English Short cut maths

1.ADDING CONSECUTIVE NUMBERS 

                 Rule: (Add the smallest number in the group to the largest number in the group, multiply the result by the amount of numbers in the group, and divide the resulting product by 2.)

Suppose we want to find the sum of all numbers from 33 to 41. First, add the smallest number to the largest number.

33 + 41 =  74

Since there are nine numbers from 33 to 41, the next step is

74 x 9 = 666

Finally, divide the result by 2.

666 / 2 = 333 Answer

The sum of all numbers from 33 to 41 is therefore 333.

2.ADDING CONSECUTIVE NUMBERS STARTING FROM 1

                                               Consider the problem of adding a group of consective numbers such as: 1, 2, 3, 4, 5, 6, 7, 8, and 9.  How would you go about finding their sum ? 

This group is certainly easy enough to add the usual way. 

But if you're really clever you might notice that the first number, 1, added to the last number , 9, totals 10 and the second number, 2, plus the next to last number,  8,  also totals  10. 

In fact, starting from both ends and adding pairs, the total in each case is 10. We find there are four pairs, each adding to 10; there is no pair for the number 5.  

Thus 4 x 10 =  40 ;   40 + 5 = 45

Going a step further, we can develop a method for finding the sum of as many numbers in a row as we please

 Going a step further, we can develop a method for finding the sum of as many numbers in a row as we please

                         Rule : ( Muliply the amount of numbers in the group by one more than their number , and divide by 2.)

As an example , suppose we are asked to find the sum of all numbers from 1 to 99. There are 99 intergers in this series : one more than this is 100 . thus

99 X 100 = 9,900

9,900 / 2 = 4,950 Answer

The sum of all nimbers from 1 to 99 is therefore 4,950.

3. FINDING THE SUM OF ALL ODD NUMBERS STARTING FROM 1

Rule : Square the amount of numbers from 1 to 100  will be calculated. There are 50 odd numbers in this group.

Therefore

50 x 50 = 2,500 Answer

This is the sum of all odd numbers from 1 to 100. As a check , we can compare this  answer with the answers found in Short Cuts 2 and 4.