MULTIPLICATION:
Multiplication using multiples
12 x 15
= 12 x 5 x 3
= 60 x 3
= 180
Multiplication by distribution
12 x 17
= (12 x 10) + (12 x 7) —> 12 is multiplied to both 10 & 7
= 120 + 84
= 204
Multiplication by “giving and taking”
12 x 47
= 12 x (50 – 3)
= (12 x 50) – (12 x 3)
= 600 – 36
= 564
Multiplication by 5 –> take the half(0.5) then multiply by 10
428 x 5
= (428 x 1/2) x 10 = 428 x 0.5 x 10
= 214 x 10
= 2140
Multiplication by 10 —> just move the decimal point one place to the right
14 x 10
= 140 —> added one zero
Multiplication by 50 —> take the half(0.5) then multiply by 100
18 x 50
= (18/2) x 100 = 18 x 0.5 x 100
= 9 x 100
= 900
Multiplication by 100 —> move the decimal point two places to the right
42 x 100
= 4200 —> added two zeroes
Multiplication by 500 —> take the half(0.5) then multiply by 1000
21 x 500
= 21/2 x 1000
= 10.5 x 1000
= 10500
Multiplication by 25 —> use the analogy $1 = 4 x 25 cents
25 x 14
= (25 x 10) + (25 x 4) —> 250 + 100 —> $2.50 + $1
= 350
Multiplication by 25 —> divide by 4 then multiply by 100
36 x 25
= (36/4) x 100
= 9 x 100
= 900Multiplication by 11 if sum of digits is less than 10
72 x 11
= 7_2 —> the middle term = 7 + 2 = 9
= place the middle term 9 between 7 & 2
= 792Multiplication by 11 if sum of digits is greater than 10
87 x 11
= 8_7 —> the middle term = 8 + 7 = 15
because the middle term is greater than 10, use 5 then
add 1 to the first term 8, which leads to the answer of
= 957
solve 37 x 3
multiply 7 by 3 = 21, knowing the last digit (1), just combine two more 1’s giving the triple digit answer 111solve 37 x 9
multiply 7 by 9 = 63, knowing the last digit (3), just combine two more 3’s giving the triple digit answer 333
solve 37 x 21
multiply 7 by 21 = 147, knowing the last digit (7), just combine two more 7’s giving the triple digit answer 777
Multiplication of the “dozen teens” group of numbers —
(i.e. 12, 13, 14, 15, 16, 17, 18, 19) by ANY of the numbers within the group:
solve 14 x 17
4 x 7 = 28; remember 8, carry 2
14 + 7 = 21
add 21 to whats is carried (2)
giving the result 23
form the answer by combinig 23 to what is remembered (8)
giving the answer 238
45 x 35
first term = [(4 + 1) x 3] = 15; (4 is the first digit of 45 and 3 is the first digit of 35 –> add 1 to the higher first digit which is 4 in this case, then multiply the result by 3, giving 15)
last term = 75
combining the first term and last term,
= 157575 x 85
first term = (8 + 1) x 7 = 63
last term = 75
combining first and last terms,
= 6375
15 x 25
= 375
Multiplication of numbers ending in 5 with the same first terms (square of a number)
25 x 25
first term = (2 + 1) x 2 = 6
last term = 25
answer = 625 —> square of 25
75 x 75
first term = (7 + 1) x 7 = 56
last term = 25
answer = 5625 —> 75 squared
DIVISION:
Division by parts —> imagine dividing $874 between two persons
874/2
= 800/2 + 74/2
= 400 + 37
= 437
Division using the factors of the divisor: “double division”
70/14
= (70/7)/2 —> 7 and 2 are the factors of 14
= 10/2
= 5
Division by using fractions:
132/2
= (100/2 + 32/2) —> break down into two fractions
= (50 + 16)
= 66
Division by 5 —> divide by 100 then multiply by 20
1400/5
= (1400/100) x 20
= 14 x 20
= 280Division by 10 —> move the decimal point one place to the left
0.5/10
= 0.05 —> 5% is 50% divided by ten
Division by 50 —> divide by 100 then multiply by 2
2100/50
= (2100/100) x 2
= 21 x 2
= 42700/50
= (700/100) x 2
= 7 x 2
= 14
Division by 100 —> move the decimal point two places to the left
25/100
= 0.25
Division by 500 —> divide by 100 then multiply by 0.2
17/500
= (17/100) x 0.2
= 0.17 x 0.2
= 0.034
Division by 25 —> divide by 100 then multiply by 4
500/25
= (500/100) x 4
= 5 x 4
= 20
750/25
= (750/100) x 4
= 7.5 x 2 x 2
= 30
ADDITION:
Addition of numbers close to multiples of ten (e.g. 19, 29, 89, 99 etc.)
116 + 39
= 116 + (40 – 1)
