Solving Mathematical Sign Puzzle

The problem asks us to find the correct combination of mathematical signs that, when placed sequentially in the equation 68 * 138 * 23 * 54 * 20, will make the equation true. We are given four options, each providing a sequence of four signs.

We need to test each option by replacing the asterisks (*) from left to right with the signs provided in the option and then evaluating the resulting expression. Remember the order of operations (BODMAS/PEMDAS): Brackets, Orders (powers/roots), Division, Multiplication, Addition, Subtraction.

Evaluating Option 1: +, ÷, −, =

Let's replace the asterisks with the signs from Option 1: +, ÷, −, and =.

The equation becomes: 68 + 138 ÷ 23 − 54 = 20

Now, we evaluate the left side of the equation following the order of operations:

1. Division: Calculate 138 ÷ 23.

   \( 138 \div 23 = 6 \)

   The equation is now: 68 + 6 − 54 = 20

2. Addition/Subtraction (from left to right):

   Calculate 68 + 6.

   \( 68 + 6 = 74 \)

   The equation is now: 74 − 54 = 20

   Calculate 74 − 54.

   \( 74 − 54 = 20 \)

So, the left side evaluates to 20.

The equation is 20 = 20, which is true.

This means Option 1 is the correct combination of signs.

Evaluating Other Options (for completeness)

Let's briefly check the other options to confirm.

Evaluating Option 2: ×, +, −, =

Equation: 68 × 138 + 23 − 54 = 20

Evaluate the left side:

1. Multiplication: \( 68 \times 138 \). This will result in a large number (\( 68 \times 138 = 9384 \)).
2. Addition/Subtraction: \( 9384 + 23 - 54 \). This will be approximately 9300.

The equation becomes \( 9353 = 20 \), which is false.

Evaluating Option 3: =, ×, +, ÷

Equation: 68 = 138 × 23 + 54 ÷ 20

Evaluate the right side:

1. Multiplication: \( 138 \times 23 \). This is a large number (\( 138 \times 23 = 3174 \)).
2. Division: \( 54 \div 20 \). \( 54 \div 20 = 2.7 \)
3. Addition: \( 3174 + 2.7 = 3176.7 \)

The equation becomes \( 68 = 3176.7 \), which is false.

Evaluating Option 4: ×, +, =, ÷

Equation: 68 × 138 + 23 = 54 ÷ 20

Evaluate the left side:

1. Multiplication: \( 68 \times 138 = 9384 \)
2. Addition: \( 9384 + 23 = 9407 \)

Evaluate the right side:

1. Division: \( 54 \div 20 = 2.7 \)

The equation becomes \( 9407 = 2.7 \), which is false.

Based on our evaluation, only Option 1 results in a true mathematical statement.

Final Answer Derivation

By systematically testing each option and applying the correct order of operations, we found that replacing the asterisks with +, ÷, −, and = in that sequence satisfies the equation:

\( 68 + 138 \div 23 - 54 = 20 \)

\( 68 + 6 - 54 = 20 \)

\( 74 - 54 = 20 \)

\( 20 = 20 \)

Therefore, the correct combination of mathematical signs is +, ÷, −, and =.

Revision Table

Additional Information on Order of Operations

The order of operations is crucial when solving mathematical expressions involving multiple operations. A commonly used mnemonic is BODMAS or PEMDAS.

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

Both mnemonics represent the same hierarchy. Division and Multiplication have equal priority, and are performed from left to right as they appear. Similarly, Addition and Subtraction have equal priority and are performed from left to right.

In the problem we solved, we first performed the division (138 ÷ 23) because division has higher priority than addition and subtraction. Then, we performed the addition and subtraction from left to right.