Select the correct combination of the mathematical signs to replace Xs in the below equation and thereby balance it. 9 X 3 X 6 X 2 X 9
÷, ×, =, ×
Understanding Equation Balancing
The problem asks us to find the correct set of mathematical signs to replace the 'X' symbols in the expression 9 X 3 X 6 X 2 X 9 so that the resulting equation is balanced. Balancing an equation means that the value of the expression on the left side of the equals sign must be equal to the value of the expression on the right side.
Available Mathematical Signs Combinations
We are given four different combinations of mathematical signs to test:
- ÷, ×, =, ×
- ÷, ×, =, -
- ×, =, -, ÷
- ×, ÷, +, =
We need to substitute each combination into the equation 9 X 3 X 6 X 2 X 9 and check if the equation holds true.
Testing Option 1: ÷, ×, =, ×
Let's substitute the signs from Option 1 into the expression:
9 ÷ 3 × 6 = 2 × 9
Now, we need to evaluate both sides of the potential equation following the order of operations (BODMAS/PEMDAS: Brackets, Orders, Division and Multiplication (from left to right), Addition and Subtraction (from left to right)).
Evaluating the Left Side (LHS)
The left side is 9 ÷ 3 × 6.
- First, perform the division: 9 ÷ 3 = 3.
- Next, perform the multiplication: 3 × 6 = 18.
So, the value of the left side is \(18\).
Evaluating the Right Side (RHS)
The right side is 2 × 9.
- Perform the multiplication: 2 × 9 = 18.
So, the value of the right side is \(18\).
Comparing LHS and RHS
We found that LHS = \(18\) and RHS = \(18\).
Since LHS = RHS (\(18 = 18\)), the equation is balanced with the signs from Option 1.
Testing Other Options (Optional but helpful)
Let's quickly look at another option to see why it doesn't balance the equation.
Testing Option 2: ÷, ×, =, -
Substitute the signs: 9 ÷ 3 × 6 = 2 - 9.
- LHS: 9 ÷ 3 × 6 = 3 × 6 = \(18\).
- RHS: 2 - 9 = \(-7\).
LHS (\(18\)) is not equal to RHS (\(-7\)). So, Option 2 does not balance the equation.
Testing Options 3 and 4 similarly would also show that the left and right sides of the equation do not evaluate to the same value.
Conclusion on Balancing the Equation
Only the combination of mathematical signs from Option 1 (\(\div\), \(\times\), \(=\), \(\times\)) successfully balances the given equation 9 X 3 X 6 X 2 X 9, resulting in the true statement \(18 = 18\).
Revision Table: Testing Sign Combinations
| Option | Signs | Equation | LHS | RHS | Balanced? |
|---|---|---|---|---|---|
| 1 | ÷, ×, =, × | \(9 \div 3 \times 6 = 2 \times 9\) | \(18\) | \(18\) | Yes |
| 2 | ÷, ×, =, - | \(9 \div 3 \times 6 = 2 - 9\) | \(18\) | \(-7\) | No |
| 3 | ×, =, -, ÷ | \(9 \times 3 = 6 - 2 \div 9\) | \(27\) | \(6 - \frac{2}{9} = \frac{52}{9}\) | No |
| 4 | ×, ÷, +, = | \(9 \times 3 \div 6 + 2 = 9\) | \(27 \div 6 + 2 = 4.5 + 2 = 6.5\) | \(9\) | No |
Additional Information on Order of Operations
When evaluating mathematical expressions that involve multiple operations, it is crucial to follow a specific order to ensure consistency and accuracy. A common mnemonic for the order of operations is BODMAS or PEMDAS.
- BODMAS: Brackets, Orders (powers and roots), Division and Multiplication (from left to right), Addition and Subtraction (from left to right).
- PEMDAS: Parentheses, Exponents (powers and roots), Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
In this problem, we only had division, multiplication, addition, subtraction, and equality. According to BODMAS/PEMDAS, multiplication and division are performed before addition and subtraction, working from left to right if they appear at the same level (like division and multiplication, or addition and subtraction).
Understanding the order of operations is fundamental for correctly solving problems involving mathematical expressions and balancing equations.
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