Question

By rearranging the given numbers, evaluate: \[21 + 22 + 23 + 24 + 25 + 75 + 76 + 77 + 78 + 79\].

Answer 1

Hint: Here, in the given question, we are asked to evaluate the sum of the given series by re-arranging the numbers. To find the sum of the given series, we will use associative property of addition. As the word “associative” suggests, this property is related to grouping.

Complete step-by-step solution:
To evaluate the given series:\[21 + 22 + 23 + 24 + 25 + 75 + 76 + 77 + 78 + 79\]
We will use associative property of addition. Let us first understand the property.
Associative property of addition states that the sum of three or more numbers does not change if the numbers are grouped or the grouping of numbers changes. Or, in other words, we can say that the result of sum of the three or more numbers remains the same irrespective of order of the terms. For example, if we have to add four numbers, \[2 + 3 + 8 + 7\] the result will be the same if we change the order or we group the numbers so that the addition becomes easy. Hence\[2 + 3 + 8 + 7\] will be same as\[\left( {2 + 8} \right) + \left( {3 + 7} \right)\]
Now, using the associative property of addition in the given problem, it becomes,
\[\left( {21 + 79} \right) + \left( {22 + 78} \right) + \left( {23 + 77} \right) + \left( {24 + 76} \right) + \left( {25 + 75} \right)\]
Adding each group separately, we get,
\[100 + 100 + 100 + 100 + 100\]
\[ = 500\]

Note: While using associative property of addition, we group the numbers such that the sum of two or more numbers becomes easy. Like, in the above case, we have made groups of two numbers each such that the sum of each group comes out to be\[100\]. Grouping does not mean re-arranging the given numbers and not getting any benefit from it. It will make no sense.
Additional Information: Associative property can be used in addition as well as multiplication but not for subtraction and division.