Understanding Operator Substitution Problems

This question involves substituting given symbols with standard mathematical operators and then evaluating the resulting expression. We need to be careful to follow the correct order of operations (BODMAS/PEMDAS) when evaluating the expression after substitution.

Defining the Operator Substitutions

The question provides the following rules for symbol substitution:

• '@' means 'addition' ($\text{+}$)
• '%' means 'multiplication' ($\times$)
• '$' means 'division' ($\div$)
• '#' means 'subtraction' ($-$)

Evaluating the Expression Step-by-Step

The given expression is: 126 $ 7 % 3 @ 19 # 21

First, let's substitute the symbols with the corresponding mathematical operators:

126 $\div$ 7 $\times$ 3 $\text{+}$ 19 $-$ 21

Now, we evaluate this expression following the BODMAS/PEMDAS rule, which dictates the order of operations:

• Brackets (Parentheses)
• Orders (Exponents, Roots)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

Let's apply the rule:

Step 1: Division

Perform the division first, as it appears before multiplication from left to right.

$\frac{126}{7} = 18$

The expression becomes:

18 $\times$ 3 $\text{+}$ 19 $-$ 21

Step 2: Multiplication

Next, perform the multiplication.

$18 \times 3 = 54$

The expression becomes:

54 $\text{+}$ 19 $-$ 21

Step 3: Addition and Subtraction

Finally, perform addition and subtraction from left to right.

First, Addition:

$54 + 19 = 73$

The expression becomes:

73 $-$ 21

Now, Subtraction:

$73 - 21 = 52$

The final value of the expression is 52.

Comparing with Options

Let's compare our calculated value with the given options:

• Option 1: 52
• Option 2: 18
• Option 3: 23
• Option 4: 4

Our calculated value, 52, matches Option 1.

Revision Table: Operator Substitution

Additional Information: Order of Operations (BODMAS/PEMDAS)

The order of operations is a set of rules used to clarify which procedures should be performed first in a given mathematical expression. The acronyms BODMAS and PEMDAS are helpful mnemonics.

• BODMAS: Brackets, Orders (powers, roots), Division, Multiplication, Addition, Subtraction.
• PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.

Division and Multiplication have the same priority and should be performed from left to right. Similarly, Addition and Subtraction have the same priority and should be performed from left to right.

In the expression 126 $\div$ 7 $\times$ 3 $\text{+}$ 19 $-$ 21, we performed division (126 $\div$ 7) first because it appeared before multiplication (18 $\times$ 3) when reading from left to right. Then we performed the multiplication. After completing all divisions and multiplications, we moved to addition and subtraction, again working from left to right (54 + 19, then 73 - 21).