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.

63 Comments

Posted January 23, 2007, under Guides, Math
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Related Posts:

Explanations for everyone.

The Rule of 72

A Quirky Introduction To Number Systems

Demystifying the Natural Logarithm (ln)

Techniques for adding the numbers 1 to 100

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Comments

nice blog…

ajay — January 23, 2007 @ 1:37 am
Two words: Metric System.

bocasfx — January 24, 2007 @ 2:21 pm
I like the metric system as much as the next guy, but 1 foot per nanosecond just works out so nicely

Kalid — January 24, 2007 @ 2:54 pm
Speed of light in metric: 3 decimeters per nanosecond (source). For those who don’t know, a decimeter is 1/10 of a meter; thus, another way of saying it would be 0.3 meters per nanosecond.

Peter Hosey — January 25, 2007 @ 3:02 am
whew! nice math shortcuts!

janice — March 9, 2007 @ 4:04 am
This line is not correct:
10,000 = hundred hundred

Steven Marzuola — July 29, 2007 @ 10:13 pm
how to multiply in shortcuts

Gladys and Jean — July 29, 2007 @ 11:16 pm
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.

Kalid — July 30, 2007 @ 8:08 am
This line is not correct:
10,000 = hundred hundred

yes it is

Neil Ramroop — August 1, 2007 @ 7:32 pm
Electrical signals in semiconductors do not travel at the speed of light. Your 1GHz clock distance
calculation is wrong.

John — August 7, 2007 @ 10:03 am
Hi John, thanks for the info. Yes, it seems that the speed of electricity varies depending on the conductor.

http://howthingswork.virginia.edu/page1.php?QNum=1267

If there’s a factor of .66c, then it would be 8 inches per nanosecond (rather than 1 foot). I’ll update the article.

Kalid — August 7, 2007 @ 12:18 pm
two more words: METRIC SYSTEM

rumba — August 7, 2007 @ 1:27 pm
I love the metric system too, but 1 foot per nanosecond just works out well, don’t you think?

30 centimeters per second doesn’t have quite the same ring to it .

Kalid — August 7, 2007 @ 3:58 pm
It’s misleading to say that a 16 bit number somehow equates to 64kb. A 16 bit number can REPRESENT any of 64 thousand different integers. But it is only made up of TWO bytes (8 bit bytes).

Josh — August 8, 2007 @ 9:41 pm
Good point. a 16-bit number can address up to 64Kb of memory, but doesn’t take 64kb of room. I’ll clarify.

Kalid — August 9, 2007 @ 3:08 am
Kalid, great site! In your derivation for work hours, the third line should read:

days per year = weeks * days

Cheers!

PS Perhaps, in a future article you might be inclined to explain UNIX load averages (and different kinds of averages in general, like here)?

Marc — September 23, 2007 @ 5:06 pm
Hi Marc, thanks for comment & catch — should be fixed now! That’s a good topic suggestion, I didn’t realize there could be so many intricacies in a “simple” performance metric

Kalid — September 24, 2007 @ 11:45 am
The last paragraphs discussing 2^10, 2^20, etc, may be a good place to introduce the concept of KiB, MiB, GiB nomenclature, (http://en.wikipedia.org/wiki/Kibibyte)

Mr Nerd — December 10, 2007 @ 9:24 pm
Hi, thanks for the suggestion. I think I’ll put in a link for the “new” notation.

Kalid — December 11, 2007 @ 7:45 pm
could you do fraction and other end of the year work for math in 1 secondary

nik — January 22, 2008 @ 11:46 am
i love you

love — February 1, 2008 @ 4:04 am
Here’s one:
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.

Michael Duplantier — March 21, 2008 @ 2:48 pm
How to calculate x^y manually, is there any short cuts?

Raj — March 26, 2008 @ 3:59 am
@Michael: Cool, thanks for the tip, it’s a nice rule of thumb. It’d be neat if there was a way to account for different terms/interest rates too.

@Raj: There’s shortcuts above for powers of 2, but otherwise I think you’d need to just multiply it out.

Kalid — March 26, 2008 @ 8:16 am
How do you find multiples of 1-12 easily like if u add the digits of the number and it is a multiple of 3 the number is a multiple of 3?

Debrish — March 31, 2008 @ 3:56 pm
do anyone knows shortcuts for solving number systems i.e. octal, binary please mail me at agarwalsagar@ymail.com

sagar — July 23, 2008 @ 1:02 am
thanks for info. its raly usefull to students and staff

saravanan — August 31, 2008 @ 10:20 pm
Thanks saravanan.

Kalid — August 31, 2008 @ 11:25 pm
Great explained. congrulation you have to be really mathematician

Ales Matematik — September 20, 2008 @ 8:42 pm
Thanks Ales!

Kalid — September 21, 2008 @ 8:55 am
What’s 43% of 200? Same as 200% of 43: 86.
Nice

Anonymous — September 28, 2008 @ 2:02 pm
Thanks, that’s my favorite trick of the bunch.

Kalid — September 29, 2008 @ 2:56 pm
Thank you so much! Very helpful. I got a 98 on my quiz thanks to this.

Talia Montgomery — October 20, 2008 @ 1:38 am
Awesome, glad it was helpful.

Kalid — October 20, 2008 @ 11:32 am
nicE bLog,.,!

pauline dianne miranda — November 27, 2008 @ 5:18 am
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)

Leilani Estrada — December 1, 2008 @ 2:56 am
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…

Anonymous — January 17, 2009 @ 8:38 am
@Anonymous: Hi there, 200% of 43 is

(200/100) * 43
2 * 43
86

Hope this helps.

Kalid — January 31, 2009 @ 4:47 pm
Dear Kalid,Please take this into account and resolve the issue ASAP:c≈299,792.458 Km/s, not c≈300K Km/s.Calvin

Calvin — March 5, 2009 @ 10:13 pm
…Why did my comment get butchered?

Calvin — March 5, 2009 @ 10:15 pm
your formula for doubling time should be close to 70/X% the correct number is ln2×100/X%

J — March 16, 2009 @ 8:04 pm
your formula for doubling time should be closer to 70/X% the correct number is ln2×100/X%

J — March 16, 2009 @ 8:05 pm
@Calvin: Not sure I understand the difference; 300,000 Km/sec is a decent approximation.

@J: Yep, the “Rule of 70″ is more accurate by 72 has more divisors so can be easier to work with (one reason it’s well known). I have another article on this.

Kalid — March 17, 2009 @ 2:36 am
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
______
1 4 4 1 is direct as second line
thank u

vimal mistry — May 9, 2009 @ 1:16 am
Hi,
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

Help me with other math shortcuts.

Anonymous — June 8, 2009 @ 12:28 pm
please give me shortcut tricks of pipes and cisterns

manish nigam — July 4, 2009 @ 2:18 am
[i][b] hooooooooooooooooooooooooooooo
hoooooooooooooooooooooooooooo
i say after doing mathematics because it is hard sub

[i][b] aviral tiwari — July 24, 2009 @ 2:52 am
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

aviral tiwari — July 24, 2009 @ 3:00 am
Nice ones. I posted my own multiplication shortcuts on Discrete Ideas.

The Count — August 14, 2009 @ 11:19 am
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)

Srivarshani — October 15, 2009 @ 10:55 am

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