After interchanging which two signs will the value of the given expression be ‘37’? 27 + 3 ÷ 81 × 93 – 57
× and +
Solving the Mathematical Sign Interchange Problem
The question asks us to find which two signs in the given mathematical expression, when interchanged, will make the expression evaluate to the value '37'. The expression is:
27 + 3 ÷ 81 × 93 – 57
To solve this problem, we need to test each given option. For each option, we will interchange the specified signs in the expression and then evaluate the new expression following the order of operations (BODMAS/PEMDAS: Brackets, Orders/Exponents, Division and Multiplication (from left to right), Addition and Subtraction (from left to right)).
Step-by-Step Evaluation of Options
Option 1: Interchange + and ÷
The expression becomes: 27 ÷ 3 + 81 × 93 – 57
Let's evaluate:
\(27 \div 3 + 81 \times 93 - 57\)
Perform Division first:
\(9 + 81 \times 93 - 57\)
Perform Multiplication:
\(9 + 7533 - 57\)
Perform Addition:
\(7542 - 57\)
Perform Subtraction:
\(7485\)
This value (7485) is not 37.
Option 2: Interchange – and +
The expression becomes: 27 – 3 ÷ 81 × 93 + 57
Let's evaluate:
\(27 - 3 \div 81 \times 93 + 57\)
Perform Division first:
\(27 - \frac{3}{81} \times 93 + 57\)
\(27 - \frac{1}{27} \times 93 + 57\)
Perform Multiplication:
\(27 - \frac{93}{27} + 57\)
\(27 - \frac{31}{9} + 57\)
Perform Addition/Subtraction from left to right:
\(84 - \frac{31}{9}\)
\(\frac{84 \times 9 - 31}{9}\)
\(\frac{756 - 31}{9}\)
\(\frac{725}{9}\)
This value (\(\frac{725}{9}\)) is not 37.
Option 3: Interchange × and +
The expression becomes: 27 × 3 ÷ 81 + 93 – 57
Let's evaluate:
\(27 \times 3 \div 81 + 93 - 57\)
Perform Multiplication and Division from left to right:
\(81 \div 81 + 93 - 57\)
\(1 + 93 - 57\)
Perform Addition and Subtraction from left to right:
\(94 - 57\)
\(37\)
This value (37) matches the required value.
Option 4: Interchange ÷ and –
The expression becomes: 27 + 3 – 81 × 93 ÷ 57
Let's evaluate:
\(27 + 3 - 81 \times 93 \div 57\)
Perform Multiplication and Division from left to right:
\(27 + 3 - 81 \times \frac{93}{57}\)
\(27 + 3 - 81 \times \frac{31}{19}\)
\(27 + 3 - \frac{81 \times 31}{19}\)
\(27 + 3 - \frac{2511}{19}\)
Perform Addition/Subtraction from left to right:
\(30 - \frac{2511}{19}\)
\(\frac{30 \times 19 - 2511}{19}\)
\(\frac{570 - 2511}{19}\)
\(\frac{-1941}{19}\)
This value (\(\frac{-1941}{19}\)) is not 37.
Based on the evaluation of each option, interchanging the '×' and '+' signs results in the expression evaluating to 37.
Revision Table: Key Concepts in Expression Evaluation
| Concept | Description | Example |
|---|---|---|
| BODMAS/PEMDAS | An acronym to remember the order of operations: Brackets, Orders (or Exponents), Division/Multiplication (left to right), Addition/Subtraction (left to right). | \(10 + 2 \times 5 = 10 + 10 = 20\) (Multiplication before Addition) |
| Interchanging Signs | Swapping the positions of two different arithmetic operators within an expression. | Replacing '+' with '×' and '×' with '+' in an expression like \(a+b \times c\) gives \(a \times b + c\). |
| Mathematical Expression | A combination of numbers, variables, and arithmetic operations. | \(2x + 5y - 3\) |
Additional Information: Understanding Order of Operations
The order of operations, often remembered by acronyms like BODMAS or PEMDAS, is crucial for consistently evaluating mathematical expressions. Without a standard order, an expression could have multiple different values depending on which operation is performed first. Here's a breakdown:
- Brackets/Parentheses: Operations inside brackets are always performed first. If there are nested brackets, work from the innermost ones outwards.
- Orders/Exponents: Next, evaluate anything involving powers or roots.
- Division and Multiplication: These operations are performed next. They have equal priority. If both appear in an expression, work from left to right.
- Addition and Subtraction: These are the last operations to be performed. They also have equal priority. If both appear, work from left to right.
Following this specific order ensures that everyone arrives at the same correct answer for any given mathematical expression. In problems involving interchanging signs, applying this order correctly after swapping the signs is key to finding the solution.
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?
