In such type of questions some relationships are shown with the help of certain symbols/notations and/ or mathematical signs. Each symbol or sign is defined clearly in the question statement itself. In other words, each symbol or sign is accorded two values –one real value and another assigned value of each symbol or sign and then solve the questions accordingly.

For example, Suppose the triangle (∆) means addition.

We know that triangle is a plane figure but here it has been assigned the value of addition (+).

Thus, 3 ∆ 5 ⇒ 3 + 5 = 8

In this way, two work out such questions substitute the assigned/ implied meanings of the symbol or sign and proceed accordingly.

### How to Solve the questions

To solve this type of questions, substitute the real signs in the given expression and then solve the expression according to the BODMAS rule.

**Order of Operations – BODMAS**

- 1st. B – Brackets, do all the maths contained in brackets first
- 2nd. O – Orders, square roots, powers and anything else not listed
- 3rd. D – Division, do your divisions now
- 4th. M – Multiplication
- 5th. A – Addition
- 6th. S – Subtraction

**Example: If + means ÷, × means –, ÷ means × and – means +, then, 8 + 6 × 4 ÷ 3 – 4 = ?**

- – 12
- – 20/3
- 12
- 20/3

**Solution. (3):** Using the given symbols, we have:

Given expression: = 8 ÷ 6 – 4 × 3 + 4 = 4/3 – 4 × 3 + 4

= 4/3 – 12 + 4 = -20/3.

### Type 1: Value of the Given Expression

**Example 1: If ‘÷’ means ‘+’, ‘–’ means ‘÷’, ‘×’ means ‘–’ and ‘+’ means ‘×’ then, 62 ÷ 8 – 4 × 12 + 4 = ?**

- 16
- 26
- 1/16
- 6

**Solution:** (1) Given expression, 62 ÷ 8 – 4 × 12 + 4 = ?

According to question, after replacement of mathematical sign

62 + 8 ÷ 4 – 12 × 4 = ?

= 64 – 48 = 16

Hence, ? ⇒ 16

### Type 2: Identification of Correct Equation

**Example 2: If ‘–’ means ‘+’, ‘+’ means ‘–’, ‘×’ means ‘÷’ and ‘÷’ means ‘×’; then which of the given equations is correct?**

- 30 + 5 – 4 ÷ 10 × 5 = 58
- 30 + 5 ÷ 4 – 10 × 5 = 22
- 30 – 5 + 4 ÷ 10 × 5 = 62
- 30 × 5 – 4 ÷ 10 + 5 = 41

**Solution. (4):** From option 4:

30 × 5 – 4 ÷ 10 + 5 = 41

According to question, after replacement of mathematical sign

30 ÷ 5 + 4 × 10 – 5 = 41 = 6 + 40 – 5 = 41 = 46 – 5 = 41

Hence option (4) is correct.

## Solved Examples

**Example 1: If ‘M’ means ‘÷’, R means ‘+’, T means ‘-’, and ‘K’ means ‘×’ then what will be the value of the following expression?**

20 R 16 K 5 M 10 T8 = ?

- 36
- 20
- 36.5
- 12

**Solution. (2):** ? = 20 + 16 × 5 ÷ 10 – 8

or ? = 20 + 16 ×5/10– 8

or ? = 20 + 8 – 8 = 20

**Example 2: Of the two subjects offered to a class in their final year, 32 students in all are studying Psychology while a total of 26 students are studying Sociology. If 16 students have opted to specialize in both, what is the strength of the class?**

- 74
- 58
- 42
- Date inadequate

**Solution. (3):** Venn diagram of given information would be as follows.

Total strength of the class = 16 + 16 + 10 = 42

**Example 3: How many numbers would remain if the numbers which are divisible by 2 and also those having ‘2’ as only one of the digits are dropped from numbers 1 to 30?**

- 14
- 17
- 15
- 10

**Solution. (4):**

Mathematical Operation is an important concept of reasoning that is usually asked in various competitive exams. This topic is asked to test the analytical abilities of the candidates. It shows how good you are at observing things and then implying it to solve the questions. To score full marks on this topic, you must practice enough questions and get acquainted with the concept behind it. Here we are providing you with the method to solve mathematical operations questions along with examples.

