Certainly! Let's work through the problem step-by-step to verify the grouping property.
### Problem (a):
We are given two different groupings of the same set of numbers:
(i) \(31,244 + 23,419 + 41,695\)
(iii) \(23,419 + 41,695 + 31,244\)
1. Calculate the sum for (a)(i):
[tex]\[
31,244 + 23,419 + 41,695
\][/tex]
2. Calculate the sum for (a)(iii):
[tex]\[
23,419 + 41,695 + 31,244
\][/tex]
3. Verify if both sums are equal:
[tex]\[
31,244 + 23,419 + 41,695 = 96,358
\][/tex]
[tex]\[
23,419 + 41,695 + 31,244 = 96,358
\][/tex]
Thus, the two sums in part (a) are equal.
### Problem (b):
We are given two different groupings of another set of numbers:
(i) \(53,206 + 24,714 + 6,692\)
(iii) \(24,714 + 53,206 + 6,692\)
1. Calculate the sum for (b)(i):
[tex]\[
53,206 + 24,714 + 6,692
\][/tex]
2. Calculate the sum for (b)(iii):
[tex]\[
24,714 + 53,206 + 6,692
\][/tex]
3. Verify if both sums are equal:
[tex]\[
53,206 + 24,714 + 6,692 = 84,612
\][/tex]
[tex]\[
24,714 + 53,206 + 6,692 = 84,612
\][/tex]
Thus, the two sums in part (b) are equal.
### Summary:
In both parts (a) and (b), we observe that regardless of how the numbers are grouped, the sums remain the same. This demonstrates the grouping property (associative property) of addition which states that the way in which numbers are grouped in an addition operation does not affect the sum.
Final results:
- (a) \(31,244 + 23,419 + 41,695\) and \(23,419 + 41,695 + 31,244\) both equal \(96,358\).
- (b) [tex]\(53,206 + 24,714 + 6,692\)[/tex] and [tex]\(24,714 + 53,206 + 6,692\)[/tex] both equal [tex]\(84,612\)[/tex].
