Ranking, Order and Mathematical Operations Notes for Reasoning

Ranking, order and mathematical operations are scoring topics in the reasoning section of SSC, banking, railways, defence and state examinations. Ranking and order questions ask you to find the position of a person from the left, right, top or bottom, or to arrange people by height, age or marks. Mathematical operations questions replace the usual signs (+, -, ×, ÷) with other symbols or ask you to solve an equation after swapping signs or numbers. Both topics are based on a few simple formulas and the BODMAS rule, and once you learn them, each question takes less than a minute. This note explains the formulas, methods and solved examples in simple language.

Part 1: Ranking and Order Test

In ranking questions, people stand in a row or are placed in order of marks, height or age. You are asked for the total number of people, a person's position from the other end, or the number of people between two persons. The most important idea is that the positions from the two ends overlap on the same person, so we subtract one from the sum.

Basic Formulas of Ranking

Solved Examples on Ranking

• Example 1: Ravi is 12th from the left and 15th from the right in a row. Total = 12 + 15 - 1 = 26 persons.
• Example 2: In a class of 40 students, Meena's rank from the top is 8. Her rank from the bottom = 40 - 8 + 1 = 33.
• Example 3: In a queue, Amit is 7th from the front and Sunil is 19th from the front. People between them = 19 - 7 - 1 = 11.
• Example 4: In a row of 30, Neha is 10th from the left and Pooja is 8th from the right. People between them = 30 - (10 + 8) = 12.
• Example 5: In a row, A is 9th from the left. When A and B exchange their places, B becomes 20th from the right. B now stands at A's old place, which is 9th from the left. Total = 9 + 20 - 1 = 28 persons.

Order and Comparison Questions

These questions give you comparisons: "A is taller than B, B is taller than C" and ask who is the tallest or the shortest. Convert the statements into a chain using the signs > (greater) and < (smaller). Example: A > B, C < B, D > A means D > A > B > C. So D is the tallest and C is the shortest. If the information is not enough to connect two persons, the answer may be "cannot be determined". Always draw the chain on paper and place each new person in the correct position, and never rely on memory.

For rank questions with marks, remember that "Rank from the top" and "rank from the bottom" can be converted using the same formula as above. If two students have the same marks, the question will state how the ranks are given. A useful point is that the number of persons above a person is rank minus one, and the number of persons below is total minus rank.

Part 2: Mathematical Operations

In these questions, the arithmetic signs are given new meanings, and you must solve the expression after replacing the symbols with the correct signs. The question may also ask you to swap two numbers or two signs and then find the value. The key is to read the instruction carefully, make the replacements correctly and then solve with the BODMAS rule.

BODMAS Rule

BODMAS gives the order of operations: Brackets, Orders (powers and roots), Division, Multiplication, Addition, Subtraction. Division and multiplication have the same priority and are solved from left to right; addition and subtraction also have the same priority and are solved from left to right. Within brackets, the order is: round brackets ( ), then curly brackets { }, then square brackets [ ].

Types of Mathematical Operations Questions

1. Sign replacement: "If + means ×, - means ÷, × means - and ÷ means +, find the value of 12 + 3 - 4 × 2 ÷ 5." Replace the symbols: 12 × 3 ÷ 4 - 2 + 5 = 36 ÷ 4 - 2 + 5 = 9 - 2 + 5 = 12.
2. Number swapping: "If 4 and 6 are interchanged, find the value of 4 × 5 + 6." After swapping: 6 × 5 + 4 = 30 + 4 = 34.
3. Sign swapping: "If + and × are interchanged, find 6 + 3 × 4." After swapping: 6 × 3 + 4 = 18 + 4 = 22.
4. Fictitious operations: "If a ★ b = 2a + b, find 3 ★ 4." Put the values: 2(3) + 4 = 10.
5. Finding the missing sign: "Which signs fill the blanks in 8 _ 4 _ 2 = 4?" Try combinations: 8 ÷ 4 × 2 = 4. So the signs are ÷ and ×.
6. Letter-symbol type: "If A means +, B means - and C means ×, find 8 C 3 A 4 B 2." That is 8 × 3 + 4 - 2 = 24 + 4 - 2 = 26.
7. Pattern-based equations: "If 2 ★ 3 = 12 and 3 ★ 4 = 24, then 4 ★ 5 = ?" Test simple rules on every given example. Here a × b × 2 gives 2 × 3 × 2 = 12 and 3 × 4 × 2 = 24, so the rule fits both. Then 4 ★ 5 = 4 × 5 × 2 = 40. Always test squares, products, sums and differences until one rule fits all the examples.

