Understanding the Symbol Substitution Problem

This question requires us to solve a mathematical expression where the usual meaning of arithmetic symbols is changed. We are given a set of rules for symbol substitution, and we need to apply these rules first before performing the calculations according to the standard order of operations.

Applying the New Rules to the Equation

The original equation is: \(54 \times 6 – 7 \div 8 + 2\)

The symbol substitution rules are:

• ‘÷’ means ‘addition’ (\(+)\)
• ‘+’ means ‘subtraction’ (\(-\))
• ‘-‘ means ‘multiplication’ (\(\times\))
• ‘×’ means ‘division’ (\(\div\))

Let's replace the symbols in the original equation based on these rules:

• The symbol ‘×’ between 54 and 6 becomes ‘÷’ (division).
• The symbol ‘–’ between 6 and 7 becomes ‘×’ (multiplication).
• The symbol ‘÷’ between 7 and 8 becomes ‘+’ (addition).
• The symbol ‘+’ between 8 and 2 becomes ‘–’ (subtraction).

So, the new equation after applying the symbol substitution is:

\(54 \div 6 \times 7 + 8 - 2\)

Solving the Mathematical Expression with Order of Operations

Now we need to solve the new equation \(54 \div 6 \times 7 + 8 - 2\). We follow the order of operations, often remembered by acronyms like BODMAS or PEMDAS.

• B/P: Brackets/Parentheses (None in this equation)
• O/E: Orders/Exponents (None in this equation)
• DM: Division and Multiplication (from left to right)
• AS: Addition and Subtraction (from left to right)

Let's solve step-by-step:

1. Perform division: \(54 \div 6 = 9\). The equation becomes \(9 \times 7 + 8 - 2\).
2. Perform multiplication: \(9 \times 7 = 63\). The equation becomes \(63 + 8 - 2\).
3. Perform addition: \(63 + 8 = 71\). The equation becomes \(71 - 2\).
4. Perform subtraction: \(71 - 2 = 69\).

The final value of the equation after symbol substitution and calculation is 69.

Final Answer Calculation Summary

Here is a summary of the steps:

The value of the given equation after applying the symbol substitution rules is 69.

Revision Table: Symbol Substitution and Arithmetic

Additional Information: Order of Operations

The order of operations is crucial in solving mathematical expressions correctly. It ensures that everyone gets the same result from the same expression. The widely used order is:

• First, calculate anything inside Brackets or Parentheses.
• Next, calculate Exponents or Orders (like square roots, powers).
• Then, perform Division and Multiplication, working from left to right.
• Finally, perform Addition and Subtraction, working from left to right.

Remembering this order helps avoid errors in calculations, especially when multiple operations are involved, such as in this symbol substitution problem.