Simple Interest and Compound Interest

Principal: The money borrowed or given for a certain period of time is called the principal.

Interest: Interest is the amount you pay to use other people's money.

Simple Interest: Simple interest is defined as the uniform amount of money used to repay on the amount borrowed for a given period of time.

The formula for simple interest is, 

                                                SI = (P*T*R)/100

Where, 

              P = Principle (Actual Money Borrowed),

              T = Time (in years), 

              R = Rate of interest (%) per annum,

              SI = Simple Interest.

We can calculate the total using the following formula:

                         Total = Principal + Interest

Here, the sum (A) is equal to the amount repaid at the end of the borrowing period (T).

So, Obviously, 

                            I = (P*T*R)/100

                            P = 100*I/T*R

                            T = 100*I/P*R

                            R = 100*I/P*T

Note: One ordinary year = 365 days

            Leap Year = 366 days

Compound Interest: Compound interest is defined as the interest calculated on the original and the interest accrued on the previous term.

        If interest is not paid to the lender for a period of one year or a certain period of time, the sum at the end of this period becomes principal for the next period. This process is repeated until the final amount is received.

1. Compound Interest formula, when the interest is compounded annually, 

                                    Amount (A) = P(1+R/100)ⁿ 

Where, 

              P = Principle (Actual Money Borrowed),

              n = Time (in years), 

              R = Rate of interest (%) per annum,

              A = Amount.

2. Compound Interest formula, when the interest is compounded Halfyearly,

 3. Compound Interest formula, when the interest is compounded quarterly,

4. Compound Interest formula, when the interest is compounded annually, but time is in fractions, i.e., 

the amount will be, 

5. Present worth of Rs. 'X' due 'n' years hence will be

6. When rates of interest are different for different years, say R1, R2, R3 ... for 1st, 2nd, 3rd .... years respectively. Then amount will be

Note: 

1) If the interest is payable half yearly, then time (n) is multiplied by 2 and rate (R) is divided by 2.

2) If the interest is payable quarterly, then time is multiplied by 4 and rate is divided by 4.

AVERAGE

1. 

Sum of observations = Average X Number of Observations

2. Average Speed: If a man covers a certain distance at X  km/h and an equal distance at Y km/h. Then, the average speed during the whole journey is 

3. Consecutive even numbers (or) odd numbers =  x, x+2, x+4, x+6 ..,,,

4. For a set of non-zero numbers, if we add/subtract/multiply/divide with some non-zero quantity, then the result average also increased/decreased/multiplied/divided by the same non-zero quantity.

5. Average of first 'n' natural numbers = (n+1)/2

6. Average of first 'n' odd numbers = n

                                                      Sum = n²

     Ex: We take 1, 3, 5, 7, 9,  then find their Average and Sum?

     Sol:   In the above question, there are first 5 odd numbers (i.e., n = 5)

               Their  Average = (1 + 3 + 5 + 7 + 9)/5  = 25/5 = 5

                              Sum  = 1 + 3 + 5 + 7 + 9 = 25  (i.e., n = 5, n² = 25)

7. Average of first 'n' even numbers = n + 1

                                                        Sum = n*(n + 1)

8. For Even (or) Odd numbers in sequence 

            First term = Average - (n - 1)

            Last term = Average + (n + 1)

9. For consecutive natural numbers 

              First term = Average - {(n - 1)/2}

               Last term = Average + {(n - 1)/2}          

9. Sum of 'n' consecutive natural numbers = 1, 2, 3, .......n  =   {n * (n+1)}/2

10. Sum of squares of 'n' consecutive natural numbers = 1², 2², 3², .......n² 

                                                                                                    = { n*(n+1)*(n+2)}/6

11. Sum of cubes of 'n' consecutive natural numbers =1³, 2³, 3³, .......n³ 

                                                                                                = { n²*(n+1)²}/4  

                                                                                                               (or)

                                                                                                  = [{n*(n+1)}/2]²

నదులు -- మారు పేర్లు

 

