Solving Operator Substitution Problems

This problem requires us to substitute the given arithmetic operators with new ones and then evaluate the resulting expression following the standard order of operations.

Understanding the Operator Changes

The rules for changing the operators are:

• + means - (Subtraction)
• - means × (Multiplication)
• × means ÷ (Division)
• ÷ means + (Addition)

Applying the Substitution to the Expression

The given expression is:

\(16 + 4 - 3 \div 8 \times 6\)

Now, let's replace the original operators with the new ones according to the rules:

• Replace + with -
• Replace - with ×
• Replace ÷ with +
• Replace × with ÷

The expression becomes:

\(16 - 4 \times 3 + 8 \div 6\)

Evaluating the New Expression Using Order of Operations (BODMAS/PEMDAS)

We need to evaluate the expression \(16 - 4 \times 3 + 8 \div 6\) following the order of operations:

1. Brackets/Parentheses (None in this expression)
2. Orders/Exponents (None in this expression)
3. Division and Multiplication (from left to right)
4. Addition and Subtraction (from left to right)

Step 1: Perform Division and Multiplication

First, let's perform the multiplication \(4 \times 3\):

\(4 \times 3 = 12\)

The expression is now:

\(16 - 12 + 8 \div 6\)

Next, perform the division \(8 \div 6\):

\(8 \div 6 = \frac{8}{6} = \frac{4}{3}\)

The expression is now:

\(16 - 12 + \frac{4}{3}\)

Step 2: Perform Addition and Subtraction from Left to Right

First, perform the subtraction \(16 - 12\):

\(16 - 12 = 4\)

The expression is now:

\(4 + \frac{4}{3}\)

Finally, perform the addition \(4 + \frac{4}{3}\). To add these, we find a common denominator, which is 3:

\(4 + \frac{4}{3} = \frac{4 \times 3}{3} + \frac{4}{3} = \frac{12}{3} + \frac{4}{3}\)

Now add the fractions:

\(\frac{12}{3} + \frac{4}{3} = \frac{12 + 4}{3} = \frac{16}{3}\)

Final Result

The value of the expression after substitution and evaluation is \(\frac{16}{3}\).

Revision Table: Operator Substitution Rules

Additional Information: Understanding BODMAS/PEMDAS

BODMAS and PEMDAS are acronyms used to remember the order of operations in mathematical expressions. Following this order ensures that everyone gets the same result when evaluating an expression.

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

In our problem, we first performed division (\(8 \div 6\)) and multiplication (\(4 \times 3\)) before doing subtraction (\(16 - 12\)) and addition (\(4 + 4/3\)). Division and multiplication are at the same level, as are addition and subtraction, so you perform them as they appear from left to right in the expression.