Which of the mathematical signs should be interchanged in the given equation to make it mathematically correct? 31 ÷ 14 × 9 + 3 − 7 = 66
÷ and +
Interchanging Mathematical Signs in Equations
This problem asks us to identify which pair of mathematical signs, when swapped in the given equation, will make the equation mathematically correct. The given equation is:
\begin{equation*} 31 \div 14 \times 9 + 3 - 7 = 66 \end{equation*}
Currently, this equation is not correct. We need to test the options provided by interchanging the specified signs and then evaluate the resulting equation using the order of operations (BODMAS or PEMDAS).
Testing the Options for Sign Interchange
Let's examine each option and see if swapping the signs makes the equation valid.
Option 1: Interchange ÷ and +
If we swap the division (÷) and addition (+) signs, the equation becomes:
\begin{equation*} 31 + 14 \times 9 \div 3 - 7 \end{equation*}
Now, let's evaluate this expression using the BODMAS/PEMDAS rule:
- Brackets / Parentheses (None in this equation)
- Orders / Exponents (None in this equation)
- Division and Multiplication (from left to right)
- Addition and Subtraction (from left to right)
Following the rule:
- First, perform the division: $9 \div 3 = 3$. The equation is now: $31 + 14 \times 3 - 7$
- Next, perform the multiplication: $14 \times 3 = 42$. The equation is now: $31 + 42 - 7$
- Then, perform the addition: $31 + 42 = 73$. The equation is now: $73 - 7$
- Finally, perform the subtraction: $73 - 7 = 66$.
The result of the evaluation is 66. This matches the Right Hand Side (RHS) of the original equation (= 66). Therefore, interchanging the ÷ and + signs makes the equation mathematically correct.
The corrected equation is:
\begin{equation*} 31 + 14 \times 9 \div 3 - 7 = 66 \end{equation*}
Testing Other Options (Verification)
While we have found the correct interchange, let's briefly consider why other options would not work. We would follow the same process: swap the signs, then evaluate using BODMAS/PEMDAS.
- Option 2: Interchange ÷ and ×
Equation becomes: $31 \times 14 \div 9 + 3 - 7$. Calculating $14 \div 9$ will result in a fraction or a non-integer, making it unlikely to reach 66. - Option 3: Interchange − and ×
Equation becomes: $31 \div 14 - 9 + 3 \times 7$. Again, $31 \div 14$ results in a fraction. - Option 4: Interchange − and +
Equation becomes: $31 \div 14 \times 9 - 3 + 7$. Once more, $31 \div 14$ results in a fraction.
Only the interchange of ÷ and + leads to a straightforward integer calculation that results in 66.
Conclusion on Sign Interchange
Based on the evaluation using the BODMAS/PEMDAS rule, interchanging the division (÷) and addition (+) signs makes the given equation mathematically correct.
Revision Table: Mathematical Signs Interchange
| Original Equation | $31 \div 14 \times 9 + 3 - 7 = 66$ |
|---|---|
| Option 1 Swap (÷ and +) | $31 + 14 \times 9 \div 3 - 7$ |
| Calculation Steps | $31 + 14 \times 3 - 7$ $31 + 42 - 7$ $73 - 7 = 66$ |
| Result | 66 |
| Correct? | Yes |
Additional Information: Understanding BODMAS/PEMDAS Rule for Equation Solving
The BODMAS or PEMDAS rule is a set of rules that dictate the order in which mathematical operations should be performed when evaluating an expression. This ensures that everyone gets the same result for a given expression.
- BODMAS: Brackets, Orders (powers, roots), Division, Multiplication, Addition, Subtraction.
- PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.
Division and Multiplication have the same priority, and should be performed from left to right. Similarly, Addition and Subtraction have the same priority and should be performed from left to right.
Applying this rule is crucial for correctly solving problems involving multiple mathematical operations or when interchanging mathematical signs to correct an equation.
Important Questions from Logical Puzzle
Below is given a figure, in which a horse is placed on one of the boxes in a square grid of size $5 \times 5$. The horse can move in either of the following ways in a single move:
(i) two boxes to either left or right side and one box to either up or down side
(ii) two boxes to either up or down side and one box to either left or right side
(iii) the horse cannot step on a grey box. The horse has to stay within the grid and any move to out of the grid is NOT legal.
How many white boxes are there which cannot be covered by horse within 3 moves?
- Six persons Raj, Rohit, Rachit, Rajat, Rishu, and Rinku are working in an office. Each of them has a favorite car viz. P, Q, R, S, T, and U, but not necessarily in the same order. Each of them got a different salary - 1000, 2000, 2500, 3000, 3500, and 4000, but not necessarily in the same order. Rajat gets a salary of 2500 and his favorite car is U. The one who likes car Q gets a salary of 3000. Rachit likes car P and gets a salary of 3500. The one who gets a salary of 1000 does not like car R or S. Rinku does not like car Q or S but gets a salary of 2000. Rishu gets a salary of 4000 and does not like a car R. Rohit does not get a salary of 1000. Who gets a salary of 3000?
- Six friends P, Q, R, S, T, and U are sitting at different positions in a row of the company, and each of them get a different salary viz – 500, 800, 1000, 1500, 1800 & 2000, but not necessary in the same order. The friend who gets a salary of 2000 does not sit at the extreme end of the row. Q sits immediately before R and gets a salary of 800. T sits between U and S. Q is second to the left of S. P sits at the left end of the row and gets a salary of 1000. T gets a salary of 1500. R gets a salary neither of 2000 nor of 1800. How much salary does U get?
- Seven men A, B, C, D, E, F and G are standing in a queue, one behind the other in that order. That is, A is first in queue while G brings up the rear. Each of them is wearing a hat of a different colour of the rainbow - Violet, Indigo, Blue, Green, Yellow, Orange, and Red, but not necessarily in that order. It is also known that:
• D can see in front of him green and blue hats, but not the violet hat.
• E can see violet and yellow hats, but not the red hat.
• G can see hats of all the colours except Orange.
• E is wearing the Indigo coloured hat.
What colour hat is worn by F? - Who amongst the females arrived third?