నదులు	మారు పేర్లు
యాంగ్ సికియాంగ్ (చైనా)	బ్లూ రివర్
హూ యాంగ్ హూ	చైనా దుఃఖదాయని, ఎల్లో రివర్
బ్రహ్మపుత్ర	అస్సాం దుఃఖదాయిని,  భారతదేశ ఎరుపు నది
దామోదర్ నది	బెంగాల్ దుఃఖదాయిని
బ్రాహ్మణి	ఒడిశా దుఃఖదాయిని
కోసీ	బీహార్ దుఃఖదాయిని
బుడమేరు	ఆంధ్ర ప్రదేశ్ దుఃఖదాయిని
గోదావరి	దక్షిణ భారత గంగ, ఇండియన్ రైన్ నది
డాన్యూబ్	అంతర్జాతీయ నది

SQUARES AND SQUARE ROOTS

 

What is a Square number?

          When you multiply a whole number (or) Integer (not a fraction) by itself, the result is called the Square of that number.

Here are the Squares of 1 to 30 Numbers.,

 

SQARES 1 to 30
1 X 1 = 1	11 X 11 = 121	21 X 21 =441
2 X 2 = 4	12 X 12 = 144	22 X 22 = 484
3 X 3 = 9	13 X 13 = 169	23 X 23 = 529
4 X 4 = 16	14 X 14 = 196	24 X 24 = 576
5 X 5 = 25	15 X 15 = 225	25 X 25 = 625
6 X 6 = 36	16 X 16 = 256	26 X 26 = 676
7 X 7 = 49	17 X 17 = 289	27 X 27 = 729
8 X 8 = 64	18 X 18 = 324	28 X 28 = 784
9 X 9 = 81	19 X 19 = 361	29 X 29 = 841
10 X 10 = 100	20 X 20 = 400	30 X 30 = 900

 Note: 

1) If a number has 1 or 9 in its unit's place then its square ends with 1 in units place.   (We can cross check it by above table).

2) Square of a even number is always even, and square of an odd number is always Odd. 

        Ex: 2² = 4,  6² = 36,   7² = 49,   9² = 81

3) Three natural numbers x, y, z is called a Pythagorean triplet if X² + y² = z².

        Ex: (3, 4, 5),  (6, 8, 10), (7, 24, 25) etc., 

        since 3² + 4² = 9 + 16 = 25, and  5².= 25

For any natural number 'n' which is greater than 1, the Pythagorean triplet is  given by (2n, n²-1, n²+1 )

        Ex: Find the other two Pythagorean triplets, one number which is 10.

        Sol: For any natural number 'n' the Pythagorean triplet is = 2n, n²-1, n²+1

                consider n as 5 as per given above

                2n = 2 *5 = 10,

                n²-1 = 5² - 1 = 25 - 1 = 24, and 

                n²+1 = 5² + 1 = 25 + 1 = 26.

                So, the other two numbers are 24 and 26.

4) The square of a natural number 'n' is equal to the sum of the first n odd numbers.

        Ex: Thus   1² = 1, sum of the first odd number,

                           2² = 4 = 1 + 3, sum of the first 2 odd numbers,

                           3² = 9 = 1 + 3 + 5, sum of the first 3 odd numbers,

                           4² = 16 = 1+3+5+7 =16, sum of the first 4 odd numbers,    etc.,

5) For any natural number 'n' which is greater than 1, (n + 1) X (n - 1) = n²-1.

6) Finding the square of a number which is ending with 5 (last digit as 5) is as follows: 

Ex: 25² = 2 X 3 = 6, 5² = 25 so the answer is 625.

       What ever may be the earlier part (n),  multiply it with one more than it self (n+1) that will become over answer's first part, the second part of the number will be 25 only.  

        35² = 3 X 4 = 12, 5² = 25 so the answer is 1225.

        45² = 4 X 5 = 20, 5² = 25 so the answer is 2025.

        55² = 5 X 6 = 30, 5² = 25 so the answer is 3025.

        135² = 13 X 14 = 182, 5² = 25 so the answer is 18225.

        155² = 15 X 16 = 240, 5² = 25 so the answer is 24025.

Square Root: