", I gathered a huge amount of information. The book tells about dozens of tricks that simplify the usual mathematical operations. It turned out that multiplication and division in a column is the last century, and it is not clear why this is still taught in schools.
Multiplying three-digit numbers by single-digit numbers is a very simple operation. All you have to do is break the big task down into smaller ones.
Example: 320×7
Squaring two-digit numbers is not much more difficult. You need to split the number by two and get an approximate answer.
Example: 41^2
The key rule here is to turn the desired number into a pair of other numbers that are much easier to multiply. For example, for the number 41 these are the numbers 42 and 40, for the number 77 - 84 and 70. That is, we subtract and add the same number.
With squares of numbers ending in 5, there is no need to strain at all. All you have to do is multiply the first digit by the number that is one more and add 25 to the end of the number.
Example: 75^2
Division in the mind is quite a useful skill. Think about how often we divide numbers each day. For example, a bill in a restaurant.
Example: 675: 8
We do not get the most accurate answer (the correct answer is 84.375), but you will agree that even such an answer will be more than enough.
To quickly find out 15% of any number, you must first calculate 10% of it (moving the comma one character to the left), then divide the resulting number by 2 and add it to 10%.
Example: 15% off 650
Perhaps we all stumbled upon this trick:
Think of any number. Multiply it by 2. Add 12. Divide the sum by 2. Subtract the original number from it.
You got 6, right? No matter what you think, you will still get 6. And here's why:
This trick is based on the elementary rules of algebra. Therefore, if you ever hear that someone is guessing it, put on your most haughty grin, make a contemptuous look and tell everyone the solution. :)
This trick has been around for centuries.
Write down any three-digit number whose digits are in decreasing order (for example, 765 or 974). Now write it in reverse order and subtract it from the original number. Add it to the answer you received, only in reverse order.
Whatever number you choose, the result will be 1,089.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 1 | 8 | 27 | 64 | 125 | 216 | 343 | 512 | 729 | 1 000 |
»
Once you remember these values, finding the cube root of any number will be elementary easy.
Example: cube root of 19,683
Note: the trick only works when the original number is a cube the whole numbers.
To find the number of years it takes to double your money, divide 70 by the annual interest rate.
Example: the number of years it takes to double money at an annual interest rate of 20%.
70:20 = 3.5 years
To find the number of years it takes for money to triple, divide 110 by the annual interest rate.
Example: the number of years it takes to triple money at an annual interest rate of 12%.
110:12 = 9 years old
Mathematics is a magical science. I'm even a little embarrassed that such simple tricks could surprise me, and I can't even imagine how many more mathematical tricks you can learn.
"Pure mathematics is in its way the poetry of the logical idea".
Albert Einstein
Perhaps, in the era of loans and installments, the most relevant mathematical skill can be called a virtuoso mental calculation of interest. The fastest way to calculate a certain percentage of a number is to multiply the given percentage by this number and then discard the last two digits in the resulting result, because the percentage is nothing but one hundredth.
How much is 20% of 70? 70 × 20 = 1400. We discard two digits and get 14. When you rearrange the factors, the product does not change, and if you try to calculate 70% of 20, then the answer will also be 14.
This method is very simple in the case of round numbers, but what if you need to calculate, for example, a percentage of the number 72 or 29? In such a situation, you will have to sacrifice accuracy for the sake of speed and round the number (in our example, 72 is rounded up to 70, and 29 to 30), and then use the same trick with multiplying and discarding the last two digits.
Can 408 candies be divided equally between 12 children? It is easy to answer this question without the help of a calculator, if we recall the simple signs of divisibility that we were taught back in school.
A number is divisible by 2 if its last digit is divisible by 2.
A number is divisible by 3 if the sum of the digits that make up the number is divisible by 3. For example, take the number 501, represent it as 5 + 0 + 1 = 6. 6 is divisible by 3, which means that the number 501 itself is divisible by 3 .
A number is divisible by 4 if the number formed by its last two digits is divisible by 4. For example, take 2340. The last two digits form the number 40, which is divisible by 4.
A number is divisible by 5 if its last digit is 0 or 5.
