Certainly! Let's simplify the expression [tex]\(50 + 20 + 37\)[/tex] using the associative property of addition. The associative property states that the way in which numbers are grouped does not affect their sum. Let's evaluate the options given:
Option A:
[tex]\[10(5+2) + 37 = 10(7) + 37 = 70 + 37 = 107\][/tex]
This option uses the distributive property incorrectly and applies unnecessary complexity. Hence, this is not a correct simplification strictly using the associative property.
Option B:
[tex]\[(50 + 20) + 37 = 70 + 37 = 107\][/tex]
This option is correct. Grouping [tex]\(50\)[/tex] and [tex]\(20\)[/tex] first gives [tex]\(70\)[/tex]. Adding [tex]\(37\)[/tex] to [tex]\(70\)[/tex] results in [tex]\(107\)[/tex].
Option C:
[tex]\[50 + 37 + 20 = 87 + 20 = 107\][/tex]
This option reorders the numbers, which is actually using the commutative property of addition, not strictly the associative property. However, since addition is both associative and commutative, the final sum is still correct: [tex]\(50 + 37 + 20 = 107\)[/tex].
Option D:
[tex]\[50 + 20 + 37 = 50 + 57 = 107\][/tex]
This option groups the second and third numbers first: [tex]\(20 + 37 = 57\)[/tex]. Adding [tex]\(50\)[/tex] to [tex]\(57\)[/tex] indeed gives [tex]\(107\)[/tex]. This uses the associative property correctly.
Based on the detailed examination:
- Option A involves an incorrect application.
- Option B uses the associative property correctly.
- Option C uses both associative and commutative properties for a correct sum.
- Option D also uses the associative property correctly.
So, all of the options except for Option A correctly simplify the expression [tex]\(50 + 20 + 37\)[/tex] to [tex]\(107\)[/tex].
Therefore, the correct simplifications are provided by Options B, C, and D.
