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
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.
4)
FINDING THE SUM OF ALL EVEN NUMBERS STARTING FROM 2
RULE:
( MULTIPLY THE AMOUNT OF NUMBERS IN THE GROUP BY ONE MORE THAN THEIR NUMBER )
We shall use this rule to find the sum of all even numbers from 1 to 100. Hall of the numbers will be even and half will be odd, which means there are 50 even numbers from 1 to 100.
Applying the rule,
50x 51 = 2,550
Thus the sum of all even numbers from 1 to 100 is 2,550.In Short Cut 2 the sum of all the numbers from 1 to 99 is found to be 4,950 : consequently the sum of all numbers from 1 to 100 is 5,050.In Short Cut 3 the sum of all odd numbers from 1 to 100 is found to be 2,500.Our answer for the sum of all the even numbers from 1 to 100 is therefore in agreement
Sum of all numbers 5,050 – Sum of all odd numbers 2,500 = Sum of all even numbers 2,550
5)
ADDING A SERIES OF NUMBERS WITH A COMMON DIFFERENCE
Sometimes it is necessary to add a group of numbers that have a common difference. No matter what the common difference is and no matter how many numbers are being added, only one addition, multiplication, and division will be necessary to obtain the answer.
Rule:
( Add the smallest number to the largest number, multiply the sum by the amount of numbers in the group, and divide by 2 )
As an example, let us find the sum of the following numbers
87, 91, 95, 99, and 103
Notice that the difference between adjacent numbers is always 4. This short-cut method can therefore be used. Add the smallest number, 87, to the largest number, 103.
Multiply the sum, 190, by 5, since there are five numbers in the group.
190 x 5= 950
Divide by 2 to obtain the answer.
950 / 2 = 475 Answer
Thus 87+ 91 95 + 99 +103 = 475.
(Naturally, this is exactly the same as the rule in Shortcut 1, because there we were simply adding a series of numbers with a common difference of one. So, for case of remembering, you can combine Short Cuts 1 and 5.)
6)
ADDING A SERIES OF NUMBERS HAVING A COMMON RATIO
Rule:
(Multiply the ratio by itself as many times as there are numbers in the series. Subtract 1 from the product and multiply by the first number in the series. Divide the result by one less than the ratio.)
This rule is best applied when the common ratio is a small number or when there are few numbers in the series. If there are many numbers and the ratio is large, the necessity of multiplying the ratio by itself many times diminishes the ease with which this short cut can be applied.
But suppose we are given the series:
53, 106, 212, 424
Here each term is twice the preceding term, and there are four terms in the series. The ratio, 2, is therefore multiplied four times.
2 x 2 x 2 x 2 = 16
Subtract 1 and multiply by the first number.
16 - 1 = 15; 15 x 53 = 795
The next step is to divide by one less than the ratio; however, since the ratio is 2, we need divide only by 1.
Thus the sum of our series is
53 + 106 + 212 + 424 = 795 Answer
SHORT CUTS IN ADDITION - MODEL QUESTION
Practice Exercises for Short Cuts 1 through 6
Find the sum in each case
1) All odd numbers from 1 to 23 =
2) 3 + 6 + 12 + 24 + 48 + 96 =
3) All numbers from 84 to 105 =
4) 56 + 59 + 62 + 65 =
5) 24 +72 + 216 =
6) 14 +15 + 16 + 17 + 18 + 19 + 20 + 21 =
7) All numbers from 1 to 1,000 =
8) All even numbers from 1 to 50 =
9) 132 +137 + 142 +147 =
10) 197 + 198 + 199 + 200+ 201 + 202 + 203 =
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