Example.
Find the product of algebraic fractions and .
Solution.
Before multiplying fractions, we factorize the polynomial in the numerator of the first fraction and the denominator of the second. The corresponding abbreviated multiplication formulas will help us with this: x 2 +2·x+1=(x+1) 2 and x 2 −1=(x−1)·(x+1) . Thus, .
Obviously, the resulting fraction can be reduced 
(we discussed this process in the article reducing algebraic fractions).
All that remains is to write the result in the form algebraic fraction, for which you need to multiply the monomial by the polynomial in the denominator: 
.
Usually the solution is written without explanation as a sequence of equalities: 
Answer:
.
Sometimes with algebraic fractions that need to be multiplied or divided, you need to perform some transformations to make the operation easier and faster.
Example.
Divide an algebraic fraction by a fraction.
Solution.
Let's simplify the form of an algebraic fraction by getting rid of the fractional coefficient. To do this, we multiply its numerator and denominator by 7, which allows us to make the main property of an algebraic fraction, we have 
.
Now it has become clear that the denominator of the resulting fraction and the denominator of the fraction by which we need to divide are opposite expressions. Let's change the signs of the numerator and denominator of the fraction, we have 
.
Counting in your head, according to many of us, is no longer relevant in our time. There is a calculator in every smartphone, and even more so on a computer and laptop. However, you can’t constantly reach into a calculator before every action, step or sneeze, but you need to count constantly and a lot. - a very necessary skill even in our high-tech age of gadgets and electronic computing systems. A simple example illustrating these theoretical calculations is the behavior of buyers and sellers in a store: you need to act quickly, because there is a long line behind you, and if you do not know how to count in your head, the seller may shortchange you - by mistake or intentionally. Children most often make their first independent “forays” into the store, so mental counting will be very useful for them.
is not an innate skill in humans, and very young children do not yet have an idea of numbers, quantity, or actions with groups of objects (adding one group to another, subtracting, etc.). The primitive peoples of Asia, Africa and America also have undeveloped ideas about numbers and arithmetic operations: most often their number system consists of the concepts of “one”, “two” and “many”; Some tribes can count to five, some to seven, but then they all follow the constant “many.” From this we can conclude that counting in general is a rather complex function for human consciousness.
So how can you teach your child the first manipulations with numbers? Before mastering the ability to operate with abstract numbers, children must understand counting through visual examples. First, you need to tell your child about numbers, at least up to the first ten, and count with him various items that can be seen around: birds in the trees, flowers in the garden, people on the street, cars in the parking lot, and so on. Gradually the baby will understand “ appearance» specific quantities – be it one, five or ten items. With undeveloped abstract thinking Young children have a very developed visual memory; they quickly remember shapes and colors. You can practice counting with him, showing bright pictures.
The main thing is to understand that Small child perceives everything as a game. And learning to count must also be submitted to game form so that he would be interested. With the right approach, the baby will grasp information very quickly, since at this age his brain absorbs everything new very actively. You can’t sit him down at the table and give him a long boring “lecture” about arithmetic operations - the child will only lose interest in learning. You need to count with him in different places and situations, during walks, games and other joint activities. You can offer to cook something tasty together, and the child can help determine, for example, how many eggs are needed to knead the dough.
After ideas about quantity are more or less formed, the game can be complicated. Teach your child the first arithmetic operations - addition and subtraction. For example, take a toy house (an ordinary large box can act as it) and figures of people or animals (you can use ordinary cubes, which we will call, for example, “gnomes”). Place one little man in the house and ask the child how many little men live in the house. He must answer that he is alone. Then put another figurine in the house and ask how many people there are. Let the child think and say the correct answer. At first it will take him a few minutes to do this, he will make mistakes; You shouldn’t rush him or scold him. When he says the correct answer, he must open the house and make sure that there are exactly two people. The abstract model that the child reproduced from memory was confirmed by a clear example. Add and subtract little people from the total number of “inhabitants” of the house, which will strengthen and develop your child’s mental counting skills.
