Mathematical Operations

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Reasoning Ability Mathematical Operations Competitive Exams

Mathematical Operations questions test your ability to understand symbols, replace symbols with actual arithmetic operations, and solve expressions correctly using the proper order of operations.

What are Mathematical Operations?

In reasoning exams, Mathematical Operations questions use ordinary arithmetic operations such as addition, subtraction, multiplication, and division. However, the symbols may be interchanged or replaced by new symbols.

Your task is to decode the meaning of the given symbols and then solve the expression accurately. These questions check both mathematical accuracy and logical interpretation.

Quick idea: First identify what each symbol means. Then replace the symbols with actual operations and solve using BODMAS.

Type	Meaning	Example
Direct Operation	Normal arithmetic calculation	$12 + 8 \times 2$
Symbol Replacement	Symbols are given special meanings	If * means $+$, solve *4 5**
Interchanged Signs	Two or more signs are swapped	$+$ means $-$, and $-$ means $+$
Missing Operator	Find the operation that makes the equation correct	$8 \ ? \ 4 = 12$
True / False Equation	Check which option satisfies the equation	Choose signs to make $6 \ ? \ 3 \ ? \ 2 = 12$

“Mathematical operation questions become simple when symbols are decoded before solving.”

Reasoning Tip

Key Points

Read symbol meanings carefully.
Replace symbols before calculation.
Apply BODMAS after replacement.
Do not solve from left to right blindly.
Check whether signs are interchanged.
Verify the final answer with the original condition.

symbols BODMAS logic calculation

Order of Operations: BODMAS

After replacing symbols with the actual operations, solve the expression using BODMAS.

Letter	Meaning	Priority
B	Brackets	First
O	Of / Powers / Roots, where applicable	Second
D	Division	Third
M	Multiplication	Third, along with division from left to right
A	Addition	Fourth
S	Subtraction	Fourth, along with addition from left to right

Important: Division and multiplication are solved from left to right. Addition and subtraction are also solved from left to right.

Common Symbol Replacement Patterns

In many questions, ordinary signs are replaced by unusual signs. First prepare a small conversion table and then solve.

1. One Symbol Replaced

Only one symbol has a new meaning.

If * means $+$, then
8 * 5 becomes $8 + 5 = 13$.

2. Two Signs Interchanged

Two arithmetic signs are swapped.

If $+$ means $\times$,
and $\times$ means $+$.

3. Multiple Symbols Replaced

More than two symbols may have special meanings.

@ means $+$, # means $-$,
$ means $\times$, & means $\div$.

4. Equation Balancing

Choose signs that make the equation correct.

$6 \ ? \ 3 \ ? \ 2 = 12$
Try suitable signs.

Tip: Never calculate until you have replaced every symbol correctly.

Step-by-Step Solving Method

Step	Action	Example
Step 1	Read the meaning of each symbol carefully.	* means $+$, @ means $\times$
Step 2	Write the original expression separately.	*4 @ 5 6**
Step 3	Replace symbols with actual operations.	$4 \times 5 + 6$
Step 4	Apply BODMAS.	$20 + 6 = 26$
Step 5	Verify the answer with the options.	Answer = $26$

Important: In symbol-based questions, one small replacement mistake can change the entire answer.

Worked Example 1: Symbol Replacement

If * means $+$, @ means $\times$, and # means $-$, find the value of:

8 @ 3 * 5 # 4

Replacement Table

Given Symbol	Actual Operation
*	$+$
@	$\times$
#	$-$

Original expression: 8 @ 3 * 5 # 4
After replacement: $8 \times 3 + 5 - 4$

\[ 8 \times 3 + 5 - 4 = 24 + 5 - 4 = 25 \]

Therefore, the answer is 25.

