Understanding Symbol Substitution Problems

This question asks us to evaluate a mathematical expression after changing the meaning of the operators. We are given specific rules for how each symbol should be interpreted differently from its usual mathematical meaning. This is a common type of problem designed to test careful reading and application of rules, along with basic arithmetic and the order of operations.

Step-by-Step Symbol Substitution

First, let's list the given symbol substitution rules:

• The symbol '+' now means 'subtraction' (-).
• The symbol '-' now means 'multiplication' ($\times$).
• The symbol '÷' now means 'addition' (+).
• The symbol '×' now means 'division' (÷).

The original expression is:

$$11 - 14 + 9 \times 3 \div 104$$

Now, we replace each original symbol with its new meaning according to the rules:

• Replace '-' with '$\times$'.
• Replace '+' with '-'.
• Replace '×' with '÷'.
• Replace '÷' with '+'.

Applying these substitutions, the expression becomes:

$$11 \times 14 - 9 \div 3 + 104$$

Evaluating the Modified Expression using BODMAS/PEMDAS

Now we need to calculate the value of the new expression: $$11 \times 14 - 9 \div 3 + 104$$

We must follow the standard order of operations, often remembered by the acronyms BODMAS or PEMDAS:

• B/P: Brackets or Parentheses
• O/E: Orders or Exponents
• D/M: Division or Multiplication (from left to right)
• A/S: Addition or Subtraction (from left to right)

In our expression, we have multiplication, division, subtraction, and addition. We perform multiplication and division first, from left to right.

First, the multiplication:

$$11 \times 14$$

Calculating this gives:

$$11 \times 14 = 154$$

Next, the division:

$$9 \div 3$$

Calculating this gives:

$$9 \div 3 = 3$$

Now, substitute these results back into the expression:

$$154 - 3 + 104$$

Finally, we perform addition and subtraction from left to right.

First, the subtraction:

$$154 - 3$$

Calculating this gives:

$$154 - 3 = 151$$

Finally, the addition:

$$151 + 104$$

Calculating this gives:

$$151 + 104 = 255$$

So, the value of the given expression after applying the symbol substitutions is 255.

Summary of Calculation Steps

The final value of the expression is 255.

Revision Table: Key Concepts

Additional Information on Order of Operations

Understanding the order of operations is fundamental in mathematics. BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) or PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) ensure that everyone gets the same result when evaluating an expression.

• Operations within Brackets/Parentheses are always done first.
• Powers or Exponents come next.
• Then, Division and Multiplication are done from left to right. It doesn't matter which comes first in the expression, you work from left to right.
• Finally, Addition and Subtraction are done from left to right. Similarly, perform these from left to right as they appear.

In this problem, after substitution, we had $11 \times 14 - 9 \div 3 + 104$. Following BODMAS/PEMDAS:

• No Brackets or Orders.
• Division and Multiplication: We perform $11 \times 14 = 154$ and $9 \div 3 = 3$.
• The expression becomes $154 - 3 + 104$.
• Addition and Subtraction: Work from left to right. $154 - 3 = 151$. Then $151 + 104 = 255$.

This systematic approach helps avoid errors in calculation.