Find Which Numbers to Interchange in Expression

The problem asks us to identify which two numbers in the given mathematical expression must be interchanged so that the final value of the expression becomes exactly 81.

The original expression is: $$798 + 27 – 719 \div 6 \times 5$$

We need to evaluate this expression after interchanging the numbers specified in each option, following the standard order of operations (BODMAS/PEMDAS):

• Brackets first
• Orders (powers and square roots)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

Testing Option 1: Interchange 5 and 27

If we interchange 5 and 27, the expression becomes:

$$798 + 5 – 719 \div 6 \times 27$$

Let's evaluate this step by step:

1. Perform Division: $$719 \div 6 \approx 119.833$$
2. Perform Multiplication: $$119.833 \times 27 \approx 3235.491$$
3. Perform Addition: $$798 + 5 = 803$$
4. Perform Subtraction: $$803 – 3235.491 \approx -2432.491$$

The result is approximately -2432.491, which is not 81.

Testing Option 2: Interchange 27 and 6

If we interchange 27 and 6, the expression becomes:

$$798 + 6 – 719 \div 27 \times 5$$

Let's evaluate this step by step:

1. Perform Division: $$719 \div 27 \approx 26.6296$$
2. Perform Multiplication: $$26.6296 \times 5 \approx 133.148$$
3. Perform Addition: $$798 + 6 = 804$$
4. Perform Subtraction: $$804 – 133.148 \approx 670.852$$

The result is approximately 670.852, which is not 81.

Testing Option 3: Interchange 798 and 719

If we interchange 798 and 719, the expression becomes:

$$719 + 27 – 798 \div 6 \times 5$$

Let's evaluate this step by step:

1. Perform Division: $$798 \div 6 = 133$$
2. Perform Multiplication: $$133 \times 5 = 665$$
3. Perform Addition: $$719 + 27 = 746$$
4. Perform Subtraction: $$746 – 665 = 81$$

The result is exactly 81. This matches the target value.

Testing Option 4: Interchange 6 and 5

If we interchange 6 and 5, the expression becomes:

$$798 + 27 – 719 \div 5 \times 6$$

Let's evaluate this step by step:

1. Perform Division: $$719 \div 5 = 143.8$$
2. Perform Multiplication: $$143.8 \times 6 = 862.8$$
3. Perform Addition: $$798 + 27 = 825$$
4. Perform Subtraction: $$825 – 862.8 = -37.8$$

The result is -37.8, which is not 81.

Conclusion

Based on the evaluation of each option, interchanging the numbers 798 and 719 results in the expression evaluating to 81.

Revision Table: Expression Evaluation

Let's summarize the results of interchanging numbers on the expression value:

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

The order of operations is a set of rules used to clarify which procedures should be performed first in a given mathematical expression. Without this order, expressions could have multiple interpretations, leading to different results.

• BODMAS: Brackets, Orders, Division/Multiplication, Addition/Subtraction.
• PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.

These acronyms represent the same hierarchy. Division and Multiplication are performed at the same level, from left to right as they appear in the expression. Similarly, Addition and Subtraction are performed at the same level, also from left to right.

In the given expression, $$798 + 27 – 719 \div 6 \times 5$$, we first perform division ($$\div$$) and multiplication ($$\times$$) from left to right, and then addition ($$+$$) and subtraction ($$–$$) from left to right.

Example of order for $$719 \div 6 \times 5$$: First $$719 \div 6$$, then multiply the result by 5.

Example of order for $$719 \div 5 \times 6$$: First $$719 \div 5$$, then multiply the result by 6.