Worked Example 2: Interchanged Signs

If $+$ means $\times$, $\times$ means $-$, and $-$ means $+$, find the value of:

\[ 7 + 4 \times 3 - 2 \]

Replace each sign according to the condition:

\[ 7 + 4 \times 3 - 2 = 7 \times 4 - 3 + 2 \]

Now apply BODMAS:

\[ 7 \times 4 - 3 + 2 = 28 - 3 + 2 = 27 \]

Common Types of Mathematical Operation Questions

Find the Value

Decode symbols and calculate the expression.

5 @ 4 * 2
8 # 3 @ 2
12 * 4 # 5

Find Correct Sign

Identify the sign that makes an equation true.

$8 \ ? \ 4 = 12$
$9 \ ? \ 3 = 27$
$20 \ ? \ 5 = 4$

Interchanged Operators

Signs are swapped before solving.

$+$ and $-$ interchanged
$\times$ and $\div$ interchanged
Multiple signs changed

Equation Checking

Choose the option that satisfies the equation.

Check options one by one
Use BODMAS carefully
Verify both sides

Rule: Decode first, calculate later. Do not apply BODMAS before replacing the symbols.

Solved Examples

Question	Method	Answer
If * means $+$, find *7 8**.	*7 8** becomes $7 + 8 = 15$	15
If @ means $\times$, find 6 @ 5.	6 @ 5 becomes $6 \times 5 = 30$	30
If # means $-$, find 25 # 9.	25 # 9 becomes $25 - 9 = 16$	16
If $+$ means $\times$, find $4 + 6 - 3$, where $-$ is unchanged.	$4 \times 6 - 3 = 24 - 3 = 21$	21
If * means $+$, @ means $\times$, find *5 @ 4 3**.	*5 @ 4 3** becomes $5 \times 4 + 3 = 20 + 3 = 23$	23
If $+$ and $\times$ are interchanged, find $8 + 2 \times 5$.	$8 \times 2 + 5 = 16 + 5 = 21$	21
Which sign makes $9 \ ? \ 3 = 27$ true?	$9 \times 3 = 27$	$\times$
Which sign makes $20 \ ? \ 5 = 4$ true?	$20 \div 5 = 4$	$\div$

Note: In operator replacement questions, write the converted expression before solving.

Common Traps and Shortcuts

Common Traps

Solving before replacing the signs.
Ignoring BODMAS after replacement.
Changing only one occurrence of a symbol.
Forgetting that two signs may be interchanged.
Solving strictly from left to right in all cases.
Confusing $+$ with $\times$ in symbol-based questions.

Useful Shortcuts

Create a small symbol replacement table first.
Rewrite the expression after replacing all signs.
Use BODMAS only after replacement.
Check options by substitution when signs are missing.
For equations, solve both left and right sides separately.
Verify the final answer once before marking.

Exam approach: Identify whether the question is based on symbol replacement, interchanged signs, missing operators, or equation checking.

Practice

A) Multiple Choice Questions

If * means $+$, find 12 * 8.
18 20 24 96
If @ means $\times$, find 7 @ 6.
13 42 36 48
If $+$ and $\times$ are interchanged, find $5 + 4 \times 3$.
17 23 35 60
Which sign makes $16 \ ? \ 4 = 4$ true?
+ - × ÷
If * means $+$, @ means $\times$, find 9 @ 2 * 5.
23 28 32 45

B) Solve the Higher-Order Problems

If * means $+$, # means $-$, and @ means $\times$, find: 6 @ 5 * 4 # 8 Hint: Replace symbols first, then apply BODMAS.
If $+$ means $\times$, $\times$ means $-$, and $-$ means $+$, find $8 + 3 \times 4 - 2$. Hint: Convert every sign before solving.
Which two signs should replace the blanks to make the equation true? $10 \ ? \ 5 \ ? \ 3 = 47$ Hint: Try multiplication before subtraction.
If A * B means $A + B$, and A @ B means $A \times B$, find: 4 @ 5 * 6 Hint: Treat the operation between two numbers according to the symbol.
If $+$ and $\div$ are interchanged, find: $24 + 6 \div 3$ Hint: Replace $+$ with $\div$, and $\div$ with $+$.

