Solving Mathematical Sign Replacement Problems

The question asks us to find the correct sequence of mathematical signs from the given options that, when substituted for the asterisks (*) in the equation, will make the equation true.

The given equation is: 55 * 63 * 9 * 8 * 3 = 38

We need to test each option by replacing the asterisks with the signs in the order given in the option and then evaluate the expression according to the order of operations (BODMAS/PEMDAS).

Testing the Options

Let's examine each option systematically.

Option 1: ÷, ×, -, +

Replacing the asterisks with these signs in sequence, the equation becomes:

\(55 \div 63 \times 9 - 8 + 3\)

Let's evaluate this expression:

• First, perform division: \(55 \div 63\) is not a whole number. This suggests this option might not be the correct one for typical aptitude problems involving integers.

Calculating the approximate value: \(55 \div 63 \approx 0.87\). Then \(0.87 \times 9 \approx 7.83\). So, the expression is approximately \(7.83 - 8 + 3\).

\(7.83 - 8 = -0.17\)

\(-0.17 + 3 = 2.83\)

\(2.83 \neq 38\). So, Option 1 is incorrect.

Option 2: ×, ÷, +, -

Replacing the asterisks with these signs in sequence, the equation becomes:

\(55 \times 63 \div 9 + 8 - 3\)

Let's evaluate this expression following BODMAS/PEMDAS:

• First, perform division: \(63 \div 9 = 7\).
• The equation becomes: \(55 \times 7 + 8 - 3\).
• Next, perform multiplication: \(55 \times 7 = 385\).
• The equation becomes: \(385 + 8 - 3\).
• Now, perform addition and subtraction from left to right:
• \(385 + 8 = 393\).
• \(393 - 3 = 390\).

\(390 \neq 38\). So, Option 2 is incorrect.

Option 3: +, -, ×, ÷

Replacing the asterisks with these signs in sequence, the equation becomes:

\(55 + 63 - 9 \times 8 \div 3\)

Let's evaluate this expression following BODMAS/PEMDAS:

• First, perform multiplication and division from left to right:
• \(9 \times 8 = 72\).
• The equation becomes: \(55 + 63 - 72 \div 3\).
• Next, perform division: \(72 \div 3 = 24\).
• The equation becomes: \(55 + 63 - 24\).
• Now, perform addition and subtraction from left to right:
• \(55 + 63 = 118\).
• \(118 - 24 = 94\).

\(94 \neq 38\). So, Option 3 is incorrect.

Option 4: +, ÷, -, ×

Replacing the asterisks with these signs in sequence, the equation becomes:

\(55 + 63 \div 9 - 8 \times 3\)

Let's evaluate this expression following BODMAS/PEMDAS (Brackets, Orders, Division/Multiplication, Addition/Subtraction):

• First, perform Division and Multiplication from left to right:
• Division: \(63 \div 9 = 7\).
• The equation becomes: \(55 + 7 - 8 \times 3\).
• Multiplication: \(8 \times 3 = 24\).
• The equation becomes: \(55 + 7 - 24\).
• Now, perform Addition and Subtraction from left to right:
• Addition: \(55 + 7 = 62\).
• The equation becomes: \(62 - 24\).
• Subtraction: \(62 - 24 = 38\).

\(38 = 38\). The equation balances.

Conclusion

The sequence of mathematical signs +, ÷, -, × correctly balances the given equation.

Revision Table: Key Concepts

Additional Information: Order of Operations Explained

Understanding the correct order of operations is crucial for solving problems like this. Without it, evaluating an expression like \(55 + 63 \div 9 - 8 \times 3\) could yield different results depending on which operation is performed first. The standard order is followed to ensure consistency.

• B/P (Brackets/Parentheses): Operations inside brackets or parentheses are always performed first.
• O/E (Orders/Exponents): Next, evaluate any powers or roots.
• DM (Division and Multiplication): These operations are performed next, from left to right in the expression. They have equal priority.
• AS (Addition and Subtraction): These operations are performed last, from left to right in the expression. They also have equal priority.

In the case of \(55 + 63 \div 9 - 8 \times 3\), we first perform the division \(63 \div 9\) and the multiplication \(8 \times 3\) because D and M come before A and S. Since division appears before multiplication when reading from left to right after addressing higher priority operations (though in this specific case they are separated by other terms, the rule applies to pairs with equal priority), we get \(55 + 7 - 24\). Then we perform addition and subtraction from left to right: \(55 + 7 = 62\), then \(62 - 24 = 38\).

This systematic approach guarantees the correct outcome when evaluating mathematical expressions with multiple operations.