**How to solve questions based on Mathematical Operations?**

The type of questions based on Mathematical operations are:

- Whether the given equations are correct
- Based on Symbols equivalent to signs
- Interchanging the signs
- Balancing the equation
- Solve the equation

For every type of Mathematical operations question, you must know only one rule i.e. BODMAS. **It is “Brackets, Orders, Division, Multiplication, Addition, and Subtraction. It means you must solve any equation in the BODMAS order. First, open the brackets, then solve the powers or roots, then perform Division followed by multiplication, Addition and subtraction.**

**Q1. If × stands for -, ÷ stands for +, + stands for ÷ and – stands for ×, which one of the following equations is correct? **(a) 15 – 5 ÷ 5 × 20 + 16 = 6

(b) 8 ÷ 10 – 3 + 5 × 6 = 8

(c) 6 × 2 + 3 ÷ 12 – 3 = 15

(d) 3 ÷ 7 – 5 × 10 + 3 = 10**Ans.(b)**

Sol. Using the proper signs, we get:

Expression in (a) = 15 × 5 + 5 – 20 ÷ 10 = 15 × 5 + 5 – 2 = 75 + 5 – 2 = 78

Expression in (b) = 8 + 10 × 3 ÷ 5 – 6 = 8 + 10 × 3/5 – 6 = 8 + 6 – 6 = 8

Expression in (c) = 6 – 2 ÷ 3 + 12 × 3 = 6 – 2/3 + 36 = 42 – 2/3=124/3

Expression in (d) = 3 + 7 × 5 – 10 ÷ 3 = 3 + 7 × 5 – 10/3=3+35-10/3=104/3

∴ Statement (b) is true.**Q2. If ‘<’ means ‘minus’, ‘>’ means ‘plus’, ‘=’ means ‘multiplied by’ and ‘$’ means ‘divided by’, then what would be the value of 31 > 81 $ 9 < 7? **

(a) 32

(b) 33

(c) 36

(d) None of these**Ans.(b)**

Sol. Using the correct symbols we have:

Given expression = 31 + 81 ÷ 9 – 7 = 31 + 9 – 7 = 33**Q3. If × means ÷, – means ×, ÷ means + and + means -, then (4 – 15 ÷ 12) × 8 + 9 = ? **

(a) -1

(b) 2

(c) 0

(d) 1**Ans.(c)**

Sol. Using the correct symbols, we have:

Given expression = (4 × 15 + 12) ÷ 8 – 9 = 72 ÷ 8 – 9 = 9 – 9 = 0

**Q4. If Q means ‘add to’, J means ‘multiply by’, T means ‘subtract from’ and K means ‘divide by’, then 26 K 2 Q 3 J 6 T 4 = ? **

(a) 10

(b) 28

(c) 30

(d) 27

**Ans.(d)**

Sol. Using the correct symbols, we have:

Given expression = 26 ÷ 2 + 3 × 6 – 4 = 13 + 18 – 4 = 27

**Q5. If ‘-’ stands for ‘division’, ‘+’ for ‘multiplication’, ‘÷’ for ‘subtraction’ and ‘×’ for ‘addition’, which one of the following equations is correct? **

(a) 6 + 20 – 12 ÷ 7 – 1 = 38

(b) 6 – 20 ÷ 12 × 7 + 1 = 57

(c) 6 + 20 – 12 ÷ 7 × 1 = 62

(d) 6 ÷ 20 × 12 + 7 – 1 = 70

**Ans.(d)**

Sol. Using the proper notations in (d), we get the statement as:

6 – 20 + 12 × 7 ÷ 1 = 6 – 20 + 84 = 90 – 20 = 70

**Q6. If L denotes ÷, M denotes ×, P denotes + and Q denotes -, then which of the following statements is true? **

(a) 32 P 8 L 16 Q 4 =-2/3

(b) 6 M 18 Q 26 L 13 P 7 =173/13

(c) 11 M 34 L 17 Q 8 L 3 =38/3

(d) 9 P 9 L 9 Q 9 M 9 = -71

**Ans.(d)**

Sol. Using the proper notations in (d), we get the statement as:

9 + 9 ÷ 9 – 9 × 9 = 9 + 1 – 9 × 9 = 9 + 1 – 81 = 10 – 81 = -71.

**Q7. Which one of the four interchanges in signs and numbers would make the given equation correct?**

3 + 5 – 2 = 4

(a) + and –, 2 and 3

(b) + and –, 2 and 5

(c) + and –, 3 and 5

(d) None of these

**Ans.(c)**

Sol. By making the interchanges given in (a)

2 – 5 + 3 = 4 or 0 = 4, which is false.

By making the interchanges given in (b)

3 – 2 + 5 = 4 or 6 = 4, which is false.

By making the interchanges given in (c)

5 – 3 + 2 = 4 or 4 = 4, which is true.

So, the answer is (c).

