Understanding the Equation and the Problem

The question asks us to find which pair of numbers, when swapped in the given equation, makes the equation mathematically correct. The original equation is:

$$78 \div 48 \times 8 + (26 \times 7) – 39 + (45 + 15) = 210$$

To solve this, we need to test each given option. Each option suggests interchanging two numbers. We will substitute the numbers as suggested by each option and then evaluate the equation using the order of operations (BODMAS/PEMDAS) to see if the left side equals 210.

Checking Option 1: Interchange 26 and 15

If we interchange 26 and 15, the equation becomes:

$$78 \div 48 \times 8 + (15 \times 7) – 39 + (45 + 26)$$

Let's evaluate this expression:

• First, calculate the values inside the parentheses: $15 \times 7 = 105$ and $45 + 26 = 71$.
• The equation is now: $$78 \div 48 \times 8 + 105 – 39 + 71$$
• Next, perform division and multiplication from left to right: $78 \div 48 = 1.625$.
• The equation is now: $$1.625 \times 8 + 105 – 39 + 71$$
• Next, multiplication: $1.625 \times 8 = 13$.
• The equation is now: $$13 + 105 – 39 + 71$$
• Finally, perform addition and subtraction from left to right:
• $$13 + 105 = 118$$
• $$118 – 39 = 79$$
• $$79 + 71 = 150$$

The result is 150, which is not equal to 210. So, interchanging 26 and 15 does not make the equation correct.

Checking Option 2: Interchange 45 and 48

If we interchange 45 and 48, the equation becomes:

$$78 \div 45 \times 8 + (26 \times 7) – 39 + (48 + 15)$$

Let's evaluate this expression:

• First, calculate the values inside the parentheses: $26 \times 7 = 182$ and $48 + 15 = 63$.
• The equation is now: $$78 \div 45 \times 8 + 182 – 39 + 63$$
• Next, perform division and multiplication from left to right: $78 \div 45 \approx 1.733$.
• The equation is now: $$1.733 \times 8 + 182 – 39 + 63$$
• Next, multiplication: $1.733 \times 8 \approx 13.864$.
• The equation is now: $$13.864 + 182 – 39 + 63$$
• Finally, perform addition and subtraction from left to right:
• $$13.864 + 182 = 195.864$$
• $$195.864 – 39 = 156.864$$
• $$156.864 + 63 = 219.864$$

The result is approximately 219.864, which is not equal to 210. So, interchanging 45 and 48 does not make the equation correct.

Checking Option 3: Interchange 48 and 39

If we interchange 48 and 39, the equation becomes:

$$78 \div 39 \times 8 + (26 \times 7) – 48 + (45 + 15)$$

Let's evaluate this expression:

• First, calculate the values inside the parentheses: $26 \times 7 = 182$ and $45 + 15 = 60$.
• The equation is now: $$78 \div 39 \times 8 + 182 – 48 + 60$$
• Next, perform division and multiplication from left to right: $78 \div 39 = 2$.
• The equation is now: $$2 \times 8 + 182 – 48 + 60$$
• Next, multiplication: $2 \times 8 = 16$.
• The equation is now: $$16 + 182 – 48 + 60$$
• Finally, perform addition and subtraction from left to right:
• $$16 + 182 = 198$$
• $$198 – 48 = 150$$
• $$150 + 60 = 210$$

The result is 210, which is equal to the right side of the equation. So, interchanging 48 and 39 makes the equation correct.

Checking Option 4: Interchange 78 and 45

If we interchange 78 and 45, the equation becomes:

$$45 \div 48 \times 8 + (26 \times 7) – 39 + (78 + 15)$$

Let's evaluate this expression:

• First, calculate the values inside the parentheses: $26 \times 7 = 182$ and $78 + 15 = 93$.
• The equation is now: $$45 \div 48 \times 8 + 182 – 39 + 93$$
• Next, perform division and multiplication from left to right: $45 \div 48 = 0.9375$.
• The equation is now: $$0.9375 \times 8 + 182 – 39 + 93$$
• Next, multiplication: $0.9375 \times 8 = 7.5$.
• The equation is now: $$7.5 + 182 – 39 + 93$$
• Finally, perform addition and subtraction from left to right:
• $$7.5 + 182 = 189.5$$
• $$189.5 – 39 = 150.5$$
• $$150.5 + 93 = 243.5$$

The result is 243.5, which is not equal to 210. So, interchanging 78 and 45 does not make the equation correct.

Conclusion: Finding the Correct Interchange

After checking all options, we found that interchanging 48 and 39 results in a correct equation ($210 = 210$). Therefore, the numbers 48 and 39 should be interchanged.

Revision Table: Checking Equation Interchanges

Additional Information: Order of Operations (BODMAS/PEMDAS)

When solving mathematical equations like this, it is essential to follow the correct order of operations. This ensures that everyone gets the same result for the same expression. The common mnemonics for the order of operations are BODMAS or PEMDAS.

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

In the given equation, we first evaluated the expressions within the parentheses, then performed division and multiplication from left to right, and finally addition and subtraction from left to right.