Solving all the problems related to numbers and techniques.

Here are the solutions to all the problems related to numbers from 5th to 10th class. Here you will find some important information for 11th-12th class. Notice the numbers below. Here are the solutions based on almost all the numbers and share the short technique to remember.

An easy way to keep the basic numbers from 1-100

Shortcuts: - 44 -22 -322-321

★ 1 to 100 prime numbers = 25

Prime numbers from 1 to 10 = 4 2,3,5,7

★ Prime numbers from 11 to 20 = 4 11,13,17,19

Prime numbers from 21 to 30 = 2 23,29

★ Prime numbers from 31 to 40 = 2 31.37

Prime numbers from 41 to 50 = 3 41,43,47

★ Prime numbers from 51 to 60 = 2 53.59

Prime numbers from 61 to 70 = 2 61,67

Prime numbers from 71 to 80 = 3 71,73,79

★ Prime numbers from 81 to 90 = 2 83,89

★ Prime numbers from 91 to 100 = 1 97

The prime numbers from 1 to 100 are 25

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97

The sum of the prime numbers from 1-100 1060.

Parallel series and multiplication series.

Sum of parallel series serial numbers-

(When the number starts from 1) 1 + 2 + 3 + 4 + ...... + n then the sum of such series = [n (n + 1) / 2]

n = last number or term number s = sum

Q: 1 + 2 + 3 + .... + 100 =?

Solution: [n (n + 1) / 2]

= [100 (100 + 1) / 2]

= 5050

 

 In case of parallel class addition method, -

The sum of the squares of the first n terms

S = [n (n + 1) 2n + 1) / 6]

(When 1² + 2² + 3² + 4² ........ + n²)

★ Question: (1² + 3² + 5² + ....... + 31²) What is equal?

Solution: S = [n (n + 1) 2n + 1) / 6]

= [31 (31 + 1) 2 × 31 + 1) / 6]

= 31

 

In case of parallel series condensing method-

The sum of the cubes of the first n term is S = [n (n + 1) / 2] 2

(When 1³ + 2³ + 3³ + ............. + n³)

Q: 1³ + 2³ + 3³ + 4³ + ………… + 10³ =?

★ Solution: [n (n + 1) / 2] 2

= [10 (10 + 1) / 2] 2

= 3025

 

In determining the sum of the number of terms and the number of terms:

Number of terms N = [(last term - first term) / increment per term] +1

Q: 5 + 10 + 15 + + 50 =?

★ Solution: Number of terms = [(last term - first term) / increment per term] +1

= [(50 - 5) / 5] + 1

= 10

 

So the sum of the number of terms

= [(5 + 50) / 2] × 10

= 275

nth term = a + (n-1) d

Here, n = term, a = 1st term, d = common interval

Question: 5 + 8 + 11 + 14 + ....... Which term of section 302?

★ Solution: Hold, nth term = 302

Or, a + (n-1) d = 302

Or, 5+ (n-1) 3 = 302

Or, 3n = 300

Or, n = 100

★★ 6) Sum of parallel sequential odd numbers - S = M- here, M = in = (1st number + last number) / 2

Q: 1 + 3 + 5 + ....... + 19 = how much?

Solution: S = M²

= {(1 + 19) / 2}

= (20/2)

= 100

8 squares

(1) ² = 1, (11) ² = 121, (111) ² = 12321, (1111) ² = 1234321, (11111) ² = 123454321

★★ As many as 1 will be squared side by side, the square will have to be written from 1 to that number one after the other and then from that number onwards will have to be written in consecutive numbers and will end in 1 number.

3 (3) ² = 9, (33) ² = 1089, (333) ² = 110889, (3333) ² = 11108889, (33333) ² = 1111088889

★★ As many as 3 will be squared side by side, the square will result in a single cell with a lower number of 9 and 9 to the left of it (as many as 3 will be), then a 0 to the left and an equal number of 8 to the left.

