Answer :
We begin by noting that subtraction is not associative; that is, how you group the terms in a subtraction expression can change the final result.
Let's look at the two different evaluations.
–––––––
Frederick’s Calculation
He evaluates the expression grouped as
[tex]$$
\left[(53.7+37.2)-(3^3+3.8)\right]-8.6.
$$[/tex]
Step 1. Calculate the sum inside the first parentheses:
[tex]$$
53.7 + 37.2 = 90.9.
$$[/tex]
Step 2. Calculate the sum inside the second parentheses. (Remember that [tex]$3^3$[/tex] means [tex]$3 \times 3 \times 3$[/tex].)
[tex]$$
3^3 = 27, \quad \text{so} \quad 27 + 3.8 = 30.8.
$$[/tex]
Step 3. Subtract the second sum from the first:
[tex]$$
90.9 - 30.8 = 60.1.
$$[/tex]
Step 4. Finally, subtract [tex]$8.6$[/tex]:
[tex]$$
60.1 - 8.6 = 51.5.
$$[/tex]
–––––––
Lana’s Calculation
She evaluates the expression without any extra grouping as
[tex]$$
53.7 + 37.2 - 3^3 + 3.8 - 8.6.
$$[/tex]
Step 1. As always, compute the exponent first:
[tex]$$
3^3 = 27.
$$[/tex]
Step 2. Now perform the addition and subtraction from left to right:
1. First, add:
[tex]$$
53.7 + 37.2 = 90.9.
$$[/tex]
2. Subtract [tex]$27$[/tex]:
[tex]$$
90.9 - 27 = 63.9.
$$[/tex]
3. Add [tex]$3.8$[/tex]:
[tex]$$
63.9 + 3.8 = 67.7.
$$[/tex]
4. Finally, subtract [tex]$8.6$[/tex]:
[tex]$$
67.7 - 8.6 = 59.1.
$$[/tex]
–––––––
Conclusion
Frederick obtains [tex]$51.5$[/tex], while Lana obtains [tex]$59.1$[/tex]. Both are mathematically correct because they followed different groupings:
- Frederick explicitly grouped the terms as
[tex]$$
\left[(53.7+37.2)-(3^3+3.8)\right]-8.6,
$$[/tex]
meaning he combined [tex]$3^3$[/tex] and [tex]$3.8$[/tex] together before subtracting from the sum of [tex]$53.7$[/tex] and [tex]$37.2$[/tex].
- Lana evaluated the expression in a left-to-right fashion (after computing the exponent), effectively grouping the operations differently.
Since subtraction is not associative, different groupings lead to different results. Thus, both answers are correct based on their interpretations of the expression.
Let's look at the two different evaluations.
–––––––
Frederick’s Calculation
He evaluates the expression grouped as
[tex]$$
\left[(53.7+37.2)-(3^3+3.8)\right]-8.6.
$$[/tex]
Step 1. Calculate the sum inside the first parentheses:
[tex]$$
53.7 + 37.2 = 90.9.
$$[/tex]
Step 2. Calculate the sum inside the second parentheses. (Remember that [tex]$3^3$[/tex] means [tex]$3 \times 3 \times 3$[/tex].)
[tex]$$
3^3 = 27, \quad \text{so} \quad 27 + 3.8 = 30.8.
$$[/tex]
Step 3. Subtract the second sum from the first:
[tex]$$
90.9 - 30.8 = 60.1.
$$[/tex]
Step 4. Finally, subtract [tex]$8.6$[/tex]:
[tex]$$
60.1 - 8.6 = 51.5.
$$[/tex]
–––––––
Lana’s Calculation
She evaluates the expression without any extra grouping as
[tex]$$
53.7 + 37.2 - 3^3 + 3.8 - 8.6.
$$[/tex]
Step 1. As always, compute the exponent first:
[tex]$$
3^3 = 27.
$$[/tex]
Step 2. Now perform the addition and subtraction from left to right:
1. First, add:
[tex]$$
53.7 + 37.2 = 90.9.
$$[/tex]
2. Subtract [tex]$27$[/tex]:
[tex]$$
90.9 - 27 = 63.9.
$$[/tex]
3. Add [tex]$3.8$[/tex]:
[tex]$$
63.9 + 3.8 = 67.7.
$$[/tex]
4. Finally, subtract [tex]$8.6$[/tex]:
[tex]$$
67.7 - 8.6 = 59.1.
$$[/tex]
–––––––
Conclusion
Frederick obtains [tex]$51.5$[/tex], while Lana obtains [tex]$59.1$[/tex]. Both are mathematically correct because they followed different groupings:
- Frederick explicitly grouped the terms as
[tex]$$
\left[(53.7+37.2)-(3^3+3.8)\right]-8.6,
$$[/tex]
meaning he combined [tex]$3^3$[/tex] and [tex]$3.8$[/tex] together before subtracting from the sum of [tex]$53.7$[/tex] and [tex]$37.2$[/tex].
- Lana evaluated the expression in a left-to-right fashion (after computing the exponent), effectively grouping the operations differently.
Since subtraction is not associative, different groupings lead to different results. Thus, both answers are correct based on their interpretations of the expression.
