to be 9, then there is no remainder*
Squaring numbers ending in 5
65 X 65 =
Multiply first digit by next number in the numer line.
6 X 7 = 42
All answers end in 25
65 X 65 = 4225
Example 2
35 X 35 = ?
3 X 4 = 12
35 X 35 = 1225
Advanced Tricks:
Estimating by 111 (This is like 11s)
2-digit numbers
76 X 111 = ?
Add the first digit to the whole number, write this number down.
7 + 76 = 83
76 X 111 = 83
Now add two zeros
76 X 111 = 8300
This will provide an answer within the acceptable range for number 11-111. The
only number that the estimate is too low for is 19.
Estimating 111 by 3 or more digits
716 X 111 = ?
Add the first digit to the first two, write this number down.
7 + 71 = 78
716 X 111 = 78
After first two digits, write the remaining digits you have not used and add 2
zeros.
716 X 111 = 78600
Example 2:
111 X 1428 = ?
Add first digit to the first two, write this number down.
1 + 14 = 15
After first two digits, write the remaining digits you have not used and add 2
zeros.
111 X 1428 = 152800
Estimating 125s
125 X 32 = ?
This is like 25s, except we divide by 8 since 8 X 125 = 1000, and we add three 0s
instead of two.
32 ÷ 8 = 4, therefore
125 X 32 = 4000
Example 2:
125 X 34 = ?
If there is a remainder, use this as the next number in place of a zero.
34 ÷ 8 = 4 remainder 2 therefore
125 X 34 = 4200
Example 3:
125 X 139 = ?
139 ÷ 8 = 17 remainder 3, therefore
125 X 139 = 17300