what is math formula for 1+1=2,2+2=4...up to 240

12 Math Tricks to Aid You lot Solve Problems Without a Calculator

Working it out in your head

Addition + Subtraction

1. Improver

The first pull a fast one on is to simplify your problem past breaking it into smaller pieces. For instance, we can rewrite

          567 + 432            
= 567 + (400 + 30 + 2)
= 967 + 30 + two            
= 997 + 2            
=            999

Switch

It's oftentimes easier to piece of work with adding a smaller number, so instead of 131 + 858, bandy the numbers

          858 + 131            
= 858 + 100 + 30 + i            
=            989

2. Subtraction

Using the complement of a number can assistance brand subtraction easier. The complement is the difference between the original number and a round number — say 100, g.

Here are some examples with the number and its complement compared with 100:

          67:33, 45:55, 89:xi, 3:97

Observe that the 2nd digits add upward to x, and the first digit adds up to nine.

Here is how this is helpful

          721–387            
# the complement of 87 is 13, and so nosotros can swap 387 with 400 – 13
-> 721 — (400 - thirteen)            
= 321 - -13            
= 321 + thirteen
=            334

Another method is to write out the larger number then it ends in 99. With the aforementioned example:

          721 -> (699 + 22)
= 699 – 387 + 22            
= 312 + 22            
=            334

Multiplication

3. Elevens

For a two-digit number, add the digits and put the respond in the middle of the number y'all are multiplying:

          35 x 11            
-> 3_5            
-> three+5 = 8            
-> iii85

If the sum is greater than ten, add the tens digit to the next cavalcade to the left, and write the ones digit in the answer. For example, iv+eight = 12, write down two and comport the 1 into the side by side column.

          48 x eleven            
-> 4_8            
-> 4+8 = 12            
-> 4,12,eight            
->            528

The process is a niggling more complicated for three-digit and greater numbers, simply it works in a similar way. This time go on the showtime and terminal digit and sum the digits in pairs

          725 X 11            
-> 7__5            
-> 7_,(7+2=ix), (ii+5=vii), _5            
->            7975                    51973 x 11            
-> 5__3            
-> 5_,(5+1=vi),(one+9=x), (9+7=16), (seven+3=ten), _3            
# where the sum is greater than x we move the tens digit into the next column            
-> 5,(6+one),(0+one),(6+1),(0),three            
->            571703

4. Nines

Multiplying by nines can be simplified by multiplying past x and subtracting the original number

          799 x 9            
= 799 x (x -one)            
= 7990 – 799            
=            7191

Use the aforementioned method for annihilation ending in 9

          72 x 89            
= 72 x (90–i)            
= (70 x xc) + (2 x 90) — 72            
= 6300 + 180–72            
=            6408

5. How to solve squares

Y'all can re-write a square equation into numbers that are easier to bargain with using this formula

          n^two = (north+d)(north-d) + d^two

where n is the number to exist squared, and d is the divergence

Here's an instance

          57^2            
= (57+3)(57–3) + 3^ii
# we add 3 to 57, every bit sixty is easier to multiply than 57, and subtract three from the 2nd 57
-> 60 x 54 + ix            
= 3000 + 240 + 9            
=            3249

The ultimate example is when you are squaring a number ending in five, so round 1 number up to the nearest 10, the other number down to the nearest 10, and add 25.

          65^2            
= (threescore 10 70) + five^ii            
= 4200 + 25            
=            4225

vi. Shut together method

A similar method works for multiplying numbers that are close together. The formula works for all numbers, just it doesn't simplify well unless the numbers are similar.

Here'southward the formula. n is the "base" number

          (n+a)(n+b) = north(north + a + b) + ab

An example:

          47 x 43            
= (40 + 7)(40 + three)            
= forty 10 (40 + three + 7) + (7 x 3)
= (40 x 50) + (vii ten iii)            
= 2000 + 21            
=            2021

In this example, the ones digits add upwards to ten, and then our "base" number and the multiplier are circular numbers (forty and 50).

Here'south some other example. Reduce the smaller number to reach the nearest circular number — our base of operations number, in this case, 40. Add together the difference to the larger number. Multiple the base and larger number. Finally, add together the production of the difference betwixt the original numbers and the base of operations number.

          47 x 42            
= (40 + vii) x (40 + 2)            
= (twoscore + 7 + two) ten 40 + (vii 10 ii)
= (49 x 40) + (7 x 2)            
= (40 ten forty) + (40 x 9) + (7 x two)
= 1600 + 360 + 14            
=            1974

You tin also round upwards to the base number. As the original numbers are smaller than the base, we add the product of 2 negative numbers.

          47 x 42            
= (50 10 39) + (-3 10 -viii)            
= (l ten thirty) + (50 ten 9) + (-3 x -8)
= 1500 + 450 + 24            
=            1974

This works for three-digit numbers too. In this case, the base of operations number is betwixt our numbers, and so the product is a negative number.

          497 10 504            
= (500 – 3) x (500 + 4)
= (500) x (500 + four - 3) + (-three x 4)
= 500 x 501 - 12
= 250,000 + 500 – 12
=            250,488

Division

7. Simplify Calculations

You lot tin can simplify some equations before y'all even first. For example, divide both the divisor and dividend by two.

          898 / 4            
= 449 / ii            
= 224 and ½

Note with this method, you have to write the remainder as a fraction:

          898/4 has a residuum of 2 — divided by 4
449/two has a remainder of 1 — divided by 2

The fraction is the same, but the absolute number is unlike.

When dividing by v, change the equation by multiplying past 2. It is much easier to divide by 10. For example:

          1753/five            
= 3506 / x
=            350.6

viii. Test for Divisibility

There are many means to quickly whether a number is a factor.