Step-by-Step Method for Operation Questions

1. Read the instruction twice and write the replacements in a small table.
2. Rewrite the whole expression with the real signs, without solving.
3. Apply BODMAS carefully; first brackets, then orders, then division and multiplication, then addition and subtraction.
4. For pattern-based equations, test the rule on every given example before applying it to the question.

Common Mistakes

In ranking, the common mistake is forgetting to subtract one while adding the two ranks, or subtracting one when finding the persons between two ranks from the same end. Another mistake is confusing "between" with "after". In order questions, students sometimes assume equality where it is not stated. In mathematical operations, the main error is solving the expression before replacing all the signs, or solving addition before multiplication. Another trap is changing only the signs and not the numbers when the question asks for interchange of numbers. Check your answer by putting it back into the original instruction.

Practice Questions With Answers

Try these: (1) A boy is 14th from the left and 9th from the right. How many boys are there? Answer: 14 + 9 - 1 = 22. (2) In a class of 45, Rita is 12th from the top. What is her rank from the bottom? Answer: 45 - 12 + 1 = 34. (3) A is 5th from the left and B is 20th from the left. How many are between them? Answer: 20 - 5 - 1 = 14. (4) If P > Q, R < Q and S > P, who is the smallest? Answer: R (S > P > Q > R). (5) If + means -, - means ×, × means ÷ and ÷ means +, find 20 ÷ 4 × 2 - 3 + 1. Answer: 20 + 4 ÷ 2 × 3 - 1 = 20 + 2 × 3 - 1 = 20 + 6 - 1 = 25. (6) If 3 and 5 are interchanged, find 3 × 4 + 5. Answer: 5 × 4 + 3 = 23.

Quick Revision

• Total = left rank + right rank - 1; other-end rank = total - rank + 1.
• Persons between two ranks from the same end = difference - 1.
• Draw a comparison chain for order questions.
• Replace the signs first, then use BODMAS.
• Test the pattern on all examples in fictitious operation questions.

With regular practice of ten questions a day, these topics can become your quickest marks in the reasoning section.

Related Notes to Read Next

Continue your preparation with these related notes from the same subject:

• Puzzles Notes for Reasoning
• Statement and Assumptions Notes for Reasoning
• Statement and Conclusions and Course of Action Notes
• All Reasoning notes

Frequently Asked Questions (FAQ)

What is the formula for total persons in a ranking question?

Total number of persons equals the rank from the left plus the rank from the right minus one.

How do I find a rank from the other end?

Rank from the other end equals the total number of persons minus the given rank plus one.

How do I find the number of persons between two people?

If both ranks are from the same end, subtract the smaller rank from the larger and subtract one more.

What is the BODMAS rule?

BODMAS is the order of operations: brackets, orders, division, multiplication, addition and subtraction, solved in that order.

Which has higher priority, multiplication or division?

Both have the same priority, so they are solved from left to right in the order in which they appear.

How do I solve sign replacement questions?

Write down the new meanings of the symbols, rewrite the expression with the real signs, and then solve it using BODMAS.

What if two persons exchange their positions?

Use the new ranks and the old ranks to find how many places they moved, and use the shift to find the total or the gap.

How do I solve order and comparison questions?

Convert each statement into a chain using greater than and less than signs and join the chains to find the tallest, smallest or heaviest person.

What does cannot be determined mean in order questions?

It means the given statements do not give enough information to compare the two persons in question.

What are fictitious operations?

They are operations defined by a new rule such as a ★ b = 2a + b, and you apply the rule to the numbers given in the question.

What is the most common mistake in ranking questions?

Forgetting to subtract one while adding left and right ranks, or while counting persons between two ranks, is the most common mistake.

How can I practise ranking and mathematical operations?

Solve ten questions daily, write each step on paper, learn the formulas by heart, and review the mistakes in a notebook.