Mental Math Shortcuts

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. Hi Steven, feel free try it out: 100 * 100 = 10,000 (hundred hundred is similar to “two hundred (200)”, “fifteen hundred (1500)”, “forty-seven hundred (4700)” or “hundred hundred (10,000)”. We don’t often say “hundred hundred” though .

Gladys/Jean, I may do a follow up with mental multiplication tricks.

Kalid8:08 am

If you are looking to purchase a car and want to quickly assess what the monthly payment on a regular purchase (not a lease) will cost you, do the following. Take the bottom line (price + tax, title, license), multiply by 2 and chop off 2 places. Example: You want to finance a 25000 car (remember, this is after sales tax). Monthly payment is ~$500/month. See? Now, this does assume 60 months at 8% interest, but these are typical.

1. 1)Write the 25th term in the number sequence: 2,3,5,7,11,13,17…..(please explain using short-cut method). Thanks.

2)If 2+4+6+….+198+200= 10 100, what is the value of 1+3+5+…+197+199? (Solve using short-cut method)

2. What’s 43% of 200? Same as 200% of 43: 86

Can some people tell me how this work? To me its 43% of 200 which is 86, vs 200% of 43 which is 172, and 172:86 is not 86 but 2.

No matter how I do this I can’t get it to be

43% of 200 =86= 200% of 43: 86, which is basically what you are saying…

3. @Anonymous: Hi there, 200% of 43 is

(200/100) * 43

2 * 43

86

.

4. hi, i request u please all mathematical shortcut formula needs from u. for example 12 quare root 144 so shortcut formula is 12*12=144

1 2

1 +2 1 + 1 = 2*2=4

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1 4 4 1 is direct as second line

thank u

5.

6. I have a gd shortcut for multiplying 3 digit numbers:

109 x 105 = _ _ _ _ _

1 _ _ (add last 2 digits) then _ _ (multiply last two digits)

Answer 1 14 45 or 11445

7. to multiply any three number by 11just dooooooo this thing e.g

453

last no 3 will come in last then

2and3 no is 5 3 we will add=8

now.we will add 1and2 no is 4 5 we will add=9

now right the first no 4

ans ==========4983

(453*11=4983)magic )

Aviral Tiwari

8. Hai Ido not about shortcuts But there are formulae to find the sum OR term in arithmetic progression First term=a difference=d n=No of terms l=last term Then n=((l-a)/d)+One

t(n)=a+(n-one)d ,sum(n)=n/2(2a+(n-one)d)OR

s(n)=n/2(a+l) For odd nos The sum is n to the power 2 (to Leilani)

9.

10. I was sitting at dinner and, being the quiet type, contemplated the relation between numbers squared. Here are some thoughts and observations:

-13^2= 169, while 31^2= 961, it would seem every number (except the exponent) is visually flipped.

-14^2= 196, while 41^2= 1681, this may seem somewhat of a stretch, but imagine 196 as 18(16), where 16 is one decimal place. The principle above applies.

-15^2= (14+1)^2= 14^2 + 2(14) + 1= 196 + (28 + 1)= 225, once you find a square, just add 2x+1 to its square to get the next integer squared.

-34^2= (30+4)^2= 30^2 + 2(30)4 + 4^2= 900 + 240 + 16= 1156, its easy to break it into tens and ones.

11. Just thought of something else, sorry, I’m new here. Over the years, i always wondered how 1 square foot equaled 144 square inches. I realized not only were the measurements squared, but so were their proportions. This lead to a discovery of “dimensional scaling” i call it.

-If one square has half the width AND height of another, its area is (1/2 * 1/2)= (1/2)^2= 1/4 the area of the other, yet a cube with half dimensions would be (1/2 * 1/2 * 1/2)= (1/2)^3= 1/8 the volume. Notice the pattern? The space occupied (area, volume, etc.) of a relative object is the (space occupation of the reference object) * (scale^dimensions).

-This works for circles and spheres when you consider the dimensional occupation of the object first (circles: dimensions=2).

-It can also apply to varying scales: a cube with width/2, height/3, depth/4 equals (whd)(1/2)(1/3)(1/4).

I really hopes this helps, it allowed me to answer an ACT question faster and easier.

To square a two digit number, say 74

7^2=49

(7*4)2=56

4^2=16

now add the digit other than the ones digit with previous number.

1. 49 cant be added with a previous number.

2. add the 5 of 56 with 49, 49+5=54 then simply attach the ones digit.becomes 546

3. similarly add 1 of 16 with 546 and attach 6.

you get 5476.

this is the square of 74.

i.e. 74^2=5476

To square a two digit number, say 74

7^2=49

(7*4)2=56

4^2=16

now add the digit other than the ones digit with previous number.

1. 49 cant be added with a previous number.

2. add the 5 of 56 with 49, 49+5=54 then simply attach the ones digit.becomes 546

3. similarly add 1 of 16 with 546 and attach 6.

you get 5476.

this is the square of 74.

i.e. 74^2=5476

I personally gravitate to the 68 mph is “about” the same as 100 fps is “about” the same as 30 meters per second. My logic…first off with the speed limit set at 65 mph you can generally travel at 68 mph and probably not get a ticket. So it is a velocity that we “know” and “feel”. More important however is that it makes it easier to envelop calculate the centripetal force formula of velocity squared divided by radius. Just a thought…nowhere near as profound as Khalid’s but a thought nonetheless

12. for shortkat for number of sqare for number thar have 5 at last like we take a number of 25 then

frist number we multiply subsiquint number and add the square of 5 for example of 25–

frist

(2*3)(5*5)

=625

— May 28, 2011 @ 2:56 am

13. A neighbour’s son brought this home: an explanation for the 9 x table (multiplication).

Both palms facing you, bend left- ring-finger towards yourself. It is the fourth digit from the left.. so 4 x 9 = 36. What is displayed is your l-thumb, l-index, and l-middle finger = the three; and the one l-little finger and five digets on the right showing six total digets, displaying the 6 of ‘thirty six’. Only works on the 9 xtable. I have found it very useful.

This session would cover the following:

• Special method for squaring numbers ending in 5

• Two Digit Multiplication whose base is 10

• Relation between Alphabets & Numerals

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