 # Mental Math Shortcuts or Tricks

Inspired by information in the Math is Cool thread, I thought it would be great to pull ideas together that can make our mathematical lives easier. The shortcut doesn’t have to be earth-shattering, just something that can save some time.

To multiply mentally by 11:

24 x 11 = 264. Add the two and four (ones and tens digits) to get the 6 (tens digit in the sum). Keep the ones digit the same. Move the 2 (tens digit) into the hundreds position in the sum.

If you get more than 9 when you total your digits, simply carry the one onto the hundreds digit. Ex. 84 x 11= 924.

To Multiply mentally by 5:

26 x 5 = 130. Take half of the number 26 and get 13. Move the decimal once to the right and get 130. Basically dividing by two then multiplying by ten.

Division by 5 works just the opposite way.
27/5 = 5.4 Double 27 and get 54. Move the decimal once to the right to get 5.4.

Come on all you math geeks and help me out.

To quickly transpose an hourly wage into its approximate yearly salary, multiply by two and add three zeros.

\$15.[sup]00[/sup] an hour = ~\$30,000 per annum

I’ve had HR people act all amazed when I perform this quick mental transposition. Idjits!

For small variations, the percent increase in something is equal to the percent decrease in its reciprocal.

For instance, speed and trip duration have a reciprocal relationship. Imagine you know it takes 4 hours to drive somewhere at 60 MPH. If you increase speed to 75 MPH, you’ve sped up by a factor of 1/4, because 15 is 1/4 of 60. So the trip duration will decrease by the same factor, and since 1/4 of 4 hours is 1 hour, the trip will take about 3 hours at 75 MPH.

square a two digit number ending in five
6565=4225
thinking:
6
the next highest number is 42. Ends in 25
quick multiplication of similar sized numbers
4852=2496
thinking:
this is the famous difference of two squares
(50-2)(50+2) = 50
50-2*2
. = 2500-4

this looks very impressive if you have memorised the square numbers.
eg:
2329 = 2626-3*3
. = 676-9
. = 667
pi^2 is approximately 10

heaps of others: I’ll come back to do some more later.

OOH OOH, I got one!

If you ever feel like doing basic 9 x (0 to 9) math on your fingers, hold your ten fingers up and procced as follows:

For example, if you want to multiply 9x4

Fold down the fourth finger from the left (in my case, the left ring finger, since I’m palm up). Now count the number of fingers on each side of the folded one…3 and 6. Therefore, 9x4 is 36!!

Isn’t that neat? Oh, and

“eight times eight fell on the floor,
when it got up, it was sixty four!”
: shrug :

Always remember: Eleven is one more than ten.

Ah yes, but how big is eleventeen? Add up all the digits in a # and if they add to 3, 6 or 9 it is divisible by 3. 123, 1+2+3= 6 divisible by 3, 2,874,123 adds to 27. add the 2 and the 7 and get 9, divisible by 3. I’ve never found it particuarly useful, However the aformentioned hourly wage trick blows a lot of people away, its even more impressive when you remember to add two weeks onto you quick calculation.

As a machinist I’ve got small stupid tricks to make my day easier, to get from diameters in inches to circumferance in feet (to figure out surface speed) divided by 3.8. There is a bunch of others but I can’t think of them now, they just pop out when I need them.

Personally - which drives the CS guys crazy - I like the approximation of 2^10 = 1024 ~ 1000 = 10^3.

So, when a friend of mine, who studies CS, told me that the IP addresses were running out and hence efforts are made to expand them, I said “Hey, an IP is made up of 12 numbers and 10^12 should be plenty of addresses.”

He corrected me that the 12 digits aren´t really 12 digits, but that an IP address is made up of 4 bytes. So then we had 2^32 different IP addresses. I then asked him how many that were and he had no idea.

Doing the conversion, I broke it down to 4 * 2^10 * 2^10 * 2^10, which would roughly be 4 * 10^3 * 10^3 * 10^3.