Reasoning Reminder

Mathematical operation questions are not only about calculation. They are mainly about correct decoding of symbols. Once the symbols are decoded, normal arithmetic rules apply.

Task: Create five questions using symbol replacement, sign interchange, missing sign, equation checking, and BODMAS-based calculation.

Show Suggested Answers

Multiple Choice

20
12 * 8 becomes $12 + 8 = 20$.
42
7 @ 6 becomes $7 \times 6 = 42$.
23
Since $+$ and $\times$ are interchanged:
$5 + 4 \times 3$ becomes $5 \times 4 + 3 = 20 + 3 = 23$.
÷
$16 \div 4 = 4$.
23
9 @ 2 * 5 becomes $9 \times 2 + 5 = 18 + 5 = 23$.

Higher-Order Problems

6 @ 5 * 4 # 8 becomes $6 \times 5 + 4 - 8$.
$30 + 4 - 8 = 26$
Answer = 26.
$8 + 3 \times 4 - 2$ becomes $8 \times 3 - 4 + 2$.
$24 - 4 + 2 = 22$
Answer = 22.
$10 \times 5 - 3 = 50 - 3 = 47$
Answer = $\times$, $-$.
4 @ 5 * 6 becomes $4 \times 5 + 6$.
$20 + 6 = 26$
Answer = 26.
Since $+$ and $\div$ are interchanged:
$24 + 6 \div 3$ becomes $24 \div 6 + 3 = 4 + 3 = 7$.
Answer = 7.

Clue Explanation

In mathematical operation questions, always decode the operator symbols first. After that, apply BODMAS carefully and verify the final answer.

Exam Tips

Make a symbol conversion table first.
Replace all symbols, not only the first one.
Use BODMAS after replacement.
Check whether signs are interchanged.
Try options when the missing operator is asked.
Do not rush in simple-looking expressions.

Practice MCQs

Given Symbol	Actual Operation
*	\(+\)
@	\(\times\)
#	\(-\)

Question	Method	Answer
If * means \(+\), find *7 8**.	*7 8** becomes \(7 + 8 = 15\)	15
If @ means \(\times\), find 6 @ 5.	6 @ 5 becomes \(6 \times 5 = 30\)	30
If # means \(-\), find 25 # 9.	25 # 9 becomes \(25 - 9 = 16\)	16
If \(+\) means \(\times\), find \(4 + 6 - 3\), where \(-\) is unchanged.	\(4 \times 6 - 3 = 24 - 3 = 21\)	21
If * means \(+\), @ means \(\times\), find *5 @ 4 3**.	*5 @ 4 3** becomes \(5 \times 4 + 3 = 20 + 3 = 23\)	23
If \(+\) and \(\times\) are interchanged, find \(8 + 2 \times 5\).	\(8 \times 2 + 5 = 16 + 5 = 21\)	21
Which sign makes \(9 \ ? \ 3 = 27\) true?	\(9 \times 3 = 27\)	\(\times\)
Which sign makes \(20 \ ? \ 5 = 4\) true?	\(20 \div 5 = 4\)	\(\div\)

Type	Meaning	Example
Direct Operation	Normal arithmetic calculation	\(12 + 8 \times 2\)
Symbol Replacement	Symbols are given special meanings	If * means \(+\), solve *4 5**
Interchanged Signs	Two or more signs are swapped	\(+\) means \(-\), and \(-\) means \(+\)
Missing Operator	Find the operation that makes the equation correct	\(8 \ ? \ 4 = 12\)
True / False Equation	Check which option satisfies the equation	Choose signs to make \(6 \ ? \ 3 \ ? \ 2 = 12\)

Step	Action	Example
Step 1	Read the meaning of each symbol carefully.	* means \(+\), @ means \(\times\)
Step 2	Write the original expression separately.	*4 @ 5 6**
Step 3	Replace symbols with actual operations.	\(4 \times 5 + 6\)
Step 4	Apply BODMAS.	\(20 + 6 = 26\)
Step 5	Verify the answer with the options.	Answer = \(26\)

Learning Modules