**Directions (8): In this question, ∆ means ‘is greater than’, % means ‘is lesser than’, ⃞ means ‘is equal to’, = means ‘is not equal to’, + means ‘is a little more than’, × means ‘is a little less than’. Choose the correct alternative.**

**Q8. If a ∆ b and b + c, then **

(a) a % c

(b) c + a

(c) c % a

(d) Can’t say

**Ans.(c)**

Sol. a ∆ b -> a > b and

b + c -> b is a little more than c

⇒ a > c ⇒ c < a i.e. c % a

**Directions (9-10): In each of the following questions, the Greek letters standing for arithmetical operations are given. Find the relationship which can definitely be deduced from the two relationships given at the top. **

**Operations: α is ‘greater than’, β is ‘less than’, γ is ‘not greater than’, δ is ‘not less than’, θ is ‘equal to’. ****Q9. If A α 2C and 2A θ 3B, then **

(a) C β B

(b) C δ B

(c) C α B

(d) C θ B

**Ans.(a)**

Sol. A α 2C ⇒ A > 2C

and 2A θ 3B ⇒ 2A = 3B

⇒ 2A > 4C and 2A = 3B

⇒ 3B > 4C ⇒ C < B i.e. C β B

**Q10. If B θ 2C and 3C γ A, then **

(a) B δ 2A

(b) B θ A

(c) 3B α 2A

(d) B β A

**Ans.(d)**

Sol. B θ 2C⇒ B = 2C

and 3C γ A ⇒ 3C ⊁ A

⇒B = 2C and 3C ≤ A

⇒ B = 2C < 3C ≤ A ⇒ B < A i.e. B β A

**Directions (11): In this question, α stands for ‘equal to’; β for ‘greater than’; γ for ‘less than’ and δ for ‘not equal to’. **

**Q11. If 6x α 5y and 2y β 3z, then **

(a) 2x β 3z

(b) 4x β 3z

(c) 2x γ z

(d) 4x α 3z

**Ans.(b)**

Sol. 6x α 5y ⇒ 6x = 5y

and 2y β 3z ⇒ 2y > 3z

⇒ 6x = 5y and y>3z/2

⇒ 6x = 5y and 5y >15z/2⇒6x>15z/2

⇒ 12x > 15z ⇒ 4x > 5z

⇒ 4x > 3z i.e. 4x β 3z

**Directions (12): In this question, if the given interchanges are made in signs and numbers, which one of the four equations would be correct? **

**Q12. Given interchanges : Signs + and – and number 4 and 8 **

(a) 4 ÷ 8 – 12 = 16

(b) 4 – 8 + 12 = 0

(c) 8 ÷ 4 – 12 = 24

(d) 8 – 4 ÷ 12 = 8

**Ans.(b)**

Sol. On interchanging + and – and 4 and 8 in (b), we get the equation as:

8 + 4 – 12 = 0 or 12 – 12 = 0 or 0 = 0, which is true

**Q13. If ÷ implies =, × implies <, + implies >, – implies ×, > implies ÷, < implies +, = implies -, identify the correct expression. **

(a) 1 – 3 > 2 + 1 – 5 = 3 – 1 < 2

(b) 1 – 3 > 2 + 1 × 5 = 3 × 1 > 2

(c) 1 × 3 > 2 + 1 × 5 × 3 – 1 > 2

(d) 1 – 3 > 2 + 1 × 5 + 3 – 1 > 2

**Ans.(d)**

Sol. Using the proper notations in (d), we get the statement as:

1 × 3 ÷ 2 > 1 < 5 > 3 × 1 ÷ 2 or 3/2>1<5>3/2, which is true.

**Q14. If > denotes +, < denotes -, + denotes ÷, – denotes =, = denotes ‘less than’ and × denotes ‘greater than’, find which of the following is a correct statement.**

(a) 3 + 2 > 4 = 9 + 3 < 2

(b) 3 > 2 > 4 = 18 + 3 < 1

(c) 3 > 2 < 4 × 8 + 4 < 2

(d) 3 + 2 < 4 × 9 + 3 < 3

**Ans.(c)**

Sol. Using proper notations, we have:

(a) Given statement is 3 ÷ 2 + 4 < 9 ÷ 3 – 2 or 11/2 < 1, which is not true

(b) Given statement is 3 + 2 + 4 < 18 ÷ 3 – 1 or 9 < 5, which is not true

(c) Given statement is 3 + 2 – 4 > 8 ÷ 4 – 2 or 1 > 0, which is true

(d) Given statement is 3 ÷ 2 – 4 > 9 ÷ 3 – 3 or -5/2 > 0, which is not true

So, the statement (c) is true.

**Q15. If + means ×, × means -, ÷ means + and – means ÷, then which of the following gives the result of 625 – 25 ÷ 5 + 20 × 3 + 10? **

(a) 77

(b) 95

(c) 88

(d) 137

**Ans.(b)**

Sol. Using the proper signs in the given expression, we get:

625 ÷ 25 + 5 × 20 – 3 × 10 = 25 + 5 × 20 – 3 × 10 = 25 + 100 – 30

= 125 – 30 = 95