Solving Mathematical Equations with Symbol Substitution

This question involves replacing symbols with mathematical operators and then solving the resulting equation to find the missing operator.

The given symbol mapping is:

• A denotes '+' (addition)
• B denotes '×' (multiplication)
• C denotes '−' (subtraction)
• D denotes '÷' (division)

The equation provided is:

\(196 \text{ D } (2 \text{ A } 3 \text{ B } 4) = 4 \text{ ? } 5 \text{ B } 2\)

We need to find the operator that replaces '?' to make the equation true.

Step-by-Step Solution: Evaluating the Equation

Let's substitute the symbols with the actual mathematical operators in the given equation:

\(196 \div (2 + 3 \times 4) = 4 \text{ ? } 5 \times 2\)

Evaluate the Left Side of the Equation: \(196 \div (2 + 3 \times 4)\)

We follow the order of operations (BODMAS/PEMDAS): Brackets/Parentheses first, then Orders/Exponents, then Division and Multiplication (from left to right), and finally Addition and Subtraction (from left to right).

1. First, evaluate the expression inside the parentheses: \( (2 + 3 \times 4) \).
2. Inside the parentheses, perform multiplication before addition: \(3 \times 4 = 12\).
3. So, the expression inside the parentheses becomes: \( (2 + 12) \).
4. Perform the addition inside the parentheses: \(2 + 12 = 14\).
5. Now substitute this value back into the left side of the main equation: \(196 \div 14\).
6. Perform the division: \(196 \div 14 = 14\).

So, the value of the left side of the equation is 14.

Evaluate the Right Side of the Equation: \(4 \text{ ? } 5 \times 2\)

We have the expression \(4 \text{ ? } 5 \times 2\). We need to perform the multiplication before the operation denoted by '?'.

1. Perform the multiplication: \(5 \times 2 = 10\).
2. So, the right side of the equation becomes: \(4 \text{ ? } 10\).

Equating Both Sides and Finding the Missing Operator

Now we equate the value of the left side (14) with the simplified right side (\(4 \text{ ? } 10\)):

\(14 = 4 \text{ ? } 10\)

We need to find which operation (A, B, C, or D) placed between 4 and 10 gives the result 14.

• If '?' is A ('+'): \(4 + 10 = 14\). This matches the left side.
• If '?' is B ('×'): \(4 \times 10 = 40\). This does not match 14.
• If '?' is C ('−'): \(4 - 10 = -6\). This does not match 14.
• If '?' is D ('÷'): \(4 \div 10 = 0.4\). This does not match 14.

The operation that satisfies the equation is addition '+', which is denoted by the symbol A.

Therefore, the symbol that will come in place of '?' is A.

The operation required is addition, which corresponds to the symbol A.

Revision Table: Key Concepts

Additional Information: Order of Operations Details

The order of operations ensures that everyone gets the same result when evaluating a mathematical expression. The commonly used mnemonics BODMAS and PEMDAS represent the order:

• BODMAS: Brackets, Orders (powers/roots), Division and Multiplication (left to right), Addition and Subtraction (left to right).
• PEMDAS: Parentheses, Exponents (powers/roots), Multiplication and Division (left to right), Addition and Subtraction (left to right).

In the problem \(196 \div (2 + 3 \times 4)\), we first calculated within the Brackets/Parentheses. Inside the brackets \( (2 + 3 \times 4) \), we performed Multiplication \(3 \times 4 = 12\) before Addition \(2 + 12 = 14\). Finally, we performed the Division \(196 \div 14 = 14\). Adhering to this order is vital for accuracy in solving such problems.