Solving Symbol Substitution Mathematical Expressions

This problem requires us to decode a mathematical expression by replacing symbols with the arithmetic operators they represent and then calculating the result according to the standard order of operations.

Understanding the Symbol Meanings

We are given a specific meaning for each symbol:

• Symbol I represents the division operation (\div).
• Symbol J represents the multiplication operation (\times).
• Symbol K represents the subtraction operation (-).
• Symbol L represents the addition operation (+).

Translating the Given Expression

The expression provided is:

29 J 7 K 168 I 4 L 71 K 39 L 14 J 2 = ?

Let's replace each symbol with its corresponding mathematical operator:

29 \times 7 - 168 \div 4 + 71 - 39 + 14 \times 2 = ?

Calculating the Expression Using Order of Operations (BODMAS/PEMDAS)

To solve this mathematical expression correctly, we must follow the order of operations. The common acronyms for this order are BODMAS or PEMDAS:

1. Brackets / Parentheses
2. Orders (powers, square roots, etc.) / Exponents
3. Division and Multiplication (from left to right)
4. Addition and Subtraction (from left to right)

Let's evaluate the translated expression step-by-step:

29 \times 7 - 168 \div 4 + 71 - 39 + 14 \times 2

Step 1: Perform Division and Multiplication

According to the order of operations, we first perform division and multiplication from left to right:

• Calculate the division: 168 \div 4 = 42
• Calculate the first multiplication: 29 \times 7 = 203
• Calculate the second multiplication: 14 \times 2 = 28

Substitute these values back into the expression:

203 - 42 + 71 - 39 + 28

Step 2: Perform Addition and Subtraction

Now we perform addition and subtraction from left to right:

• 203 - 42 = 161
• 161 + 71 = 232
• 232 - 39 = 193
• 193 + 28 = 221

Alternatively, you can group positive and negative numbers:

Positive terms: 203 + 71 + 28 = 302

Negative terms: -42 - 39 = -(42 + 39) = -81

Combine the results: 302 - 81 = 221

Both methods yield the same result.

Final Answer for the Expression

After substituting the symbols and performing the calculations following the order of operations, the value of the expression is 221.

Therefore, 221 will come in place of '?'.

Revision Table: Summary of Symbol Decoding

Additional Information: The Importance of Order of Operations

The order of operations is crucial in mathematics because it ensures that everyone gets the same answer when evaluating a given expression. Without a standard order, the same expression could be interpreted in different ways, leading to different results.

For example, if we calculated 2 + 3 \times 4 without following the order of operations, we might do the addition first (2 + 3 = 5) and then the multiplication (5 \times 4 = 20). However, following the order (multiplication before addition), we do the multiplication first (3 \times 4 = 12) and then the addition (2 + 12 = 14). The correct answer is 14.

Understanding and applying BODMAS/PEMDAS is fundamental for solving mathematical problems accurately.