4 Billion numbers, ok, now I could see that this might cause a problem and since the derivation from the actual number was less than 10% off, I figure it´s a solid estimation.

Somewhat related: To measure your speed whilst driving without looking at the speedometer, time the distance it takes you to travel one mile (mile markers on Interstate highways make this really easy). 60 mph = 1 mile per minute. Every second above 60 seconds subtracts one mile per hour. Every second below 60 seconds adds one mile per hour. Try this in an older (mid 80’s or older) car, compare your timed result to the speedometer, you might be suprised at how far off the speedometer is.

To figure out a 15% tip: Take the amount of the check (say \$51.98), slide the decimal point one place to the left (\$5.19). Now take half of that number ( about \$2.50) and add it to the first number (about \$7.70). Now round up to the nearest whole dollar, ya cheap bastard! Or, if you live in Ontario, just add the PST and GST which is right on your receipt Now how about some math shortcuts for doing differential equations?

Optihut’s post actually leads into the one I like.

I frequently have to translate very large binary numbers into something the norms can understand. E.g., how long it would take to do 2^80 operations. The first step is to convert to a power of 10: multiply by .3. Then remember, Pi seconds is a nanocentury. Doing this in your head in real time scares some people.

Easy, just do the odd-numbered problems. The answer’s are in the back. Even better, get the teacher’s edition, all the anwers!

No they don’t. Speed and distance have a reciprocal relationship that results in trip duration. Thus your 240 mile trip at 75mph will take 3 hours and 12 minutes.

I was going to add that the way to derive this is to say the increase in speed is 1/5 (15/75) and decrease the original trip by one fifth i.e 48 minutes.

Of course, but that was not the point. 3 hours and 12 minutes is not that far off from 3 hours, so it´s a good estimate, which is quicker to execute than calculating the actual value. Besides, you don´t accelerate from 60 to 75 instantly and probably will not drive a constant 75, so all those calculations are just guestimates. It´s reasonable to do a quick and dirty method, that is slightly inaccurate.

I don’t know how useful this will be for everyday math as it concerns prime numbers. Remember that, except for 2 and 5, all prime numbers will end in an odd number, and never a five.

The magazine Popular Science once had an article on math shortcuts. The one that sticks in my mind is how to multiply by 9 — simply shift the decimal point over by one (multiply by ten, that is) and subtract the original number. To multiply by five, shift the decimal point over (multiply by ten, again) and then take half. These “tricks” might seem as hard as the original problem, but in many cases they’re actually easier to do.

I find that it’s easier to multiply by four or by eight by doing repeated doubling than by multiplying directly.
It’s not a strictly mathematical thing, but to convert from degrees Celsius to Fahrenheit, simply double and add thirty. The correct formula is (C = 9/5 F + 32), but the “double and add thirty” gives you the temperature to within a degree or two across the human comfort range, and is a heckuva lot faster. To go from Fahrenheit to celsius, subtract 30 and take half. Useful if you’re in Canada or Europe.

In a similar vein, to convert kilometers to miles (or km/hr to miles per hour), just multiply by 0.6 (or multiply by six and shift the decomal point over one).

C --> F conversions, I use a different rule. More steps, but usually easier, and I find it easier to remember because of the symmetry in the rule.
C --> F
Add 40, divide by 5, multiply by 9, subtract 40

F --> C
Add 40, divide by 9, multiply by 5, subtract 40

If you’d like to guestimate how fast someone is going when they blitz by you on the interstate:

Take your own speed and divide it by 4 e.g. 72/4=18
As they pass you start counting at a steady rate…if you’ve got a song on the radio, count to the beat. When you reach 18 note where the other vehicle is and start counting again from one. When you reach the point where he was when you started the second count stop and multiply the number you got by 4. If it took 4 counts for you to reach the spot it would be 4x4=16. Add 16 to your own speed, 72+16=88 and you have theirs.