Ayush Agarwal Maths

Percentage -   According to mathematics, a percentage is a number or ratio expressed as a fraction of 100. It is often denoted by using the percent sign, "%" .  Percentage formula -  Percentage = (Obtained marks/Total                                marks)×100 Example :-  3/5 ×100 = 0.60×100 = 60% =                                  60percent About Percentage - 1. 10% is equal to 1/10 fraction 2.  20% is equivalent to ⅕ fraction 3.  25% is equivalent to ¼ fraction 4.  50% is equivalent to ½ fraction 5.  75% is equivalent to ¾ fraction 6.  90% is equivalent to 9/10 fraction Points to remember :- 1. A percentage is a dimensionless number, it has no unit of measurement.  Practice Questions - Q1.   Convert as a fraction and as a decimal :   40% Q2. Express the ratio 4:5 as a percentage. Q3. Find 30% of ₹800. Q4. If 10% of a number is 45, then find the number? Thank you for visiting  Keep sharing  Join us now :-  Instagram

Compound Interest - compound interest is the addition of interest to the principal sum of a loan or deposit, or in other words, interest on interest. Applications of Compound Interest  1. In increase or decrease in population.  2. The growth of bacteria when the rate of growth is known.  3. Depreciation in the values of machines.  Important Facts :  1. When time is given in days, we convert it to year by dividing it by 365. 2. When time is given in months, we convert it to year by dividing it by 12. 3. When dates are given, the day on which the sum is borrowed is not included but the day on which money is returned is included, while counting the number of days. Remark - negative sign of rate indicates the depreciation.  Practice Questions  1.Find the compound interest on ₹ 5000 at 8% per annum for 2 years, compounded annually?  2.Find the amount of ₹18000 for 3 years, compounded annually at 10% per annum. Also, find the compound interest.  3.The compound interest on Rs. 30,000 at 7% pe

Simple interest -  Simple interest  is  interest  calculated on the principal portion of a loan  to a savings account.

Direct proportion - Two quantities x and y are said to be in direct proportion if whenever the value of x increases or decreases, then the value of y increases or decreases in such a way that the ratio x/y remains constant. 

Factorisation - The process of finding two or more expressions whose product is the given expression is called factorisarion. 

Algebra-  Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols.  ALGEBRAIC EXPRESSIONS- A combination of constants and variables, connected by some or all of the operations +, -, ×, and  ÷ is known as an algebraic expression.                          Polynomials  Monomial  - Polynomial having one term.  Binomial  - Polynomial having two terms.  Trinomial  - Polynomial having three terms Zero Polynomial  - Polynomial having all its coefficient zero.  Constant Polynomial  - A polynomial having only a single term ( of real number).  Standard form  - Powers of x are either in increasing or decreasing order.  Addition of Algebraic expressions - In the addition of algebraic expression we collect the like terms and add them.  Practice Questions  1. Solve the following equation for P  : 2P/3 = 8 + 4P 2.If 8x + 5x + 2x + 4x = 114, then 7x + 8 = ?  3. If a = 3, then a 3 (a 3 -a)= ?  4. 5x + 2(x + 7) = 14x – 7. Find x ?  5. 

Rational numbers :- The numbers of the form p/q , where p and q are integers and q is not equal to zero, q≠ 0 , are called rational numbers.  0 is neither positive nor negative.  Example-  Numbers 3/5 , -1/14 is a rational number.  - Every integer is a rational number.  - Every whole number is a rational number because every whole number can be expressed as a fraction. Positive rationals - A rational number is said to be positive if its numerator and denominator are either both positive or both negative.  Example - 3/7 or -6/11 are positive rationals.  Negative rationals - A rational number is said to be negative if its numerator and denominator are of opposite signs.  Example- -7/9, 4/-11 are negative rationals.  Some important facts :- 1. The product of a rational number and its reciprocal is 1. 2.  Zero '0' has no reciprocal.  3. The reciprocal of a positive rational number is positive.  4. The reciprocal of a negative rational number is negative.  5. Every natural number N

                   CUBES AND CUBE ROOTS Cube number- A cube number is a number multiplied by itself twice, or we can say the cube of a number m is its third power.  For example- m³ =  m  ×  m 2  =  m  ×  m  ×  m . 8³ = 8 × 8 2  = 8  × 8  × 8 = 512 .  15³ = 15  ×  15 2  =  15  ×  15  ×  15 = 3375 .   Perfect cube - A natural number is said to be a perfect cube if it is the cube of some natural numbers.  Examples-   1 ³ = 1, 2³ = 8, 3³ = 27, 4³ = 64,  etc.            Thus 1, 8, 27, 64 etc.., are perfect cubes.  Facts About cube numbers:- 1. The cube of a number is that number raised to the power 3 .  2. The cube of an even natural number is even.  3. The cube of an odd natural number is odd.  4.  Numbers ending in 1, 2, 3, 4, 6, 7, 8, 9, 000, 125, 375, 625 or 875 may be cube numbers.  Practice Questions 1. Find the cube of 21 .  2. Find the least number by which 750 should be multiplied, so that it becomes a perfect cube. 3. Solve :- (i) ∛4096  (ii) ∛(729/1000 ) 4. By w

          SQUARES AND SQUARE ROOTS   Square number - The square of a number is the product of the number with the number itself.  A given number is a perfect square if its prime factors can be expressed in pairs of equal factors.  A perfect square number is never negative. For example - 8^2 = (8×8) = 64 Facts About Squares -  1. A number ending in 2, 3, 7 or 8 is never a perfect square.  2. A number ending in an odd number of zeros is never a perfect square.  3. The square of an even number is always even.  4. The square of an odd number is always odd.  5. The ending digits of a square number is 0, 1, 4, 5, 6, or 9 only.  Note:- It is not necessary that all numbers ending with digits 0, 1, 4, 5, 6, or 9 are square numbers.  Important Facts-  1. Numbers ending in an even number of zeros are not necessarily perfect squares. Eg- 3700, 290000, 47000000.   Practice Questions :-  1. Find the square of 54 ?  2. What least number must be added to 5607 to mak