How to teach your child to multiply and divide
If and are fairly easy procedures, then it is much more difficult for a child to understand. Division is even more difficult to master. Parents will also be here to help illustrative examples, toys and figures.
You need to prepare identical boxes and sets of figures. In the simplest case, the figures will be pebbles, cubes, lids from plastic bottles- you can find anything. Each box must contain an equal number of figures. Invite your child to fill one box by putting figures in it. Let him count how many items are in the box. And after that, let him fill the second box, make sure that there are the same number of objects in it, and count the total number of figures in both boxes. At first, one box should contain only a few items - two, three. In this way, you can lead your child to the idea that two times three equals six, two times two equals four, and so on. There is no need to enlarge the boxes and figures to infinity: at this stage it is important that the child understands the specific, material meaning of multiplication as the sum of several identical groups of objects. The next stage is memorizing the multiplication tables. You need to learn it by heart, like a poem. More precisely, a group of poems. The “lines” in them are examples: twice three is six, twice four is eight... You can learn only one “poem” at a time - multiplication by two, three, four, and so on. Multiplication by five also resembles a poem in appearance - its “lines” rhyme with each other, so it is the easiest to remember.
- the most difficult action for a baby, even in primary school start later than other sections of arithmetic. Division is the inverse procedure of multiplication, so to master it, the child must already know the multiplication table. However, at first, the same visual examples will do, and in this sense, division is the action that is closest and most relevant to the baby. How to divide candy among everyone so that everyone has an equal amount? After all, if someone has less than others, he will be offended. It is necessary to distribute fairly, and first this can be done by selection: first, distribute one candy, then another one... Total An adult must pick up the candy so that it is truly shared among all the children without leaving a trace. Subsequently, you can explain to the child that not all numbers can be divided by each other. In this case, division is more difficult than multiplication - after all, absolutely all numbers can be multiplied. If possible, the children are also introduced to division with a remainder: the remaining candies that cannot be distributed equally to everyone are taken by an adult (or they will go to the most obedient of the children).
How can you help a child
Performing arithmetic operations can be simplified for a child if you tell him about the properties of numbers from 2 to 10. For example, 4 is two times two; 5 can be obtained different ways– add 3 to 3 or 1 to 4. Particular attention should be paid to the number 0. To simplify counting, you need to understand round numbers: 30 is three times 10, and 5 is half of 10.
Formulas for more complex treatments
As your child gets older and already masters basic arithmetic, you can introduce him to formulas for quick addition and multiplication large numbers. There are many such formulas, and here we will give only a few.
It is enough to simply multiply two-digit numbers by 11. For example, 23*11. You just need to add up the numbers of the first factor and write down this factor in the answer, in the middle of which enter the resulting amount: 2+3=5, therefore, 23*11=253. If, when adding the digits, a two-digit number is obtained, then the first digit of this number is added to the first digit of the multiplier. For example, 38*11. 3+8=11; we add the first one to the three, and write the second in the middle of the answer: 38*11=418.
Addition of large numbers can be simplified by increasing one addend by some number, which is then subtracted from the answer. For example: 358+340=(358+2)+340-2= 360+340-2=700-2=698.
Such formulas will certainly be of interest to many adults, because they will significantly simplify the work process, counting money and other essential operations with numbers.
It is obvious that numbers with powers can be added like other quantities , by adding them one after another with their signs.
So, the sum of a 3 and b 2 is a 3 + b 2.
The sum of a 3 - b n and h 5 -d 4 is a 3 - b n + h 5 - d 4.
Odds equal powers of identical variables can be added or subtracted.
So, the sum of 2a 2 and 3a 2 is equal to 5a 2.
It is also obvious that if you take two squares a, or three squares a, or five squares a.
But degrees various variables And various degrees identical variables, must be composed by adding them with their signs.
So, the sum of a 2 and a 3 is the sum of a 2 + a 3.
It is obvious that the square of a, and the cube of a, is not equal to twice the square of a, but to twice the cube of a.