2: The number is even.

          Example 28790 is even, and so it is divisible by ii.

3: The sum of the digits is divisible by 3.

          Example: 1281 -> 1+2+8+1 = 12            
-> 12 is a multiple of 3, so 1281 is divisible past three

4: The terminal ii digits are divisible past iv. Why does this work? 100 is a multiple of 4, so we simply have to bank check the last 2 digits.

          Example: 1472, 72 is divisible by 4, then 1472 is divisible by four.

five: The number ends in 5 or 0.

          Example: 575 ends in 5, so it is divisible past zilch

6: The number is fifty-fifty, and the sum of the digits is divisible by 3. 6 is 3 x 2, so the rules of ii and three utilise.

          Example: 774 is even and 7+7+4 = eighteen            
-> 18 is divisible past three, so 774 is divisible by 6.

7: Add or subtract a multiple of 7 to your number then that it ends in goose egg. Drib the last digit with the zero and repeat the process. Proceed until you tin determine whether the event is divisible by seven.

          Instance: 2702 add together 98 (7 x 14) -> 2800, drib the zeroes
-> 28 is a multiple of seven, so 2702 is divisible by 7.

8: The last iii digits are divisible by 8.

          Example: 79256, 256 is divisible by 8, and so 79256 is divisible by eight.          (Alternate dominion: if the hundreds digit is            even, last            two            digits divisble past eight, if hundreds digit is            odd, last            ii            digits            + iv            divisible by eight)

nine: The same rule as 3, but with 9. If the sum of the digits is divisible past 9, and then the number is divisible by ix.

          Example: 13671 -> 1+3+6+7+1 = 18            
-> xviii is divisible by 9, so 13671 is divisible past 9

10: The number ends in 0.

          Example: 280 ends in 0, 280 is divisible by x

11: Similar rule to 3 and 9, start at the correct digit and alternate subtracting and adding the remaining digits. If the answer is zilch, or a multiple of 11, then the number is divisible past 11.

          Example: 12727 -> 1 - ii + 7 - 2 + 7 = xi, and then 12727 is divisible by 11.

You lot can check out some additional methods here.

9. Dividing big numbers past 9

          Example:
-> 10520/9

Write the first digit above the equation and write an "R" (for remainder) higher up the last digit. Add the number you just wrote and the number diagonally below and to the right of it. Write this new number in the second spot. Add together that number to the number diagonally below and to the right. Continue this process until you attain the R.

Sum the numbers with the same color to create the next digit

Finally, add the last digit to the number below the R to get your residuum.

          10520/9
=            1168 R8            
or 1168.889

Hither'southward some other case:

          -> 57423/9

This fourth dimension subsequently nosotros've completed the first pace, the sum of our first number and the number diagonally below and to the right is larger than ten (five+seven =12). We put a one in a higher place the first digit and subtract nine from information technology. (We are dividing past a base of operations of nine, so nosotros subtract nine rather than ten). Identify the resulting number in the 2d position (12–9 = 3). Continue with the same process.

In this example, our remainder is larger than 9 (9+3 = 12). Once again, nosotros behave a one above the previous digit and decrease 9 from the residual, leaving us with 3. At present add together the issue and the bear digits.

          57423 / 9            
=            6380 R3
            or 6380.333

Percentages and Fractions

x. Reverse the question

Percentages are associative, then sometimes reversing the question's guild makes it easier to calculate.

          Case:            
What'due south 36% of 25            
-> is the same as 25% of 36            
-> 25% is ¼            
-> 36/four = nine            
36% of 25 is            9

11. Fractions

As y'all tin see with the utilize of ¼ in the last example, it helps know fractions and how they relate to percentages.

          1/2 = 50%          one/iii = 33.33%, two/3 = 66.67%,          1/4 = 25%, 3/4= 75%          1/5 = 20%, 2/5 = twoscore% …          one/6 = 16.67%, 5/half-dozen = 83.33% (two/6 = 1/3, iii/6 = ane/2, four/6 = 2/three)          1/7 = 14.2857%, 2/seven = 28.5714%, 3/7 = 42.8571%, iv/seven = 57.1428%  (annotation the recurring .142857 pattern)          1/8 = 12.five%, 3/viii = 37.5%, 5/8 = 62.5%, 7/8 = 87.five%          i/9 = eleven.11%, 2/ix = 22.22%, 3/nine = 33.33% …          1/10 = 10%, 2/x = 20% …          1/11 = 9.09%, two/11 = 18.18%, 3/eleven = 27.27% …          one/12 = 8.33%, 5/12 = 41.67%, 7/12 = 58.33%, 11/12 = 91.67%

12. Rule of 72

The rule of 72 provides an estimate of how many years information technology volition take an investment to double in value at a given percentage render. Information technology works by dividing 72 by the percentage, with the reply as the number of years it will have to double.

          2% -> 72/two = 36, approximately 36 years to double
8% -> 72/8 = 9, approximately ix years to double

Annotation that the rule of 72 is a guideline based on the natural log of 2 — which gives 0.693. So a dominion of 69.3 would be more accurate, but 72 is easier to calculate.

There is as well a rule of 114 for tripling an investment and a rule of 144 for quadrupling your money.

Additional Resources

I found two books by Arthur Benjamin to be helpful on this topic. Many of the examples in this blog were inspired past these books. You tin check them out here.

https://www.amazon.com/Secrets-Mental-Math-Mathemagicians-Adding/dp/0307338401/

Please leave a comment if you found this helpful, or share any other useful tricks yous have come across.

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Source: https://towardsdatascience.com/12-math-tricks-to-help-you-solve-problems-without-a-calculator-704fdd663286