Solving Symbol Operator Expressions

This problem requires us to substitute given symbols with their corresponding arithmetic operators and then evaluate the expression to find the missing operator that makes the equation true.

Understanding the Operator Symbols

We are given a mapping of symbols to operators:

• 'P' means '\(\div\)' (Division)
• 'Q' means '\(+\)' (Addition)
• 'R' means '\(-\)' (Subtraction)
• 'S' means '\(\times\)' (Multiplication)

Analyzing the Given Expression

The expression is:

34 P 17 S 9 ? 13 R 11 S 3 R 4 = -6

Let's substitute the known operators into the expression:

34 \(\div\) 17 \(\times\) 9 ? 13 \(- \) 11 \(\times\) 3 \(- \) 4 = -6

We need to determine which operator should replace '?' from the given options (P, Q, R, S) to make the equation equal to -6.

Evaluating Each Option for the Missing Operator

We will test each option by substituting the symbol for '?' and evaluating the expression using the order of operations (BODMAS/PEMDAS).

Option 1: If '?' is 'P' (Division)

The expression becomes: 34 \(\div\) 17 \(\times\) 9 \(\div\) 13 \(- \) 11 \(\times\) 3 \(- \) 4

Let's evaluate:

1. Division/Multiplication (from left to right):
2. \(34 \div 17 = 2\)
3. Expression is now: \(2 \times 9 \div 13 - 11 \times 3 - 4\)
4. \(2 \times 9 = 18\)
5. Expression is now: \(18 \div 13 - 11 \times 3 - 4\)
6. \(11 \times 3 = 33\)
7. Expression is now: \(18 \div 13 - 33 - 4\)
8. \(18 \div 13 \approx 1.38\) (using fractions: \(18/13\))
9. Expression is now: \(18/13 - 33 - 4\)
10. Subtraction (from left to right):
11. \(18/13 - 33 = (18 - 33 \times 13) / 13 = (18 - 429) / 13 = -411/13\)
12. Expression is now: \(-411/13 - 4\)
13. \(-411/13 - 4 = (-411 - 4 \times 13) / 13 = (-411 - 52) / 13 = -463/13\)

\(-463/13 \neq -6\). So, Option 1 is incorrect.

Option 2: If '?' is 'Q' (Addition)

The expression becomes: 34 \(\div\) 17 \(\times\) 9 \(+\) 13 \(- \) 11 \(\times\) 3 \(- \) 4

Let's evaluate using BODMAS/PEMDAS:

1. Division/Multiplication (from left to right):
2. \(34 \div 17 = 2\)
3. Expression is now: \(2 \times 9 + 13 - 11 \times 3 - 4\)
4. \(2 \times 9 = 18\)
5. Expression is now: \(18 + 13 - 11 \times 3 - 4\)
6. \(11 \times 3 = 33\)
7. Expression is now: \(18 + 13 - 33 - 4\)
8. Addition/Subtraction (from left to right):
9. \(18 + 13 = 31\)
10. Expression is now: \(31 - 33 - 4\)
11. \(31 - 33 = -2\)
12. Expression is now: \(-2 - 4\)
13. \(-2 - 4 = -6\)

\(-6 = -6\). This matches the required result. So, Option 2 is likely the correct answer.

Option 3: If '?' is 'R' (Subtraction)

The expression becomes: 34 \(\div\) 17 \(\times\) 9 \(- \) 13 \(- \) 11 \(\times\) 3 \(- \) 4

Let's evaluate using BODMAS/PEMDAS:

1. Division/Multiplication (from left to right):
2. \(34 \div 17 = 2\)
3. Expression is now: \(2 \times 9 - 13 - 11 \times 3 - 4\)
4. \(2 \times 9 = 18\)
5. Expression is now: \(18 - 13 - 11 \times 3 - 4\)
6. \(11 \times 3 = 33\)
7. Expression is now: \(18 - 13 - 33 - 4\)
8. Addition/Subtraction (from left to right):
9. \(18 - 13 = 5\)
10. Expression is now: \(5 - 33 - 4\)
11. \(5 - 33 = -28\)
12. Expression is now: \(-28 - 4\)
13. \(-28 - 4 = -32\)

\(-32 \neq -6\). So, Option 3 is incorrect.

Option 4: If '?' is 'S' (Multiplication)

The expression becomes: 34 \(\div\) 17 \(\times\) 9 \(\times\) 13 \(- \) 11 \(\times\) 3 \(- \) 4

Let's evaluate using BODMAS/PEMDAS:

1. Division/Multiplication (from left to right):
2. \(34 \div 17 = 2\)
3. Expression is now: \(2 \times 9 \times 13 - 11 \times 3 - 4\)
4. \(2 \times 9 = 18\)
5. Expression is now: \(18 \times 13 - 11 \times 3 - 4\)
6. \(18 \times 13 = 234\)
7. Expression is now: \(234 - 11 \times 3 - 4\)
8. \(11 \times 3 = 33\)
9. Expression is now: \(234 - 33 - 4\)
10. Addition/Subtraction (from left to right):
11. \(234 - 33 = 201\)
12. Expression is now: \(201 - 4\)
13. \(201 - 4 = 197\)

\(197 \neq -6\). So, Option 4 is incorrect.

Conclusion

Based on the evaluation of each option, substituting 'Q' (Addition) for the '?' results in the correct value of -6 for the expression. Therefore, the symbol that comes in place of '?' is 'Q'.

Revision Table: Symbol Substitution and BODMAS

Additional Information: Understanding BODMAS/PEMDAS

The order of operations is crucial when evaluating mathematical expressions with multiple operators. The widely used acronyms are BODMAS and PEMDAS.

• BODMAS: Brackets, Orders (powers/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).

Both acronyms represent the same order. Division and Multiplication have the same priority and should be performed from left to right as they appear in the expression. Similarly, Addition and Subtraction have the same priority and are also performed from left to right.

In our problem, we first performed division (\(34 \div 17\)), then multiplications (\(2 \times 9\) and \(11 \times 3\)), and finally additions and subtractions from left to right (\(18 + 13 - 33 - 4\)). This systematic approach ensures the correct evaluation of the expression.