The sum of a 3 b n and 3a 5 b 6 is a 3 b n + 3a 5 b 6.
Subtraction powers are carried out in the same way as addition, except that the signs of the subtrahends must be changed accordingly.
Or:
2a 4 - (-6a 4) = 8a 4
3h 2 b 6 - 4h 2 b 6 = -h 2 b 6
5(a - h) 6 - 2(a - h) 6 = 3(a - h) 6
Multiplying powers
Numbers with powers can be multiplied, like other quantities, by writing them one after the other, with or without a multiplication sign between them.
Thus, the result of multiplying a 3 by b 2 is a 3 b 2 or aaabb.
Or:
x -3 ⋅ a m = a m x -3
3a 6 y 2 ⋅ (-2x) = -6a 6 xy 2
a 2 b 3 y 2 ⋅ a 3 b 2 y = a 2 b 3 y 2 a 3 b 2 y
The result in the last example can be ordered by adding identical variables.
The expression will take the form: a 5 b 5 y 3.
By comparing several numbers (variables) with powers, we can see that if any two of them are multiplied, then the result is a number (variable) with a power equal to amount degrees of terms.
So, a 2 .a 3 = aa.aaa = aaaaa = a 5 .
Here 5 is the power of the result of the multiplication, equal to 2 + 3, the sum of the powers of the terms.
So, a n .a m = a m+n .
For a n , a is taken as a factor as many times as the power of n;
And a m is taken as a factor as many times as the degree m is equal to;
That's why, powers with the same bases can be multiplied by adding the exponents of the powers.
So, a 2 .a 6 = a 2+6 = a 8 . And x 3 .x 2 .x = x 3+2+1 = x 6 .
Or:
4a n ⋅ 2a n = 8a 2n
b 2 y 3 ⋅ b 4 y = b 6 y 4
(b + h - y) n ⋅ (b + h - y) = (b + h - y) n+1
Multiply (x 3 + x 2 y + xy 2 + y 3) ⋅ (x - y).
Answer: x 4 - y 4.
Multiply (x 3 + x - 5) ⋅ (2x 3 + x + 1).
This rule is also true for numbers whose exponents are negative.
1. So, a -2 .a -3 = a -5 . This can be written as (1/aa).(1/aaa) = 1/aaaaa.
2. y -n .y -m = y -n-m .
3. a -n .a m = a m-n .
If a + b are multiplied by a - b, the result will be a 2 - b 2: that is
The result of multiplying the sum or difference of two numbers equal to the sum or the difference of their squares.
If you multiply the sum and difference of two numbers raised to square, the result will be equal to the sum or difference of these numbers in fourth degrees.
So, (a - y).(a + y) = a 2 - y 2.
(a 2 - y 2)⋅(a 2 + y 2) = a 4 - y 4.
(a 4 - y 4)⋅(a 4 + y 4) = a 8 - y 8.
Division of degrees
Numbers with powers can be divided like other numbers, by subtracting from the dividend, or by placing them in fraction form.
Thus, a 3 b 2 divided by b 2 is equal to a 3.
Or:
$\frac(9a^3y^4)(-3a^3) = -3y^4$
$\frac(a^2b + 3a^2)(a^2) = \frac(a^2(b+3))(a^2) = b + 3$
$\frac(d\cdot (a - h + y)^3)((a - h + y)^3) = d$
Writing a 5 divided by a 3 looks like $\frac(a^5)(a^3)$. But this is equal to a 2 . In a series of numbers
a +4 , a +3 , a +2 , a +1 , a 0 , a -1 , a -2 , a -3 , a -4 .
any number can be divided by another, and the exponent will be equal to difference indicators of divisible numbers.
When dividing degrees with the same basis their indicators are subtracted..
So, y 3:y 2 = y 3-2 = y 1. That is, $\frac(yyy)(yy) = y$.
And a n+1:a = a n+1-1 = a n . That is, $\frac(aa^n)(a) = a^n$.
Or:
y 2m: y m = y m
8a n+m: 4a m = 2a n
12(b + y) n: 3(b + y) 3 = 4(b +y) n-3
The rule is also true for numbers with negative values of degrees.
