Solving Equations with Symbol Replacement

The question asks us to substitute specific mathematical operators for given letters in an equation and then solve for the missing value, denoted by '?'.

The mapping of letters to operators is provided as:

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

The original equation given is:

(9 A 9 C 55 D 5) B ? = 10 B 3 A 5

Let's replace the letters in the equation with their corresponding mathematical operators:

$$(9 + 9 - 55 \div 5) \times ? = 10 \times 3 + 5$$

To solve this equation, we must follow the standard order of operations, often remembered by the acronyms BODMAS or PEMDAS.

• B/P: Brackets/Parentheses
• O/E: Orders/Exponents (powers, square roots)
• DM: Division and Multiplication (from left to right)
• AS: Addition and Subtraction (from left to right)

Step-by-Step Calculation

First, we evaluate the expression inside the parentheses on the left side of the equation:

$$(9 + 9 - 55 \div 5)$$

Within the parentheses, we follow the order of operations. Division comes before addition and subtraction.

Perform the division:

$$55 \div 5 = 11$$

Substitute this result back into the expression inside the parentheses:

$$(9 + 9 - 11)$$

Now, perform the addition and subtraction from left to right:

Perform the addition:

$$9 + 9 = 18$$

Perform the subtraction:

$$18 - 11 = 7$$

So, the expression inside the parentheses evaluates to 7.

The left side of the main equation simplifies to:

$$7 \times ?$$

Next, we evaluate the right side of the main equation:

$$10 \times 3 + 5$$

According to BODMAS/PEMDAS, multiplication comes before addition.

Perform the multiplication:

$$10 \times 3 = 30$$

Perform the addition:

$$30 + 5 = 35$$

So, the right side of the equation evaluates to 35.

Now, we have the simplified equation:

$$7 \times ? = 35$$

To find the value of '?', we need to isolate '?' by performing the inverse operation of multiplication, which is division. Divide both sides of the equation by 7:

$$? = \frac{35}{7}$$

$$? = 5$$

Thus, the number that comes in place of '?' is 5.

Revision Table: Operator Mapping Summary

Additional Information: Importance of Order of Operations (BODMAS/PEMDAS)

The order of operations is a fundamental concept in mathematics that ensures consistency in evaluating expressions. Without a standard order, the same expression could yield different results. BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) or PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) provides this standard.

• Always tackle operations inside brackets or parentheses first.
• Then, evaluate any exponents or orders.
• Perform multiplication and division from left to right.
• Finally, perform addition and subtraction from left to right.

Understanding and applying this rule is essential for solving various mathematical problems correctly, including algebraic equations and complex arithmetic expressions.