A number is divisible by 6 if it is divisible by 2 and 3.
A number is divisible by 9 if the sum of the digits that make up the number is divisible by 9. For example, let's take the number 6,390 and represent it as 6 + 3 + 9 + 0 = 18. 18 is divisible by 9, which means the number 6 itself 390 is divisible by 9.
A number is divisible by 12 if it is divisible by 3 and 4.
The square root of 4 is 2. Anyone can count that. What about the square root of 85?
For a quick approximate solution, we find the nearest square number to the given one, in this case it is 81 = 9^2.
Now find the next nearest square. In this case it is 100 = 10^2.
The square root of 85 is somewhere between 9 and 10, and since 85 is closer to 81 than it is to 100, the square root of that number is 9 something.
Do you want to quickly find out the time it will take for your cash deposit at a certain interest rate to double? There is also no need for a calculator, it is enough to know the “rule of 72”.
We divide the number 72 by our interest rate, after which we get the approximate period after which the deposit will double.
If the deposit is made at 5% per annum, then it will take 14-odd years for it to double.
Why exactly 72 (sometimes they take 70 or 69)? How it works? These questions will be answered in detail by Wikipedia.
In this case, the interest rate on the deposit should become a divisor of 115.
If the deposit is made at 5% per annum, then it will take 23 years for it to triple.
Imagine that you are interviewing with two employers who do not state salaries in the usual “rubles per month” format, but talk about annual salaries and hourly pay. How to quickly calculate where they pay more? Where the annual salary is 360,000 rubles, or where they pay 200 rubles per hour?
To calculate the payment for one hour of work when voicing the annual salary, it is necessary to discard the last three characters from the named amount, and then divide the resulting number by 2.
360,000 turns into 360 ÷ 2 = 180 rubles per hour. Other things being equal, it turns out that the second proposal is better.
Your fingers are capable of much more than simple addition and subtraction.
With your fingers, you can easily multiply by 9 if you suddenly forgot the multiplication table.
Let's number the fingers on the hands from left to right from 1 to 10.
If we want to multiply 9 by 5, then we bend the fifth finger from the left.
Now let's look at the hands. It turns out four unbent fingers to bent. They represent tens. And five unbent fingers after the bent one. They represent units. Answer: 45.
If we want to multiply 9 by 6, then we bend the sixth finger from the left. We get five unbent fingers before the bent finger and four after. Answer: 54.
Thus, you can reproduce the entire column of multiplication by 9.
There is an extremely easy way to lightning-fast multiply even large numbers by 4. To do this, it is enough to decompose the operation into two steps, multiplying the desired number by 2, and then again by 2.
See for yourself. Not everyone can multiply 1,223 immediately by 4 in their minds. And now we do 1223 × 2 = 2446 and then 2446 × 2 = 4892. This is much easier.
Imagine that you are taking a series of five tests, for which you need a minimum score of 92 to pass. The last test remains, and the results for the previous ones are: 81, 98, 90, 93. How to calculate the required minimum that you need to get in the last test?
To do this, we consider how many points we missed / went over in the tests already passed, denoting the shortage with negative numbers, and the results with a margin - positive.
So, 81 − 92 = −11; 98 - 92 = 6; 90 - 92 = -2; 93 - 92 = 1.
Adding these numbers, we get the adjustment for the required minimum: -11 + 6 - 2 + 1 = -6.
It turns out a deficit of 6 points, which means that the required minimum increases: 92 + 6 = 98. Things are bad. :(
The approximate value of an ordinary fraction can be very quickly represented as a decimal fraction, if you first bring it to simple and understandable ratios: 1/4, 1/3, 1/2 and 3/4.
For example, we have a fraction 28/77, which is very close to 28/84 = 1/3, but since we increased the denominator, the original number will be slightly larger, that is, slightly more than 0.33.
You can play a bit of David Blaine and surprise your friends with an interesting but very simple math trick.
The answer will always be 3.
Yes, very stupid, but often the effect exceeds all expectations.
And, of course, we could not help but insert into this post that same picture with a very cool way of multiplying.
