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Number System -2


Rules of counting numbers:

1. Sum of all first natural numbers: = {n(n+1)}/2






Ex 1 Sum od 1+2+3+....100= {100(100+1)}/2=50*101 =5050

2. Sum of first n  odd Numbers= n^2 (where "n" is numbers)

Ex. 1+3+5+...20th odd numbers= 20^2= 400


3. Sum of first n even numbers= n(n+1)

Ex. Sum of 2+4+6... 100= 50 th even numbers then =50(50+1)= 2550

4. Sum of square of first n numbers ={n(n+1)(2n+1)}/6

Ex. 1 to 10th sqare sum is {10(10+1)(2*10+1)}/6



5. Sum of Cube of first numbers= [{n(n+1)}/2]^2

Ex. 1 to 6 cube sum = [{6(6+1)}/2]^2 = 21^2= 441



In the first n counting number if n is even the number of odd numbers is n/2 and the even number is also n/2.\

If the first n counting number is odd the number of odd numbers is (n+1)/2 and the even number is (n-1)/2.

Ex. 




Rule-2 Difference between squares of all consecutive numbers is always an odd number.

Ex: 4^2 and 5^2. difference is 25-16=9 = odd number

Rule-3 Difference between squares of consecutive numbers is the sum of these numbers.

Ex. 5^2-4^2= 9=5+4

26^2-25^2= 51= 26+25

Remember a^2-b^2=(a+b)(a-b) for consecutive number (a-b) = 0

Ex 1: What is the sum of the even number of 1 to 400?

Sol: 1 to 400, so there are 200 even and 200 odd numbers. So from rule No 3 we can find the sum of even number is n(n+1)= 200*(200+1)= 40200

Ex 2 Sum of even number from 1 to 461?

Sol: total even nember= (461-1)/2 = 230. so Sum of even number is 230(230+1)=?

Division Problems:

Division= (Divisor x Quotient) + Reminder

Some Misclenious example

1. The sum of the two numbers is 14 and if the difference is 10, then what is the product of the numbers?

Let two numbers are x and y  so x+y=14 and x-y=10

by the formula, we know that 4xy= (x+y)^2 -(x-y)^2

so we have to find xy.


2. The sum of two numbers is twice their difference. If one number is 10. find other?

- lets two numbers x and y

(x+y)= 2(x-y) so we have to do 



3. The ratio between two numbers is 3:5 if 9 are substructed from each the ratio will be 12:23. find the numbers.



4. A boy was asked to find 7/9 of a fraction, he mistakenly divided it with the fraction 7/9 and got the answer which exceeds the correct answer by  8/21. find the correct answer.



5. If the numerator of a fraction increased by 12% and the denominator decreased by 2% then the fraction becomes 6/7. Find the Original fraction.



6. Sum of two-digit a number is 8. If the digit is reversed the number decreased by 54. Find the number.



7. The ratio of three numbers 3:4:5. The sum of the largest and smallest equals the sum of the third and 52. find the numbers.


8. If 40% of a number is 360. What will be 15% of 15% of the number?



9. If the ratio of the sum and differences between the two numbers is 7:1. find the ratio of two numbers.?



10. The difference between a two-digit number and the number obtained by interchanging the digits is 27. What are the sum and the difference of the two-digit number?




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