Which two signs (mathematical operators) should be interchanged to make the given equation correct? 56 – 8 + 42 ÷ 6 × 5 = 19
÷ and –
Solving Equations by Interchanging Mathematical Signs
The problem asks us to identify which two mathematical operators (signs) need to be swapped in the given equation to make it mathematically correct. The original equation is:
56 – 8 + 42 ÷ 6 × 5 = 19
We need to test the given options by interchanging the specified signs and then evaluating the new equation using the BODMAS (or PEMDAS) rule, which dictates the order of operations: Brackets, Orders (powers and roots), Division and Multiplication (from left to right), Addition and Subtraction (from left to right).
Testing the Interchange of ÷ and – signs
Let's test the option where we interchange the ÷ (division) sign and the – (subtraction) sign in the original equation. The original equation is:
56 – 8 + 42 ÷ 6 × 5 = 19
After swapping – and ÷, the new equation becomes:
56 ÷ 8 + 42 – 6 × 5
Now, we evaluate this new expression following the BODMAS rule:
- Division: Perform the division operation first.
56 ÷ 8 = 7- The equation is now:
7 + 42 – 6 × 5 - Multiplication: Next, perform the multiplication operation.
6 × 5 = 30- The equation is now:
7 + 42 – 30 - Addition: Now, perform the addition operation.
7 + 42 = 49- The equation is now:
49 – 30 - Subtraction: Finally, perform the subtraction operation.
49 – 30 = 19
The result of the expression after interchanging the ÷ and – signs is 19. This matches the right side of the original equation (which is 19).
Therefore, interchanging the signs ÷ and – makes the given equation correct.
| Step | Operation | Calculation | Equation after step |
|---|---|---|---|
| Original Eq. | –, +, ÷, × | 56 – 8 + 42 ÷ 6 × 5 |
56 – 8 + 42 ÷ 6 × 5 |
| Swap – and ÷ | ÷, +, –, × | 56 ÷ 8 + 42 – 6 × 5 |
56 ÷ 8 + 42 – 6 × 5 |
| 1 (BODMAS: Division) | ÷ | 56 ÷ 8 = 7 |
7 + 42 – 6 × 5 |
| 2 (BODMAS: Multiplication) | × | 6 × 5 = 30 |
7 + 42 – 30 |
| 3 (BODMAS: Addition) | + | 7 + 42 = 49 |
49 – 30 |
| 4 (BODMAS: Subtraction) | – | 49 – 30 = 19 |
19 |
Since the evaluation results in 19, which is the target value, the correct signs to interchange are ÷ and –.
Revision Table: Key Concepts
| Concept | Explanation |
|---|---|
| Mathematical Operators | Symbols representing mathematical operations like +, –, ×, ÷. |
| Equation | A statement that the values of two mathematical expressions are equal. |
| Interchanging Signs | Swapping the positions or roles of two different mathematical operators in an equation. |
| BODMAS/PEMDAS | Rules for the order of operations in a mathematical expression: Brackets/Parentheses, Orders/Exponents, Division/Multiplication, Addition/Subtraction. |
Additional Information: Order of Operations in Equations
Understanding the correct order of operations (BODMAS/PEMDAS) is crucial when evaluating mathematical expressions, especially when dealing with multiple operators like in this problem involving interchanging signs. Calculations must always follow this specific sequence to arrive at the correct result. If the order is not followed, the result will likely be incorrect.
When solving problems that require interchanging signs, it's important to systematically test each option by applying the BODMAS/PEMDAS rule to the modified equation. This systematic approach helps in identifying the correct pair of signs that satisfy the given condition (making the equation true).
Important Questions from Logical Puzzle
Seven persons P, Q, R, S, T, U and V like different watches namely W1, W2, W3, W4, W5, W6 and W7 (not necessarily in the same order). P and R do not like odd numbered watch. T likes W5. U does not like W2 or W3 or W6 or W7. P likes prime numbered watch. Q likes W4. S likes W2 or W7. Which of the following statement(s) is/are correct ?
I. S likes W7.
II. R likes W2.
III. U likes W1.
IV. V likes W3.
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