Solving Arithmetic Expressions with Symbolic Representation

The problem provides an arithmetic expression where standard mathematical operators (+, -, ×, ÷) are replaced by letters (C, D, B, A respectively). We need to substitute the letters with their corresponding operators and then evaluate the expression following the correct order of operations.

Understanding the Symbol Mapping

The question gives us the following mapping of letters to mathematical symbols:

Rewriting the Expression

The given expression is: 23 C 13 B 841 A 29 D 91 = ?

Replacing the letters with their corresponding symbols, the expression becomes:

\(23 + 13 \times 841 \div 29 - 91 = ?\)

Applying the Order of Operations (BODMAS/PEMDAS)

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

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

In our expression \(23 + 13 \times 841 \div 29 - 91\), we have Addition, Multiplication, Division, and Subtraction. According to BODMAS, we perform Division and Multiplication first, from left to right, followed by Addition and Subtraction, from left to right.

Step-by-Step Calculation

Let's evaluate the expression step by step:

The expression is: \(23 + 13 \times 841 \div 29 - 91\)

Step 1: Perform Division (as it appears before multiplication from left to right in the 'DM' part of BODMAS).

\(841 \div 29\)

To calculate \(841 \div 29\):

• We can estimate: \(30 \times 30 = 900\), so the result should be close to 30.
• Let's try \(29 \times 20 = 580\). Remaining is \(841 - 580 = 261\).
• How many 29s are in 261? \(29 \times 10 = 290\), so it's less than 10. Let's try 9: \(29 \times 9 = (30 - 1) \times 9 = 270 - 9 = 261\).
• So, \(841 \div 29 = 29\).

Substitute this back into the expression:

\(23 + 13 \times 29 - 91\)

Step 2: Perform Multiplication

\(13 \times 29\)

To calculate \(13 \times 29\):

• \(13 \times 29 = 13 \times (30 - 1) = (13 \times 30) - (13 \times 1) = 390 - 13 = 377\).

Substitute this back into the expression:

\(23 + 377 - 91\)

Step 3: Perform Addition and Subtraction (from left to right)

First, perform Addition:

\(23 + 377\)

\(23 + 377 = 400\)

The expression is now:

\(400 - 91\)

Finally, perform Subtraction:

\(400 - 91\)

\(400 - 91 = 309\)

Thus, the value of the expression is 309.

Revision Table: Key Concepts

Additional Information: BODMAS/PEMDAS

The Order of Operations is a fundamental concept in mathematics to ensure consistent results when evaluating expressions. Without it, expressions could have multiple interpretations and results.

• B/P: Brackets/Parentheses are always evaluated first. Any operation inside brackets takes priority.
• O/E: Orders/Exponents come next. These include powers and square roots.
• DM: Division and Multiplication are performed from left to right. If both are present, the one appearing first from the left is done first.
• AS: Addition and Subtraction are performed last, also from left to right. If both are present, the one appearing first from the left is done first.

Understanding and correctly applying the order of operations is crucial for solving any complex mathematical expression accurately.