The result of dividing a -5 by a -3 is a -2.
Also, $\frac(1)(aaaaa) : \frac(1)(aaa) = \frac(1)(aaaaa).\frac(aaa)(1) = \frac(aaa)(aaaaa) = \frac (1)(aa)$.
h 2:h -1 = h 2+1 = h 3 or $h^2:\frac(1)(h) = h^2.\frac(h)(1) = h^3$
It is necessary to master multiplication and division of powers very well, since such operations are very widely used in algebra.
Examples of solving examples with fractions containing numbers with powers
1. Reduce the exponents by $\frac(5a^4)(3a^2)$ Answer: $\frac(5a^2)(3)$.
2. Decrease the exponents by $\frac(6x^6)(3x^5)$. Answer: $\frac(2x)(1)$ or 2x.
3. Reduce the exponents a 2 /a 3 and a -3 /a -4 and bring to a common denominator.
a 2 .a -4 is a -2 the first numerator.
a 3 .a -3 is a 0 = 1, the second numerator.
a 3 .a -4 is a -1 , the common numerator.
After simplification: a -2 /a -1 and 1/a -1 .
4. Reduce the exponents 2a 4 /5a 3 and 2 /a 4 and bring to a common denominator.
Answer: 2a 3 /5a 7 and 5a 5 /5a 7 or 2a 3 /5a 2 and 5/5a 2.
5. Multiply (a 3 + b)/b 4 by (a - b)/3.
6. Multiply (a 5 + 1)/x 2 by (b 2 - 1)/(x + a).
7. Multiply b 4 /a -2 by h -3 /x and a n /y -3 .
8. Divide a 4 /y 3 by a 3 /y 2 . Answer: a/y.
9. Divide (h 3 - 1)/d 4 by (d n + 1)/h.
Division is one of the four basic mathematical operations (addition, subtraction, multiplication). Division, like other operations, is important not only in mathematics, but also in Everyday life. For example, you as a whole class (25 people) donate money and buy a gift for the teacher, but you don’t spend it all, there will be change left over. So you will need to divide the change among everyone. The division operation comes into play to help you solve this problem.
Division is an interesting operation, as we will see in this article!
Dividing numbers
So, a little theory, and then practice! What is division? Division is breaking something into equal parts. That is, it could be a bag of sweets that needs to be divided into equal parts. For example, there are 9 candies in a bag, and the person who wants to receive them is three. Then you need to divide these 9 candies among three people.
It is written like this: 9:3, the answer will be the number 3. That is, dividing the number 9 by the number 3 shows the number of three numbers contained in the number 9. The reverse action, a check, will be multiplication. 3*3=9. Right? Absolutely.
So let's look at example 12:6. First, let's name each component of the example. 12 – dividend, that is. a number that can be divided into parts. 6 is a divisor, this is the number of parts into which the dividend is divided. And the result will be a number called “quotient”.
Let's divide 12 by 6, the answer will be the number 2. You can check the solution by multiplying: 2*6=12. It turns out that the number 6 is contained 2 times in the number 12.
Division with remainder
What is division with a remainder? This is the same division, only the result is not an even number, as shown above.
For example, let's divide 17 by 5. Since the largest number divisible by 5 to 17 is 15, then the answer will be 3 and the remainder is 2, and is written like this: 17:5 = 3(2).
For example, 22:7. In the same way, we determine the maximum number divisible by 7 to 22. This number is 21. The answer then will be: 3 and the remainder 1. And it is written: 22:7 = 3 (1).
Division by 3 and 9
A special case of division would be division by the number 3 and the number 9. If you want to find out whether a number is divisible by 3 or 9 without a remainder, then you will need:
Find the sum of the digits of the dividend.
Divide by 3 or 9 (depending on what you need).
If the answer is obtained without a remainder, then the number will be divided without a remainder.
For example, the number 18. The sum of the digits is 1+8 = 9. The sum of the digits is divisible by both 3 and 9. The number 18:9=2, 18:3=6. Divided without remainder.
