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.

Posted January 23, 2007, under Guides, Math
Tags: , , , , , ,
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

    1. nice blog…

      ajay — January 23, 2007 @ 1:37 am

    2. Two words: Metric System.

      bocasfx — January 24, 2007 @ 2:21 pm

    3. 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

    4. 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

    5. whew! nice math shortcuts!

      janice — March 9, 2007 @ 4:04 am

    6. This line is not correct:
      10,000 = hundred hundred

      Steven Marzuola — July 29, 2007 @ 10:13 pm

    7. how to multiply in shortcuts

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

    8. 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

    9. This line is not correct:
      10,000 = hundred hundred

      yes it is

      Neil Ramroop — August 1, 2007 @ 7:32 pm

    10. 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

    11. 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

    12. two more words: METRIC SYSTEM

      rumba — August 7, 2007 @ 1:27 pm

    13. 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

    14. 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

    15. 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

    16. 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

    17. 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

    18. 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

    19. Hi, thanks for the suggestion. I think I’ll put in a link for the “new” notation.

      Kalid — December 11, 2007 @ 7:45 pm

    20. could you do fraction and other end of the year work for math in 1 secondary

      nik — January 22, 2008 @ 11:46 am

    21. i love you

      love — February 1, 2008 @ 4:04 am

    22. 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

    23. How to calculate x^y manually, is there any short cuts?

      Raj — March 26, 2008 @ 3:59 am

    24. @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

    25. 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

    26. 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

    27. thanks for info. its raly usefull to students and staff

      saravanan — August 31, 2008 @ 10:20 pm

    28. Thanks saravanan.

      Kalid — August 31, 2008 @ 11:25 pm

    29. Great explained. congrulation you have to be really mathematician

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

    30. Thanks Ales!

      Kalid — September 21, 2008 @ 8:55 am

    31. What’s 43% of 200? Same as 200% of 43: 86.
      Nice :)

      Anonymous — September 28, 2008 @ 2:02 pm

    32. Thanks, that’s my favorite trick of the bunch.

      Kalid — September 29, 2008 @ 2:56 pm

    33. Thank you so much! Very helpful. I got a 98 on my quiz thanks to this.

      Talia Montgomery — October 20, 2008 @ 1:38 am

    34. Awesome, glad it was helpful.

      Kalid — October 20, 2008 @ 11:32 am

    35. nicE bLog,.,!

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

    36. 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

    37. 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

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

      (200/100) * 43
      2 * 43
      86

      Hope this helps.

      Kalid — January 31, 2009 @ 4:47 pm

    39. 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

    40. …Why did my comment get butchered?

      Calvin — March 5, 2009 @ 10:15 pm

    41. 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

    42. 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

    43. @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

    44. 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

    45. 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

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