You probably haven’t had to do longhand maths in years, but you do mental maths every day. Or maybe you google maths problems ten times a day, because you’ve forgotten how to do any maths beyond your basic multiplication tables. Here are some shortcuts that will help you do more maths in your head.
Calculate percentages backward
X% of Y = Y% of X. You can always swap those percentages if doing the maths is easier the other way around. So 68% of 25 = 25% of 68 = 68/4 = 17.
That makes a lot of calculations easy, once you’ve memorized the percentages that equal basic fractions:
10% = 1/10
12.5% = 1/8
16.666...% = 1/6
20% = 1/5
25% = 1/4
33.333...% = 1/3
50% = 1/2
66.666...% = 2/3
75% = 3/4
Subtract without carrying digits
Mental subtraction is easiest when you can subtract each digit without having to carry any places. If the second number has some bigger digits than the first, it gets more complicated. To avoid carrying places, you want to get rid of those bigger digits. Here’s how:
Say you’re calculating 925-734. That tens place makes things a little complicated. It’d be easier to calculate 925-724, and then subtract that extra 10 separately: 925-724 = 201, and 201-10 = 191. There’s your answer.
Tell if a number is evenly divisible by another number
All (and only) multiples of 2 end in 0, 2, 4, 6, or 8.
All (and only) multiples of 3 have digits that add up to 3 (or another multiple of 3).
Multiples of 4: Ignore everything from the hundreds place up. Divide the remaining two-digit number in half. Then run the multiples-of-2 test.
All (and only) multiples of 5 end in 5 or 0.
Multiples of 6: Run the 2 test and the 3 test.
Multiples of 7: There are a few tests, but they’re all harder than digging out your phone. This one’s probably the easiest:
Double the units and subtract from the tens. E.g., 1365→136−(2×5)=126→12−(2×6)=0. If the chain ends in 0 or a multiple of 7, then the original number is divisible by 7.
Multiples of 8: Ignore everything from the thousands place up. Divide the remaining three-digit number in half. Then in half again. Then run the multiples-of-2 test.
All (and only) multiples of 9 have digits that add up to 9 or a multiple of 9.
All (and only) multiples of 10 end in 0.
To test divisibility by a larger number, try to factor it down to single-digit numbers, then run the tests above, keeping any repeated factors together. For example, 60 = 2*2*3*5. So all multiples of 60 are also multiples of 2*2, 3, and 5. Note the 2*2; a multiple of 60 must be divisible by 4, not just by 2. (150 is divisible by 2, but not by 4, so it’s not divisible by 60.)
Use these multiplication shortcuts
To multiply in your head, try turning the problem into an easier one. For example:
Doubling numbers tends to be easier. So when multiplying by an even number, first multiply by half that number, then by 2.
Multiply by 5: First multiply by 10, then divide by 2.
Multiply by 9: Multiply by 10 and subtract the number. So 65*9 = (65*10)-65 = 650-65 = 585.
Multiply a single-digit number x by 9: The first digit is x-1. The second digit is 9 minus the first digit. So 8*9=72.
Memorise simple arithmetic
The more basic calculations you’ve memorized, the more you can break down bigger maths problems. If you’ve forgotten your times tables, brush up on them. It feels great to recognise a multiple of 12 and realise you can split up a bigger number.
The difference between two square numbers is the product of their square roots
If you know the square of a number, you can easily find the square of the next number. Say you know that 10*10=100. So 11*11 = 100+10+11 = 121. So 12*12 = 121+11+12 = 144. So 13*13 = 144+12+13 = 169. And so on.
To square a two-digit number, round it first
Say you need to square 46. First round it to the nearest multiple of 10 (by adding 4), then subtract the same amount for a new number, so you have 50 and 42. Then multiply those two numbers, and then add the square of the amount you rounded up by: (in this case 4²). So 46² = (50*42)+4² = 2,100+16 = 2,116.
By the way, when I did this mentally, 50*42 was still a little hard for me, so I turned it into 100*21. Combining mental maths tricks really increases your power.
If you didn’t follow that, here’s a longer explanation that might do the trick.
To roughly convert from Celsius to Fahrenheit, multiply by 2 and add 30. From Fahrenheit to Celsius, subtract 30 and divide by 2. (To more precisely convert C to F, multiply by 1.8 and add 32.)
The order is important: The addition/subtraction is always closer to the Fahrenheit side of the conversion. If you forget the order, you know that 32° F = 0° C, so you can test your formula against that.
Or just memorise that room temperature is about 20–22 °C or 68–72 °F, and normal body temperature is around 36-37° C or 97-99° F, depending on several factors.
Your annual salary is about 2,000 times your hourly rate
For a full-time job, $1/hour = $2,000/year.
Your annual salary is your hourly rate, times the hours you work in a week, times 52 weeks. 40*52 is 2,080, but to calculate it mentally, you can round down to 2,000 for a ballpark figure. Double your hourly rate and add three zeroes. So $25/hour is about $50,000/year. Or do it in reverse: Take three digits off your salary and halve it, and that’s roughly your hourly rate. It’ll be two weeks low, if you get paid for every weekday of the year.
If you want to be a little more precise, take that rough total, and add your hourly rate times 100. That’ll be just two and a half workdays over your 52-week salary.
To be more precise, multiply by 2,080 (40*52): Multiply by 2,000, and set that total aside. Then multiply your hourly rate by 80 (double it, double that, double that, and add a zero). Add that to the rough estimate and you’ve got your 52-week salary.
If you want to factor in your paid holidays or other specifics, go use this working day calendar, where you can tweak the numbers and workdays until you get your actual number of work hours. But I thought you were here for mental maths.
Find more shortcuts
Listverse has some easy mental maths shortcuts. Wikipedia has many advanced shortcuts that cover arithmetic, squares and cubes, roots, and logarithms. And Better Explained lists some common unit conversions.