For example, the number 63. The sum of the digits is 6+3 = 9. Divisible by both 9 and 3. 63:9 = 7, and 63:3 = 21. Such operations are carried out with any number to find out whether it is divisible with the remainder by 3 or 9, or not.
Multiplication and division
Multiplication and division are opposite operations. Multiplication can be used as a test for division, and division can be used as a test for multiplication. You can learn more about multiplication and master the operation in our article about multiplication. Which describes multiplication in detail and how to do it correctly. There you will also find the multiplication table and examples for training.
Here is an example of checking division and multiplication. Let's say the example is 6*4. Answer: 24. Then let's check the answer by division: 24:4=6, 24:6=4. It was decided correctly. In this case, the check is performed by dividing the answer by one of the factors.
Or an example is given for the division 56:8. Answer: 7. Then the test will be 8*7=56. Right? Yes. In this case, the test is performed by multiplying the answer by the divisor.
Division 3rd grade
In third grade they are just starting to go through division. Therefore, third graders solve the simplest problems:
Problem 1. A factory worker was given the task of putting 56 cakes into 8 packages. How many cakes should be put in each package to make the same amount in each?
Problem 2. On New Year's Eve at school, children in a class of 15 students were given 75 candies. How many candies should each child receive?
Problem 3. Roma, Sasha and Misha collected 27 apples from the apple tree. How many apples will each person get if they need to be divided equally?
Problem 4. Four friends bought 58 cookies. But then they realized that they could not divide them equally. How many additional cookies do the kids need to buy so that each gets 15?
Division 4th grade
The division in the fourth grade is more serious than in the third. All calculations are carried out using the column division method, and the numbers involved in the division are not small. What is long division? You can find the answer below:
Column division
What is long division? This is a method that allows you to find the answer to dividing large numbers. If prime numbers like 16 and 4, can be divided, and the answer is clear - 4. That 512:8 in the mind is not easy for a child. And it’s our task to talk about the technique for solving such examples.
Let's look at an example, 512:8.
1 step. Let's write the dividend and divisor as follows:
The quotient will ultimately be written under the divisor, and the calculations under the dividend.
Step 2. We start dividing from left to right. First we take the number 5: 
Step 3. The number 5 is less than the number 8, which means it will not be possible to divide. Therefore, we take another digit of the dividend:
Now 51 is greater than 8. This is an incomplete quotient.
Step 4. We put a dot under the divisor.
Step 5. After 51 there is another number 2, which means there will be one more number in the answer, that is. quotient is a two-digit number. Let's put the second point:
Step 6. We begin the division operation. Largest number, divisible by 8 without a remainder to 51 – 48. Dividing 48 by 8, we get 6. Write the number 6 instead of the first dot under the divisor:
Step 7. Then write the number exactly below the number 51 and put a “-” sign:
Step 8. Then we subtract 48 from 51 and get the answer 3.
* 9 step*. We take down the number 2 and write it next to the number 3:
Step 10 We divide the resulting number 32 by 8 and get the second digit of the answer – 4.
So the answer is 64, without remainder. If we divided the number 513, then the remainder would be one.
Division of three digits
Division three-digit numbers performed by the long division method, which was explained in the example above. An example of just a three-digit number.
Division of fractions
Dividing fractions is not as difficult as it seems at first glance. For example, (2/3):(1/4). The method of this division is quite simple. 2/3 is the dividend, 1/4 is the divisor. You can replace the division sign (:) with multiplication ( ), but to do this you need to swap the numerator and denominator of the divisor. That is, we get: (2/3)(4/1), (2/3)*4, this is equal to 8/3 or 2 integers and 2/3. Let's give another example, with an illustration for better understanding. Consider the fractions (4/7):(2/5):
As in the previous example, we reverse the 2/5 divisor and get 5/2, replacing division with multiplication. We then get (4/7)*(5/2). We make a reduction and answer: 10/7, then take out the whole part: 1 whole and 3/7.