6 (6) ² = 36, (66) ² = 4356, (666) ² = 443556, (6666) ² = 44435556, (66666) ² = 4444355556

**As many as 6 will be squared side by side, so the square will have a lesser number of 5 and 6 to the left of the unit cell (as many as 6 will be), then a 3 to the left and an equal number of 5 to the left.

** (9) ² = 81, (99) ² = 9801, (999) = 998001, (9999) ² = 99980001, (99999) ² = 9999800001

As many as 9 will be squared side by side, so the square will have a lesser number of 0s to the left of 1 and 1 (as many as 9), then an 8 on the left and an equal number of 0s on the left.

Fathers of theory.

1) Numerology - Pythagoras (Pythagoras)

2) Geometry - Euclid

3) Calculus (Calculus) - Newton (Newton)

4) Matrix - Arthur Cayley

5) Trigonometry (Hipparchus)

6) Asthmatic (arithmetic) Brahmagupta (Brahmagupta)

7) Algebra (Algebra) - Muhammad ibn Musa al-Khwarizmi

8) Logarithm - John Napier

9) Set theory (set theory) - George Cantor (George Cantor)

10) Zero - Brahmagupta (ব্রহ্মগুপ্ত)

The English word for arithmetic

Arithmetic and parameters

Digit, Ratio, Basic Number: Prime number, Full square, Perfect square, Factor, Sequential Proportion: Continued proportion, Purchase-Cost price, Loss, Average, Average, Velocity, Product, C, Sa, Gu- Highest Common Factor, Power- Power, Cube Root: Cube root, Cube- Cube, Cube- Volume, Integer- Integer, Pressure- Arc, Cone- Cylinder, Chord, Even number- Even number, Constant -Constant, range-Perimeter, real-real, square root-square root, busy ratio: Inverse ratio, odd number: Odd number, selling price -selling price, algebra: Algebra, rational, middle proportional -Mean proportional, sum = Sum

Law, Sa, Gu- Lowest Common Multiple, Lob- Numerator, Percentage- Percentage, Proportion- Proportion, Proportional- Proportional, Interest- Interest, Rate- Denominator,

Geometry

Hypotenuse, Hypotenuse, Internal angle, Semi-circle, In-radius, Rectangle, Height, Corner-Diagonal, Corner-Angle, Center Cylinder, Geometry, Length, Pentagon, Breadth

Complementary Angles, Arm-Side, Circle, Circle, Radius, Diameter, Polygon, Polygon, Square, External, Cone, Cone, Right angle, Equilateral triangle, Equilateral triangle Triangle: Scalene triangle, Equilateral triangle-isosceles Triangle, Right-angled triangle Right angled triangle, Acute-angled triangle, Obtuse-angled triangle Obtuse angled triangle, Parallel, Angle

Roman numerals ≠ Roman numerals)

1: I

2: II

3: III

4: IV

5: V

6: VI

7: VII

8: VIII

9: IX

10: X

11: XI

12: XII

13: XIII

14: XIV

15: XV

16: XVI

17: XVII

18: XVIII

19: XIX

20: XX, 30: XXX, 40: XL, 50: L, 60: LX, 70: LXX, 80: LXXX

, 90: XC, 100: C, 200: CC, 300: CCC, 400: CD, 500: D, 600: DC

, 700: DCC, 800: DCCC, 900: CM, 1000: M

1. Even number + even number = even

Numbers.

For example: 2 + 6 = 8.

2. Even number + odd number =

Odd number.

E.g. 6 + 7 = 13.

3. odd number + odd number =

Even number.

E.g. 3 + 5 = 8.

4. Even number: Even number = even

Numbers.

E.g. 6 × 8 = 48.

5. Even number: even number = even

Numbers.

For example: 6 × 7 = 42

6. Odd number: Odd number =

Odd number.

For example: 3 × 9 = 27

An effective technique to divide any number without a calculator!