= 116 + 40 – 1 —> add 40 then subtract 1
= 156 – 1
= 155116 + 97
= 116 + (100 – 3)
= 116 + 100 – 3 —> add 100 then subtract 3
= 216 – 3
= 213
Addition of decimals
12.5 + 6.25
= (12 + 0.5) + (6 + 0.25)
= 12 + 6 + 0.5 + 0.25 —> add the integers then the decimals
= 18 + 0.5 + 0.25
= 18.75
SUBTRACTION:
Subtraction by numbers close to 100, 200, 300, 400, etc.
250 – 96
= 250 – (100 – 4)
= 250 – 100 + 4 —> subtract 100 then add 4
= 150 + 4
= 154250 – 196
= 250 – (200 – 4)
= 250 – 200 + 4 —> subtract 200 then add 4
= 50 + 4
= 54
216 – 61
= 216 – (100 – 39)
= 216 – 100 + 39
= 116 + (40 – 1) —> now the operation is addition, which is much easier
= 156 – 1
= 155
Subtraction of decimals
47 – 9.9
= 47 – (9 + 0.9) —> “double subtraction”
= 47 – 9 – 0.9 —> subtract the integer first then the decimal
= 38 – 0.9
= 37.1
18.3 – 0.8
= 18 + 0.3 – 0.8
= (18 – 0.8) + 0.3 —> subtract 0.8 from 18 first
= 17.2 + 0.3
= 17.5
WORKING ON PERCENTAGES:
= 10% x 3 x 120 —> it is much easier working with tens (10%)
= 10% x 120 x 3
= 12 x 3
= 36
five percent of a number: 5%
360 x 5%
= 360 x 10%/2 —> take the 10% and divide by 2
= 36/2
= 18360 x 5%
= 360 x 50%/10 —> take the half(0.5) and divide by 10
= (360/2)/10
= 180/10
= 18
ninety percent of a number: 90%
90% of 700
= (100% – 10%) x 700
= (100% x 700) – (10% x 700) —> 100% minus 10% of the number
= 700 – 70
= 630
What is 18 as a percentage of 50?
= 18/50
= (18/100) x 2 —> method: division by 50 (explained above)
= 0.18 x 2
= 0.36
= 36%
What is 132 as a percentage of 200?
= 132/200
= (132/2)/100
= [100/2 + 32/2]/100 —> solution by “double division”
= (50 + 16)/100
= 66/100
= 0.66
= 66%
What is 270 as a percentage of 300?
= 270/300
= [(270/3)/100] —> “double division” (using the factors of 300)
= 90/100
= 90%
What is 17 as percentage of 500?
= 17/500
= (17/50)/10
= (17/100) x 2/10 —> solution using the procedure: division by 50
= (0.17 x 2)/10
= 0.34/10
= 0.034
= 3.4 %
percentages close to 100:
95% of 700
= (100% – 5%) x 700
= (100% x 700) – (5% x 700)
= 700 – (10% x 700/2) ——> 5% is 10%/2
= 700 – 70/2
= 700 – 35
= 665
percentages less than 10 percent:
3% of 70
= (3/100) x 70
= (70/100) x 3 —> divide by 100 then multiply the percent value
= 0.7 x 3
= 2.1
DECIMALS:
2% = 2/100 = 0.02 = 1/50
3% = 3/100 = 0.03
4% = 4/100 = 0.04 = 1/25
5% = 5/100 = 0.05 = 1/20
6.25% = 6.25/100 = 0.0625 = 1/16
7% = 7/100 = 0.07
7.5% = 7.5/100 = 0.075
10% = 10/100 = 0.1 = 1/10
12.5% = 12.5/100 = 0.125 = 1/8
20% = 0.2 = 1/5
21% = 0.21
25% = 0.25 = 1/4
30% = 0.3 = 3/10
33.33% = 33.33/100 = 0.3333 = 1/3
37.5% = 0.375 = 3/8
40% = 0.4 = 2/5
50% = 0.5 = 1/2
60% = 0.6 = 3/5
62.5% = 0.625 = 5/8
66.66% = 66.66/100 = 2/3
75% = 0.75 = 3/4
80% = 0.8 = 4/5
87.5% = 0.875 = 7/8
100% = 1
125% = 1.25 = 1 1/4
150% = 1.5 = 1 1/2
200% = 2
FRACTIONS:
What is three quarters of 80?
= 3/4 x 80
= (80/4) x 3 —> divide by 4 then multiply by 3
= 20 x 3
= 60How many quarters in two and a half?
2.5/.25
= 10 —> there are 10 quarters in $2.50
Improper fractions:
3/2 = 1 1/2 = 1.5 = 150%
4/3 = 1 1/3 = 1.3333 = 133.33% —> useful number for volume of sphere, etc.
9/5 = 1 4/5 = 1.8 = 180% —> conversion factor for Celsius/Fahrenheit temperatures
V = 4/3 pi * r^3
where:
V = volume of sphere
r = radius of sphere
F = (1.8 C) + 32
where:
F = temperature in Fahrenheit
C = temperature in Celsius
Step1: Multiply the first digit of number, with the next to its number. Ex: 35 is the number you want to square. 3 x 4 = 12
Step2: Finally add 25 at the end of the result.
Ex: 47
Step1: Look for the nearest 10 boundary.
3 from 47 to 50
Step2: Since we went up 3 to 50, now go down 3 from 47 to 44.
Step3: Now mentally multiply 44 x 50 = 2200 – 1st answer.
Step4: 47 is 3 away from the 10 boundary 50, Square 3 as 9 – 2nd answer.
Step5: Add the first and second answer, 2200 + 9
Sequential Inputs of numbers with 8

