3. Find the values of the following expressions. For each pair, first try to guess whether they have the same value. When are the two expressions equal? (a) and (b) and (c) and

Understand the Question To evaluate expressions with parentheses, always compute the operations inside the parentheses first (BODMAS / PEMDAS rule). Associative property: Addition is associative, meaning regrouping terms does not change the result, but subtraction is not associative (). Distributing a negative sign across a sum changes the signs inside: , making . Part (a) Find the values of and , and determine if they are equal. Step 1 · Evaluate both expressions Evaluating the first expression: Evaluating the second expression: Both expressions yield the same result. Answer > (a) Both expressions equal . They have the same value. Part (b) Find the values of and , and determine if they are equal. Step 1 · Evaluate both expressions Evaluating the first expression: Evaluating the second expression: Since , the expressions are not equal. Answer > (b) and . They do not have the same value. Part (c) Find the values of and , and determine if they are equal. Step 1 · Evaluate both expressions Evaluating the first expression: Evaluating the second expression: Both expressions yield the same result. Answer > (c) Both expressions equal . They have the same value. Common Mistakes Assuming Subtraction is Associative: Incorrectly assuming . Notice in part (b) that while . Sign Errors When Removing Parentheses: Forgetting that subtracting a sum is equivalent to adding both negative terms: . Ignoring Parentheses Priority: Performing operations from left to right without resolving operations inside brackets first.

Arithmetic Expressions

Question 3

Find the values of the following expressions. For each pair, first try to guess whether they have the same value. When are the two expressions equal?

(a) $(6 + 10) - 2$ and $6 + (10 - 2)$

(b) $16 - (8 - 3)$ and $(16 - 8) - 3$

Solution

Understand the Question

To evaluate expressions with parentheses, always compute the operations inside the parentheses first (BODMAS / PEMDAS rule).
Associative property: Addition is associative, meaning regrouping terms does not change the result, but subtraction is not associative ( $a - (b - c) \neq (a - b) - c$ ).
Distributing a negative sign across a sum changes the signs inside: $-(b + c) = -b - c$ , making $a - (b + c) = a + (-b - c)$ .

(a) Find the values of $(6 + 10) - 2$ and $6 + (10 - 2)$ , and determine if they are equal.

Step 1 · Evaluate both expressions

Evaluating the first expression:

\begin{aligned} (6 + 10) - 2 &= 16 - 2 \\ &= 14 \end{aligned}

Evaluating the second expression:

\begin{aligned} 6 + (10 - 2) &= 6 + 8 \\ &= 14 \end{aligned}

Both expressions yield the same result.

Answer

(a) Both expressions equal $14$ . They have the same value.

(b) Find the values of $16 - (8 - 3)$ and $(16 - 8) - 3$ , and determine if they are equal.

Step 1 · Evaluate both expressions

Evaluating the first expression:

\begin{aligned} 16 - (8 - 3) &= 16 - 5 \\ &= 11 \end{aligned}

Evaluating the second expression:

\begin{aligned} (16 - 8) - 3 &= 8 - 3 \\ &= 5 \end{aligned}

Since $11 \neq 5$ , the expressions are not equal.

Answer

(b) $16 - (8 - 3) = 11$ and $(16 - 8) - 3 = 5$ . They do not have the same value.

(c) Find the values of $27 - (18 + 4)$ and $27 + (-18 - 4)$ , and determine if they are equal.

Step 1 · Evaluate both expressions

Evaluating the first expression:

\begin{aligned} 27 - (18 + 4) &= 27 - 22 \\ &= 5 \end{aligned}

Evaluating the second expression:

\begin{aligned} 27 + (-18 - 4) &= 27 + (-22) \\ &= 5 \end{aligned}

Both expressions yield the same result.

Answer

(c) Both expressions equal $5$ . They have the same value.

Common Mistakes

Assuming Subtraction is Associative: Incorrectly assuming $a - (b - c) = (a - b) - c$ . Notice in part (b) that $16 - (8 - 3) = 11$ while $(16 - 8) - 3 = 5$ .
Sign Errors When Removing Parentheses: Forgetting that subtracting a sum is equivalent to adding both negative terms: $-(18 + 4) = -18 - 4$ .
Ignoring Parentheses Priority: Performing operations from left to right without resolving operations inside brackets first.

Question 3

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