 An effective technique to divide any number by 5 without a calculator

1. 13/5 = 2.6 (it can be solved in just 3 seconds without calculator)

★ tekanikah

Multiply the number by dividing by 5 by 2 then place the decimal 1 cell before the right. Done !!! 13 * 2 = 26, then put 1 decimal before 1 cell 2.6.

2. 213/5 = 42.6 (213 * 2 = 426)

0.03 / 5 = 0.006 (0.03 * 2 = 0.06 which is 0.006 when the decimal is placed one room before) 333,333,333 / 5 = 66,666,666.6 (Doesn't it take a calculator to do this again!)

3. 12,121,212 / 5 = 2,424,242.4

Now try to divide any number by 5 as you wish

 An effective technique to divide any number by 25 without a calculator

1. 13/25 = 0.52 (can also be solved without calculator)

★ tekanikah

Multiply the number by dividing by 25 by 4 then place the decimal 2 cells to the right. 13 * 4 = 52, then put 2 decimal places before 0.52.

02. 210/25 = 8.40

03. 0.03 / 25 = 0.0012

04. 222,222 / 25 = 8,888.88

05. 13,121,312 / 25 = 524,852.48

 An effective technique to divide any number by 125 without a calculator

01. 7/125 = 0.056

★ tekanikah

Multiply the number you divide by 125 by 8 then place the decimal 3 cells to the right. Done! 7 * 8 = 56, then put 3 decimal places before 0.056.

02. 111/125 = 0.888

03. 600/125 = 4.800

Let's do it easily

Topic: Find the square root in 10 seconds.

Note: The square root of the numbers between 1 and 99 can be easily found in this method. The question must have a whole number. That is, if the answer is a decimal fraction, then this method will not work.

Must read carefully and practice. Otherwise forget.

But let's get started. Let's memorize the squares of numbers from 1 to 9 at the beginning. I hope everyone knows these. For convenience I write below-

1 square = 1, 2 square = 4

3 square = 9, 4 square = 16

5 square = 25, 6 square = 36

7 square = 49, 8 square = 64

9 square = 81

If you look at each square number here, you will see that in the case of the last digit of all -

The squares of 1 and 9 have the same number (1, 81)

The squares of 2 and 8 have the same number (4, 64)

The squares of 3 and 7 have the last digit match (9, 49);

The squares of 4 and 6 have the last digit match (16, 36);

And 5 single frown emoticons

If there is any problem in understanding so far, read it again.

Example: - Find the square root of 576.

Step 1: Find the unit cell number of the number that needs to be determined by the square root. In this case it is '6'.

 Step 2: From the list above, take the last digit 6 of the square of that number. In this case 4 and 6. Again, notice- the squares of 4 and 6 are 16 and 36, respectively; Whose unit cell number is '6'. Got it? If you don't understand, read it again.

 Step 3: Write 4/6 in the notebook. (We've got the unit cell number in the answer, which is 4 or 6; but which one? You'll find the answer in the eighth step, keep reading ...)

 Step 4: Look at the rest of the numbers, excluding the unit of the question and the number of decades. In this case it is 5.

Step 5: Take the square root of the square number near 5 from the list above. In this case 4, which is the square of 2. (We've got the North Decade house numbers, which is 2)

Step 6: Multiply the next number by 2. That is 2 * 3 = 6

Step 7: Find the number found in the fourth step (5) smaller or larger than the number found in the sixth step (6). If it's small, I'll take the smallest number in the third step, if it's big, it's big. (Understood? Or read again)

Step 8: In our example, 5 is less than 6, so we will take the smallest number in 4/6, that is, 4.

Ninth step: I remember, in the fifth step I got the number of the house of the decade 2, now I got the number of the house of the unit 4. So the answer would be 24

Sounds tough? Not at all, try some practice. I don't think it should take too long.

Example: - Find the square root of 4225.

Remember 5 that was alone? Your work has become much easier since he is alone. See why the last digit of the question is 5 so the unit cell of the answer must be 5.

- Subtracting the unit of the question and the house number of the decade, 42 is left.

- The nearest whole number of 42 is 36, whose square root is 6. So the answer is 65