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

BODMAS

          In The simplifications of numerical questions, the order of mathematical operations to be followed is given in the abbreviated form : "BODMAS", where

·  B stands for 'Bracket '

·  O stands for 'Of '

·  D stands for 'Division '

·  M stands for 'Multiplication '

·  A stands for 'Addition ' and

·  S stands for 'Subtraction '.

          If the expression contains brackets, you must solve the expression within the brackets First. Some problems may have specific numbers in a bracket or more than one bracket. Such problems can be solved with the help of the following rule.

Order to be followed by in case of brackets is as follows :

· ---- : Line or Vinculum (Virnaculum) or Bar,

· ( )  : Small Brackets,

· { }  : Curly Brackets and

· [ ]  : Square Brackets

          If the expression involves 'of' (as we have mentioned in one of the above examples), multiplication (or division), addition and subtraction, then 'of' should be performed first, followed by multiplication or division. Then proceeding from the left to the right, addition and subtraction are carried out in the order in which the sign of addition and subtraction are given. It should be noted that if the expression contains 'of' and 'division' always do 'of' first and then do 'division' in order to get the correct answer.

DIVISIBILITY RULES

The DIVISABILITY rule is a concise and useful way to determine whether a given integer is divisible by a fixed divisor, usually by examining its digits without performing the Division.

Let us see the divisibility rules of various numbers:

• There is no specific condition for divisibility by '1', Any integer is divisible by '1'.
• A number is divisible by ‘2’, when its unit digit is either  ‘Zero’ Or ‘Even number’ (2,4,6, or 8).

Number	Last Digit	Even or Odd	Is divisible by 2 as per rule?	Remainder after actual calculation
124	4	Even	Yes	0
67485	5	Odd	No	1
1245987	7	Odd	No	1

• A number is divisible by ‘3’, when the sum of its digits is divisible by ‘3’.

Number	Sum of the Digits	Is divisible by 3 as per rule?	Remainder after actual calculation
124	1+2+4 = 7	No	1
67485	6+7+4+8+5 = 30	Yes	0
1245987	36	Yes	0

• A number is divisible by ‘4’, when the number formed by the two extreme right end digits is either divisible by '4' or both these digits are zero’s.

Number	Number formed by last two digits	Number formed by last two digits divisible by 4	Number divisible as per rule?	Remainder after actual calculation
45	45	No	No	1
124	24	Yes	Yes	0
67485	85	No	No	1
1245987	87	No	No	3

• A number is divisible by ‘5’, when its unit digit is either 'zero' or '5'.

Number	Last Digit	Number divisible by 5 as per rule?	Remainder after actual calculation
45	5	Yes	0
124	4	No	4
67480	0	Yes	0

• A number is divisible by ‘6’, when it is divisible bye '2' as well as '3'.

Number	Number Divisible by 2?	Number Divisible by 3?	Number divisible by 6 as per rule?	Remainder after actual calculation
45	No	Yes	No	3
124	Yes	No	No	4
67482	Yes	Yes	Yes	0

Note: In the above examples if the number is not divisible any one of 2 & 3 or both the numbers then the number is not divisible by 6.

• A number is divisible is by 7, 

 

• A number is divisible by ‘8’, when the number formed by its three extreme right end digits is divisible by 8 or when these last three digits are Zero’s.

Number	Number formed by last three digits	Number formed by last two digits divisible by 8?	Number divisible as per rule?	Remainder after actual calculation
445	445	No	No	5
128	128	Yes	Yes	0
67485	485	No	No	5
1245987	987	No	No	3

• A number is divisible by ‘9’, when the sum of its digits is divisible by 9.

Number	Sum of the Digits	Is divisible by 9 as per rule?	Remainder after actual calculation
124	1+2+4 = 7	No	1
67485	6+7+4+8+5 = 30	No	3
1245987	36	Yes	0

• A number is divisible by ‘10’, when its unit digit is zero.

Number	Last Digit	Number divisible by 10 as per rule?	Remainder after actual calculation
45	5	No	5
124	4	No	4
67480	0	Yes	0

Note: If the last digit of the number is zero then the number is divisible by 10 

• A number is divisible by ‘11’, when the difference between the sums of the alternate digits is either zero or divisible by 11.

Number	Sum of the digits in Odd Positions	Sum of the digits in Even Positions	Difference between the sums	Number divisible by 11 as per rule?	Remainder after actual calculation
45	5	4	5-4 = 1	No	1
124	4+1 = 5	2	5-2 = 3	No	3
67480	0+4+6 = 10	8+7 = 15	15-10 = 5	No	5
96419818	1+9+4+9 = 23	8+8+1+6 = 23	23-23 = 0	Yes	0

• A number is divisible by ‘12’, when it is divisible by ‘3’ as well as ‘4’.
• A number is divisible by ‘13’, if sum of 4 times the digit in units place and the number in the remaining part is multiple of 13.
• If the difference of 5 times the digit in units place and the number in the remaining part is '0' or multiple of 17, then the number is divisible by 17
• If the sum of double the digit in units place of a given number and number in the remaining part is multiple of 19, then the given number is divisible of 19.
• A number is divisible by ‘25’, when its last two digits are  "00, 25, 50 or 75".
• A number is divisible by ‘50’, when its last two digits are  "00, 50".
• A number is divisible by ‘75’, when its last two digits are  "00, 25, 50 or 75", and the sum of all the digits must be divisible by 3.