Understanding Number Swapping in Equations

The question asks us to identify which two numbers in the given equation, 13 + 9 ÷ 78 × 27 - 16 = 3, should be swapped to make the equation mathematically correct.

To solve this, we need to consider the order of operations (BODMAS/PEMDAS) and test the given options by swapping the specified numbers and evaluating the resulting equation.

Applying Order of Operations (BODMAS/PEMDAS)

The order of operations tells us the sequence in which mathematical operations should be performed:

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

Testing the Swapping Option: 3 and 78

Let's swap the numbers 3 and 78 in the original equation: 13 + 9 ÷ 78 × 27 - 16 = 3

The equation becomes: 13 + 9 ÷ 3 × 27 - 16 = 78

Now, let's evaluate the left side of this new equation step-by-step using the BODMAS/PEMDAS rule:

1. Division: Perform the division operation first. $\text{9} \div \text{3} = \text{3}$. The equation is now: $13 + 3 \times 27 - 16 = 78$
2. Multiplication: Next, perform the multiplication. $\text{3} \times \text{27} = \text{81}$. The equation is now: $13 + 81 - 16 = 78$
3. Addition: Perform the addition from left to right. $\text{13} + \text{81} = \text{94}$. The equation is now: $94 - 16 = 78$
4. Subtraction: Finally, perform the subtraction. $\text{94} - \text{16} = \text{78}$. The equation is now: $78 = 78$

Since the left side of the equation ($78$) equals the right side ($78$) after swapping 3 and 78, this swap makes the equation correct.

Conclusion on Swapping Numbers

Interchanging the numbers 3 and 78 in the given equation 13 + 9 ÷ 78 × 27 - 16 = 3 results in the correct equation 13 + 9 ÷ 3 × 27 - 16 = 78. Therefore, this is the correct pair of numbers to interchange.

Revision Table: Key Concepts

Additional Information: Understanding BODMAS

BODMAS (or PEMDAS) is a crucial rule in mathematics to ensure everyone gets the same answer when evaluating expressions involving multiple operations. It provides a standard order:

• B/P: Brackets/Parentheses - Calculations inside grouping symbols first.
• O/E: Orders/Exponents - Powers and roots next.
• DM: Division and Multiplication - These are done from left to right as they appear.
• AS: Addition and Subtraction - These are done from left to right as they appear after division and multiplication.

Following this order is essential when simplifying expressions or checking equations like in this problem.