For many people, mathematics can be terrifying. If you are one of them and you don't care about math, it's not your fault. We were simply not taught at school mathematical tricks with which any calculations become elementary.
The proposed list will probably improve your general knowledge of mathematical tricks and speed up the performance of mathematical calculations in your mind.
1. Multiply by 11
We all know that when multiplying by 10, 0 is added to the number, but did you know that there is an equally simple way to multiply a two-digit number by 11? Here he is:
Take the original number and imagine the gap between two digits (in this example, we use the number 52):
5_2
Now add the two numbers and write them in the middle:
5_(5+2)_2
So your answer is: 572.
If adding the numbers in brackets results in a two-digit number, just remember the second digit, and add one to the first number:
9_(9+9)_9
(9+1)_8_9
10_8_9
1089 - this always works.
2. Fast squaring
This technique will help you quickly square a two-digit number that ends in 5. Multiply the first digit by itself +1, and add 25 at the end. That's it!
252 = (2×(2+1)) & 25
2 x 3 = 6
625
3. Multiply by 5
It's very easy for most people to memorize the multiplication table for 5, but when you have to deal with large numbers, it becomes more difficult to do this. Or not? This trick is incredibly simple.
Take any number, divide by 2 (in other words, divide in half). If the result is an integer, add 0 at the end. If not, ignore the comma and add 5 at the end. This always works:
2682 × 5 = (2682 / 2) & 5 or 0
2682 / 2 = 1341 (whole number so add 0)
13410
Let's try another example:
5887×5
2943.5 (fractional, omit comma, add 5)
29435
4. Multiply by 9
It's simple. To multiply any number between 1 and 9 by 9, look at the hands. Bend the finger that corresponds to the multiplied number (for example 9x3 - bend the third finger), count the fingers to the bent finger (in the case of 9x3 it is 2), then count after the bent finger (in our case 7). The answer is 27.
5. Multiply by 4
This is a very simple technique, although it is obvious only to some. The trick is to simply multiply by 2 and then multiply by 2 again:
58 × 4 = (58 × 2) + (58 × 2) = (116) + (116) = 232
6. Counting tips
If you need to leave a 15% tip, there is an easy way to do so. Calculate 10% (divide the number by 10), and then add the resulting number to half of it and get the answer:
15% of $25 = (10% of 25) + ((10% of 25) / 2)
$2.50 + $1.25 = $3.75
7. Complex multiplication
If you need to multiply large numbers and one of them is even, you can simply rearrange them to get the answer:
32 × 125 is the same as:
16 × 250 is the same as:
8 × 500 is the same as:
4 x 1000 = 4,000
8. Divide by 5
In fact, dividing large numbers by 5 is very simple. All you need to do is simply multiply by 2 and move the decimal point: 195 / 5
Step1: 195 * 2 = 390
Step2: Move the comma: 39.0 or just 39.
2978 / 5
Step1: 2978 * 2 = 5956
Step2: 595.6
9. Subtraction from 1000
To subtract from 1000, you can use this simple rule: Subtract all but the last digit from 9. And subtract the last digit from 10: 1000
-648
Step 1: subtract 6 from 9 = 3
Step 2: subtract 4 from 9 = 5
Step 3: subtract 8 from 10 = 2
Answer: 352
10. Systematized multiplication rules
Multiply by 5: Multiply by 10 and divide by 2.
Multiply by 6: Sometimes it's easier to multiply by 3 and then by 2.
Multiply by 9: Multiply by 10 and subtract the original number.
Multiply by 12: Multiply by 10 and add the original number twice.
Multiply by 13: Multiply by 3 and add the original number 10 times.
Multiply by 14: Multiply by 7 and then by 2.
Multiply by 15: Multiply by 10 and add the original number 5 times as in the previous example.
Multiply by 16: Multiply by 2 4 times if you like. Or multiply by 8 and then by 2.
Multiply by 17: Multiply by 7 and add the original number 10 times.
Multiply by 18: Multiply by 20 and subtract the original number twice.
Multiply by 19: Multiply by 20 and subtract the original number.
Multiply by 24: Multiply by 8 and then by 3.
Multiply by 27: Multiply by 30 and subtract the original number 3 times.
