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.

There’s more shortcuts & techniques over at Ars Calcula.

Other Posts In This Series

  1. Mental Math Shortcuts
  2. How to Develop a Sense of Scale
  3. Techniques for adding the numbers 1 to 100
  4. Easy Permutations and Combinations
  5. How To Understand Combinations Using Multiplication
  6. Navigate a Grid Using Combinations And Permutations
  7. How To Analyze Data Using the Average
Kalid Azad loves those Aha! moments when an idea finally clicks. BetterExplained is dedicated to learning with intuition, not blind memorization, and is honored to serve 250k readers each month.

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95 Comments

  1. 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.

  2. 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.

  3. Electrical signals in semiconductors do not travel at the speed of light. Your 1GHz clock distance
    calculation is wrong.

  4. 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 :).

  5. 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).

  6. 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)?

  7. 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 :)

  8. 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.

  9. @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.

  10. 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?

  11. 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)

  12. 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…

  13. Dear Kalid,Please take this into account and resolve the issue ASAP:c≈299,792.458 Km/s, not c≈300K Km/s.Calvin

  14. @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.

  15. 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

  16. 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.

  17. 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

  18. 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)

  19. Does anyone agree with me that one of the greatest roadblocks keeping 5th – 12th graders from using mental math is that they use calculators too much?

  20. I LOVE IT !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

  21. 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.

    Not sure if this is helpful, just thought it may spark some epiphanies or allow slightly quicker squaring.

  22. 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.

  23. I think they should be teaching our children to use an abacus in elementary school…I learned how to use one at that age and I have always used most of these mental math shortcuts my whole life. Not because someone showed me, I have just always been able to reduce things to simple math, and visualize the abacus…I’m just sayin…

  24. @Mike: Yep, I think having more hands-on learning techniques could be helpful in developing an intuition for math.

  25. I’ll be very much thankful if you could explain how the series / order arrived :
    1) 8 11 19 25 31 47
    2) 2 20 21 22 26 49

  26. For easy math squaring shortcuts and multiplication shortcuts contact me at iforindia123@gmail.com

    For example to show one of my methods:

    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

  27. For easy math squaring,multiplication,Division,Addition,Subtraction, percentage shortcuts contact me at iforindia123@gmail.com

    For example to show one of my methods:

    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 :-) :-) :-) :-) :-) :-)

    http://mathshortcuts1.blogspot.com/ (Under construction)

  28. 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

  29. @mark: Really cool insight! I think it’s important to have a gut feel for numbers, and I think most people can “feel” how fast 65/68mph is (i.e. you have an idea of how fast things should be zooming past you on a highway).

  30. 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

  31. 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.

  32. Having discovered KhanAcadem.org, I can’t stop promoting it. Go to their website and watch the TEDtalk by Salman Khan. It blew my mind!
    Khan Academy is a free online school. Had you heard about it before now?

  33. @Raj
    How to calculate x^y manually, is there any short cuts?
    Using logarithms and antilogarithms you can.
    ==========
    Quick calculations with a few logarithms
    If you can remember a few logarithms, you can do many calculations quite easily without the aid of calculators or computers.
    Try to remember the logarithms of just seven numbers:
    Log 2 = 0.30, log 3 = 0.48, log 7 = 0.85, log 11= 1.04, log 13= 1.11, log 17 = 1.23 and log 19=1.28.
    The logarithm of a composite number is equal to the sum of the logarithms of its prime factors; you can formulate the following table of logarithms:
    Number Logarithm Number Logarithm
    2 0.30 11 1.04
    3 0.48 12 1.08
    4 0.60 13 1.11
    5 0.70 14 1.15
    6 0.78 15 1.18
    7 0.85 16 1.20
    8 0.90 17 1.23
    9 0.95 18 1.26
    19 1.28
    Note: Logarithm of 11 is 1.04. It means log 1.1 = 0.04. Log 5 = log (10/2) = log10 – log 2.
    With the above figures in hand you can do a lot of calculations quite easily.
    1. What is 2 raised to power 37?
    Logarithm of 2 is 0.30.
    37 times 0.30 is 11.10
    Since logarithm of 1.2 is 0.08 and logarithm of 1.3 is 0.11, antilog of 0.10 is about 1.26.
    So antilog of 11.10 is 1.26 * 10^11.
    That is 2^37 = 1.26 * 10^11.
    2. What is the 31st root of a 35 digit number?
    This can be answered even without knowing the number.
    Log of 35th digit number is 34. …
    Log of 31st root = 34. …./ 31
    This lies between 34/31 and 34.99/31 or between 1.09 and 1.13.
    From the above table, we can see that in this interval, logarithm of 13 is 1.11.
    So, 31st root of a 35 digit number is 13.
    3. What is the 64th root of a 20 digit number? Answer is 2.
    4. What is the cube root of say 74567?
    Log of 74567 = 4.873 (interpolating between log 7 =0.85 and log 8 = 0.90).
    4.873 / 3 = 1.624.
    Antilog of 0.624 is 4.24 (because log 4 = 0.60 and log 5 = 0.70).
    So, cube root of 74567 is about 42.4

  34. It is so useful and so helpful for me. Thanks a lot. If you can, please send me more helpful tricks to my e-mail ID sir.

  35. “Interest rate” can be generalised to “percentage per year”, and applied to such things as population growth, crime statistics, etc. So: Years To Double = 72 / %p.a.
    (or, more accurately, Years To Double = 70 / %p.a.).

  36. I believe this line has a typo:
    Have 32-bit color? That’s 2 + 30 bits = 2^2 * 2^30 = 2^2 billion = 4 billion (4gb exactly).

    2^2 billion should be 2 * 4 billion. It should read:
    Have 32-bit color? That’s 2 + 30 bits = 2^2 * 2^30 = 2 * 2 billion = 4 billion (4gb exactly).

  37. Disregard my last comment. I believe you were expressing (2^2) billion, not 2^2,000,000,000.

  38. Kalid, Hi:

    The subtitle on the selection button for ‘Mental Math Shortcuts’ is ‘A few memorable converions’. Did you mean ‘conversions’?

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LaTeX: $$e=mc^2$$