1 x 8 + 1 = 9

12 x 8 + 2 = 98

123 x 8 + 3 = 987

1234 x 8 + 4 = 9876

12345 x 8 + 5 = 98765

123456 x 8 + 6 = 987654

1234567 x 8 + 7 = 9876543

12345678 x 8 + 8 = 98765432

123456789 x 8 + 9 = 987654321

Sequential 1’s with 9

1 x 9 + 2 = 11

12 x 9 + 3 = 111

123 x 9 + 4 = 1111

1234 x 9 + 5 = 11111

12345 x 9 + 6 = 111111

123456 x 9 + 7 = 1111111

1234567 x 9 + 8 = 11111111

12345678 x 9 + 9 = 111111111

123456789 x 9 + 10 = 1111111111

Sequential 8’s with 9

9 x 9 + 7 = 88

98 x 9 + 6 = 888

987 x 9 + 5 = 8888

9876 x 9 + 4 = 88888

98765 x 9 + 3 = 888888

987654 x 9 + 2 = 8888888

9876543 x 9 + 1 = 88888888

98765432 x 9 + 0 = 888888888

Numeric Palindrome with 1’s

1 x 1 = 1

11 x 11 = 121

111 x 111 = 12321

1111 x 1111 = 1234321

11111 x 11111 = 123454321

111111 x 111111 = 12345654321

1111111 x 1111111 = 1234567654321

11111111 x 11111111 = 123456787654321

111111111 x 111111111 = 12345678987654321

Without 8

12345679 x 9 = 111111111

12345679 x 18 = 222222222

12345679 x 27 = 333333333

12345679 x 36 = 444444444

12345679 x 45 = 555555555

12345679 x 54 = 666666666

12345679 x 63 = 777777777

12345679 x 72 = 888888888

12345679 x 81 = 999999999

Sequential Inputs of 9

9 x 9 = 81

99 x 99 = 9801

999 x 999 = 998001

9999 x 9999 = 99980001

99999 x 99999 = 9999800001

999999 x 999999 = 999998000001

9999999 x 9999999 = 99999980000001

99999999 x 99999999 = 9999999800000001

999999999 x 999999999 = 999999998000000001