Dividing numbers into classes
Let's imagine the number 148951784296, and divide it into three digits: 148,951,784,296. So, from right to left: 296 is the class of units, 784 is the class of thousands, 951 is the class of millions, 148 is the class of billions. In turn, in each class 3 digits have their own digit. From right to left: the first digit is units, the second digit is tens, the third is hundreds. For example, the class of units is 296, 6 is ones, 9 is tens, 2 is hundreds.
Division of natural numbers
Division natural numbers– this is the simplest division described in this article. It can be either with or without a remainder. The divisor and dividend can be any non-fractional, integer numbers. 
Sign up for the course "Speed up mental arithmetic, NOT mental arithmetic" to learn how to quickly and correctly add, subtract, multiply, divide, square numbers and even extract roots. In 30 days, you'll learn how to use easy tricks to simplify arithmetic operations. Each lesson contains new techniques, clear examples And useful tasks.
Division presentation
Presentation is another way to visualize the topic of division. Below we will find a link to an excellent presentation that does a good job of explaining how to divide, what division is, what dividend, divisor and quotient are. Don’t waste your time, but consolidate your knowledge!
Examples for division
Easy level
Average level
Difficult level
Games for developing mental arithmetic
Special educational games developed with the participation of Russian scientists from Skolkovo will help improve mental arithmetic skills in an interesting game form.
Game "Guess the operation"
The game “Guess the Operation” develops thinking and memory. The main point game, you need to choose a mathematical sign for the equality to be true. There are examples on the screen, look carefully and put the right sign"+" or "-" so that the equality is true. The “+” and “-” signs are located at the bottom of the picture, select the desired sign and click on the desired button. If you answered correctly, you score points and continue playing.
Game "Simplification"
The game “Simplification” develops thinking and memory. The main essence of the game is to quickly perform a mathematical operation. A student is drawn on the screen at the blackboard, and a mathematical operation is given; the student needs to calculate this example and write the answer. Below are three answers, count and click the number you need using the mouse. If you answered correctly, you score points and continue playing.
Game "Quick addition"
The game "Quick Addition" develops thinking and memory. The main essence of the game is to choose numbers whose sum is equal to a given number. In this game, a matrix from one to sixteen is given. A given number is written above the matrix; you need to select the numbers in the matrix so that the sum of these digits is equal to the given number. If you answered correctly, you score points and continue playing.
Visual Geometry Game
The game "Visual Geometry" develops thinking and memory. The main essence of the game is to quickly count the number of shaded objects and select it from the list of answers. In this game, blue squares are shown on the screen for a few seconds, you need to quickly count them, then they close. Below the table there are four numbers written, you need to select one correct number and click on it with the mouse. If you answered correctly, you score points and continue playing.
Game "Piggy Bank"
The Piggy Bank game develops thinking and memory. The main point of the game is to choose which piggy bank to use more money.In this game there are four piggy banks, you need to count which piggy bank has the most money and show this piggy bank with the mouse. If you answered correctly, then you score points and continue playing.
Game "Fast addition reload"
The game “Fast addition reboot” develops thinking, memory and attention. The main point of the game is to choose the correct terms, the sum of which will be equal to the given number. In this game, three numbers are given on the screen and a task is given, add the number, the screen indicates which number needs to be added. You select the desired numbers from three numbers and press them. If you answered correctly, then you score points and continue playing.
Development of phenomenal mental arithmetic
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Pure mathematics is, in its own way, the poetry of the logical idea. Albert Einstein
In this article we offer you a selection of simple mathematical techniques, many of which are quite relevant in life and allow you to count faster.
1. Quick interest calculation
Perhaps, in the era of loans and installment plans, the most relevant mathematical skill can be called masterly calculation of interest in the mind. The most in a fast way to calculate a certain percentage of a number is to multiply this percentage by this number and then discard the last two digits in the resulting result, because a percentage is nothing more than one hundredth.
