Sure, let's evaluate the expression and then understand why Harper's representation is incorrect.
### Step-by-Step Evaluation
1. Evaluating Harper's Expression:
- Harper wrote the expression [tex]\(15 - 6 + 7\)[/tex].
- According to the order of operations (PEMDAS/BODMAS), we should proceed with addition and subtraction from left to right.
- So, we first evaluate [tex]\(15 - 6\)[/tex]:
[tex]\[
15 - 6 = 9
\][/tex]
- Next, we add 7 to the result:
[tex]\[
9 + 7 = 16
\][/tex]
- Hence, the value of Harper's expression [tex]\(15 - 6 + 7\)[/tex] is 16.
2. Understanding the Correct Interpretation:
- Harper intended to represent "15 minus the sum of 6 and 7". Mathematically, this should be written as [tex]\(15 - (6 + 7)\)[/tex].
- To evaluate this correctly, we need to perform the addition inside the parentheses first:
[tex]\[
6 + 7 = 13
\][/tex]
- Then, we subtract this result from 15:
[tex]\[
15 - 13 = 2
\][/tex]
- Therefore, the value of the correct expression [tex]\(15 - (6 + 7)\)[/tex] is 2.
### Explanation of Why Harper's Expression is Incorrect
Harper's expression, [tex]\(15 - 6 + 7\)[/tex], does not accurately represent the phrase "15 minus the sum of 6 and 7". This is because of the absence of parentheses, which are crucial in modifying the standard order of operations.
- Without parentheses, the expression [tex]\(15 - 6 + 7\)[/tex] is interpreted naturally by performing the operations from left to right.
- With the correct use of parentheses as [tex]\(15 - (6 + 7)\)[/tex], the addition inside the parentheses is performed first, which changes the outcome dramatically.
In summary, while Harper calculated [tex]\(15 - 6 + 7 = 16\)[/tex], the intended expression [tex]\(15 - (6 + 7)\)[/tex] correctly evaluates to 2. Hence, Harper's expression is incorrect due to the missing parentheses that should group the sum of 6 and 7 together before the subtraction from 15.
