Mental Math Shortcuts

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
= 900

Multiplication 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
= 792

Multiplication 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 111

solve 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
Multiplication of numbers ending in 5 with difference of 10
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,
= 1575

75 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
= 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
= 42

700/50
= (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
= 155

116 + 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
= 154

250 - 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
= 18

360 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
= 60

How 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

Here’s a collection of time-saving math shortcuts, great for back-of-the-envelope estimates.

Time and Distance

60 mph = 1 mile per minute

Going 60 mph and the exit is in 10 miles? That’s 10 minutes.
Been driving a half hour? That’s about 30 miles at highway speeds.

Feet Per Second = MPH * 1.5
MPH = Feet Per Second * 2/3 (derivation)

60 mph is about 90 feet per second (88 exactly), so just multiply by 1.5. Or, just add half to itself (60 + 30 = 90).
Going 100 mph? That’s 150 fps.
Going 10 fps? That’s about 7 mph (10 * 2/3 is 6.666). Or, just take away 1/3 (10 – 3 = 7).

speed of light = 1 foot per nanosecond (derivation)

The US is about 3000 miles long. There’s about 5000 feet/mile, so that’s about 3000 × 5000 or 15 million feet. 15 million feet takes 15 million nanoseconds, or 15/1000, or 15 milliseconds. That’s the minimum time for a signal to go across the country.
Inside a microchip, if you have a clock cycle every nanosecond (1 GHz), your signal can only travel 1 foot at most (or less, depending on the material). Even light takes 30ns to cross a 30 foot room.

1 year = 250 work days = 2000 work hours (derivation)

Project takes 1000 man hours? That’s 6 months for 1 person.
Daily commute of 1/2 hour? That’s .5 * 250 = 125 hours in the car each year.

Money and Finance

$1/hour = $2000/year (derivation)

Earn $25/hour? That’s about 50k/year.
Make 200k/year? That’s about $100/hour. This assumes a 40-hour work week.

$20/week = $1000/year (derivation)

Spend $20/week at Starbucks? That’s a cool grand a year.

Rule of 72: Years To Double = 72/Interest Rate (derivation)

Have an investment growing at 10% interest? It will double in 7.2 years.
Want your investment to double in 5 years? You need 72/5 or about 15% interest.
Growing at 2% a week? You’ll double in 72/2 or 36 weeks. You can use this rule for any duration of time, not just years.
Inflation at 4%? It will halve your money in 72/4 or 18 years.

Mental Arithmetic

Numbers
10,000 = hundred hundred
million = thousand thousand
billion = thousand million
trillion = million million

1% of 10k is 100. The Dow is roughly 10k (it’s about 12k now). So if the dow drops 100, it’s about a 1% loss.
What’s 5k x 50k? That’s 250 * thousand * thousand or 250 million.

Visualizing numbers (read more)

12 days = 1 million seconds
1 year = 31 million seconds (about pi * 10 million)
30 years = 1 billion seconds
30,000 years = 1 trillion seconds

One “part per million” means an accuracy of 1 second every 12 days. One “part per trillion” means an accuracy of 1 second every 30,000 years.

Powers of 2
2^6 = 64 (the sixes match: six and sixty-four)
2^10 ~ thousand (1 kb)
2^20 ~ million (1 mb)
2^30 ~ billion (1 gb)

Sure, 2 to the tenth = 1024, but 1000 is good enough for government work. (Read on about KB vs KiB).
Have 32-bit color? That’s 2 + 30 bits = 2^2 * 2^30 = 2^2 billion = 4 billion (4gb exactly).
Have a 16-bit number? That’s 6 + 10 bits, or 2^6 thousand, or 64 thousand (64 kb).

a% of b = b% of a

It’s not immediately clear, but it’s true: a% of b = .01 * a * b, which is the same as b% of a (.01 * b * a).
What’s 16% of 25? The same as 25% of 16: 4
What’s 43% of 200? Same as 200% of 43: 86.

Calculus - Differential and Integral
CALCULUS: Differentiation and Integration formulas

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Wednesday 13 July 2011

MATH SHORTCUTS