Multiply by 45: Multiply by 50 and subtract 5 times the original number.
Multiply by 90: Multiply by 9 and add 0.
Multiply by 98: Multiply by 100 and subtract the original number twice.
Multiply by 99: Multiply by 100 and subtract the original number.
Bonus: Interest
Calculate 7% of 300. Sounds complicated?
Percentage: First you need to understand the meaning of the word "Percent" (Percent). The first part of the word is PRO (PER), like 10 points per listverse page. PER = FOR EACH . The second part is CENT (CENT), as 100. For example, one hundred years = 100 years. 100 CENTS in 1 dollar and so on. So, PERCENT = FOR EVERY HUNDRED.
So, it turns out that 7% of 100 will be 7. (7 for every hundred, only one hundred).
8% of 100 = 8.
35.73% of 100 = 35.73
But how can this be useful?
Let's return to the problem 7% of 300. 7% of
the first hundred is 7. 7%, the second hundred is the same 7, and 7% of the third hundred is the same 7. So, 7 + 7 + 7 = 21. If 8% of 100 = 8, then 8% of 50 = 4 (half of 8).
Split each number if you need to calculate percentages of 100, if the number is less than 100, just move the comma to the left.
EXAMPLES:
8%200 = ? 8 + 8 = 16.
8%250 = ? 8 + 8 + 4 = 20,
8%25 = 2.0 (Move decimal point to the left).
15%300 = 15+15+15 =45,
15%350 = 15+15+15+7,5 = 52,5
It's also good to know that you can always swap the numbers: 3% of 100 is the same as 100% of 3. 35% of 8 is the same as 8% of 35.
How are you doing with math? Did you like solving problems in school, or did you have problems with addition and subtraction? How confident are you in your knowledge and are you ready to upgrade your skills?
Math for Adults is the friendliest and most accessible math textbook. The book will help you understand basic terms and calculations, teach you how to apply them in life, and also talk about mathematical tricks that you can surprise your friends with. Let's focus on the tricks.
The multiplication table for the number 9 is one of the most difficult, but these days, almost every schoolchild is familiar with an elegant way of remembering.
Raise your palms in front of you and imagine that the fingers are numbered 1 to 10 from left to right. Bend the finger corresponding to the number you want to multiply by 9. Count how many fingers are to the left and right of the bent finger. This will be the answer.
Remember the next step. Suppose you need to calculate how much 6000 ÷ 200 will be. The task can be greatly simplified by removing the same number of zeros from the end of each number. That is, 6000 ÷ 200 can be simplified to 60 ÷ 2, which equals 30. It's easier that way!
Percentages repel many, but it is enough to figure it out once and everything falls into place. Take a look at the image:
The shop owner clearly wanted to lower the price back to £20, so what went wrong? When you take a certain price and perform more than one action on it, remember that 100% is the original price and all percentage calculations must be based on it. The saleswoman raised the price by 40% and the new price was 140% of the original price (20 x 140% = £28).
When the saleswoman reduced the price by 40%, she had to take 40% of the original price and subtract that amount from the new price. Then the price would go back to 100%. The mistake was that the saleswoman took the new price as 100% and took 40% of it.
Did you know that the meter was originally defined as 1/10,000,000 of the distance from the Equator to the North Pole along a line through Paris. Thus, the distance from the equator to the North Pole is 10,000 km, and the circumference of the equator is approximately 40,000 km. In reality, the Earth is not perfectly round, and the length of the equator is approximately 40,075 km.
How much is 3000 × 900? It's simple: we multiply the numbers in front (3 × 9 = 27), and then add the number of zeros at the end of both numbers and attribute them to the end of the answer. Since there are five zeros here, we get 2,700,000.
But when calculating how much 7500 × 80 will be, you need to be a little more careful. First we multiply 75 × 8 = 600. Now we add three more zeros, according to the number of zeros in both initial numbers. Answer: 600,000.
To multiply 1030 by 50, first we take 103 × 5 = 515. Then we add two zeros and get 51500. The zero between 1 and 3 in the number 1030 does not need to be taken into account, it has already played its role when multiplying 103 by 5.
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