MULTIPLICATION:
Multiplication using multiples
12 × 15
= 12 × 5 × 3
= 60 × 3
= 180
Multiplication by distribution
12 × 17
= (12 x
10) + (12 x
7) `rArr` 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 `rArr` 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 `rArr` take the half(0.5) then multiply by 100
18 x 50
= (18÷2) × 100 = 18 × 0.5 × 100
= 9 x 100
= 900
Multiplication by 100 ⇒ move the decimal point two places to the right
42 × 100
= 4200 ⇒ added two zeroes
Multiplication by 500 ⇒ take the half(0.5) then multiply by 1000
21 × 500
= 21 ÷ 2 × 1000
= 10.5 × 1000
= 10500
Multiplication by 25 ⇒ use the analogy $1 = 4 x 25 cents
25 × 14
= (25 × 10) + (25 × 4) → 250 + 100 → $2.50 + $1
= 350
Multiplication by 25 ⇒ divide by 4 then multiply by 100
36 × 25
= (36 ÷ 4) × 100
= 9 × 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
Multiplication of 37 by the 3, 6, 9 until 27 series of numbers ⇒ the “triple effect”
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 × 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
Multiplication of numbers ending in 5 with difference of 10
45 × 35
first term = [(4 + 1) × 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 × 85
first term = (8 + 1) x 7 = 63
last term = 75
combining first and last terms,
= 6375
15 × 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
= 280 Division 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:
30% of 120
= 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:
To convert or express percentages as decimals, divide by 100 or simply just move the decimal point by two places to the left of the given number, thus:1% = 1/100 = 0.01
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
Square Number
Tips 1: 2 Digit Number which ends in 5
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.
Answer: 1225
Tips 2: Any 2 Digit Number
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
Answer: 2209
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 |