Which two signs should be interchanged to make the given equation correct? 17 + 10 × 5 ÷ 9 + 18 = 53
× and ÷
Solving Mathematical Equation by Swapping Signs
The problem asks us to find which pair of mathematical signs needs to be interchanged in the given equation to make it correct. The original equation is:
\(17 + 10 \times 5 \div 9 + 18 = 53\)
Let's evaluate the original equation using the order of operations (BODMAS/PEMDAS):
\(17 + (10 \times 5) \div 9 + 18\)
\(17 + 50 \div 9 + 18\)
\(17 + (50 \div 9) + 18\)
Since \(50 \div 9\) results in a non-integer (\(\approx 5.56\)), the original equation does not result in the integer 53. We need to test each option by swapping the specified signs and re-evaluating the equation.
Testing Options for Sign Interchange
Option 1: Interchanging + and ÷
If we interchange the + and ÷ signs, the equation becomes:
\(17 \div 10 \times 5 + 9 \div 18 = 53\)
Evaluate using order of operations:
\((17 \div 10) \times 5 + (9 \div 18)\)
\(1.7 \times 5 + 0.5\)
\(8.5 + 0.5 = 9\)
This does not equal 53.
Option 2: Interchanging + and ×
If we interchange the + and × signs, the equation becomes:
\(17 \times 10 + 5 \div 9 \times 18 = 53\)
Evaluate using order of operations:
\((17 \times 10) + (5 \div 9) \times 18\)
\(170 + (5/9) \times 18\)
\(170 + (5 \times 18 / 9)\)
\(170 + (90 / 9)\)
\(170 + 10 = 180\)
This does not equal 53.
Option 3: Interchanging + and −
The original equation does not contain a − sign. Therefore, this interchange is not possible with the given equation structure.
Option 4: Interchanging × and ÷
If we interchange the × and ÷ signs, the equation becomes:
\(17 + 10 \div 5 \times 9 + 18 = 53\)
Evaluate using order of operations:
\(17 + (10 \div 5) \times 9 + 18\)
\(17 + 2 \times 9 + 18\)
\(17 + (2 \times 9) + 18\)
\(17 + 18 + 18\)
\(17 + 36 = 53\)
This equals 53. Therefore, interchanging the × and ÷ signs makes the equation correct.
Conclusion
By testing each option and applying the order of operations, we find that interchanging the × and ÷ signs correctly solves the mathematical equation, resulting in 53.
Revision Table: Checking Sign Interchanges
| Interchanged Signs | New Equation | Evaluation | Result | Correct? |
|---|---|---|---|---|
| + and ÷ | \(17 \div 10 \times 5 + 9 \div 18\) | \(1.7 \times 5 + 0.5 = 8.5 + 0.5 = 9\) | 9 | No |
| + and × | \(17 \times 10 + 5 \div 9 \times 18\) | \(170 + (5/9) \times 18 = 170 + 10 = 180\) | 180 | No |
| × and ÷ | \(17 + 10 \div 5 \times 9 + 18\) | \(17 + 2 \times 9 + 18 = 17 + 18 + 18 = 53\) | 53 | Yes |
Additional Information on Order of Operations
When solving mathematical equations involving multiple operations, it is crucial to follow a specific order. This order is commonly known by acronyms like BODMAS or 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).
In the context of this problem, Division and Multiplication are performed before Addition and Subtraction. If both Division and Multiplication are present, they are performed from left to right as they appear in the equation. Similarly, Addition and Subtraction are performed from left to right.
Understanding the correct order of operations is fundamental to solving such mathematical equation problems accurately.
Similar Questions
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Statements:
All beaches are sand.
Some deserts are sand.
All mountains are rocky.
Conclusions:
(I) At least some beaches are desert.
(II) At least some sands are rocky.
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