How much is 20% of 70? 70 × 20 = 1400. We discard two digits and get 14. When rearranging the factors, the product does not change, and if you try to calculate 70% of 20, 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, the 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 to 70, and 29 to 30), and then use the same technique with multiplication and discarding the last two digits.
2. Quick divisibility check
Is it possible to divide 408 candies equally among 12 children? It’s easy to answer this question without the help of a calculator, if you remember 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, imagine it as 5 + 0 + 1 = 6. 6 is divisible by 3, which means 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 2,340. 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, take the number 6 390, imagine it as 6 + 3 + 9 + 0 = 18. 18 is divisible by 9, which means the number itself is 6 390 is divisible by 9.
- A number is divisible by 12 if it is divisible by 3 and 4.
3. Fast square root calculation
The square root of 4 is 2. Anyone can calculate this. What about the square root of 85?
For a quick approximate solution, we find the square number closest to the given one, in this case it is 81 = 9^2.
Now we find the next closest 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 100, then Square root this number will be 9-something.
4. Quick calculation of the time after which a cash deposit at a certain percentage will double
Do you want to quickly find out the time it will take for your money deposit at a certain interest rate to double? You don’t need a calculator here either, just 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 s small years old so that it doubles.
Why exactly 72 (sometimes they take 70 or 69)? How it works? Wikipedia will answer these questions in detail.
5. Quick calculation of the time after which a cash deposit at a certain percentage will triple
In this case interest rate by contribution should become a divisor of the number 115.
If the investment is made at 5% per annum, it will take 23 years for it to triple.
6. Quickly calculate your hourly rate
Imagine that you are undergoing interviews with two employers who do not give salaries in the usual format of “rubles per month”, but talk about annual salaries and hourly wages. 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 announcing the annual salary, you need to discard the last three digits from the stated amount, and then divide the resulting number by 2.
360,000 turns into 360 ÷ 2 = 180 rubles per hour. All other things being equal, it turns out that the second offer is better.
7. Advanced math on your fingers
Your fingers are capable of much more than simple operations addition and subtraction.
Using your fingers you can easily multiply by 9 if you suddenly forget the multiplication table.
Let's number the fingers from left to right from 1 to 10.
If we want to multiply 9 by 5, then we bend the fifth finger to the left.
Now let's look at the hands. It turns out four unbent fingers before the bent one. 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 to the left. We get five unbent fingers before the bent finger and four after. Answer: 54.
In this way you can reproduce the entire column of multiplication by 9.
8. Multiply by 4 quickly
There is extremely easy way lightning-fast multiplication of even large numbers by 4. To do this, it is enough to decompose the operation into two actions, multiplying the desired number by 2, and then again by 2.
See for yourself. Not everyone can multiply 1,223 by 4 in their head. Now we do 1223 × 2 = 2446 and then 2446 × 2 = 4892. This is much simpler.
9. Quickly determine the required minimum
Imagine that you are taking a series of five tests to... successful completion which you need minimum score 92. Stayed last test, and according to the previous results: 81, 98, 90, 93. How to calculate minimum required, which should be obtained in the last test?
To do this, we count how many points we have under/overtaken in the tests we have already passed, indicating a shortage negative numbers, and the results are more than 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.
The result is a deficit of 6 points, which means that the required minimum increases: 92 + 6 = 98. Things are bad. :(
10. Quickly represent the value of a fraction
Approximate value common fraction can be very quickly represented in the form decimal, if you first reduce 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, a little more than 0.33.
11. Number guessing trick
You can play a little David Blaine and surprise your friends with an interesting, but very simple mathematical trick.
- Ask a friend to guess any integer.
- Let him multiply it by 2.
- Then he will add 9 to the resulting number.
- Now let him subtract 3 from the resulting number.
- Now let him divide the resulting number in half (in any case, it will be divided without a remainder).
- Finally, ask him to subtract from the resulting number the number he guessed at the beginning.
The answer will always be 3.
Yes, it’s very stupid, but often the effect exceeds all expectations.
Bonus
And, of course, we couldn’t help but insert into this post that same picture with a very in a cool way multiplication.
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