Topics Covered
Types of Reasoning
Number Series
Letter Series
Coding and Decoding
Blood Relations and Directions
Fractions
Time and Distance
Ratio, Proportion and Percentage
Profit and Loss
Interest and Discounting
Averages
Contents
Topic Overview — What & Why
Unit V is the arithmetic and pattern-finding unit. Nothing here is beyond school mathematics; what the paper tests is speed and care. Most questions can be solved in under a minute with the right shortcut, and most wrong options are the answers a hurried candidate gets: the simple average where a weighted one was needed, simple interest where compound was asked, a percentage of the wrong base.
- Reasoning is deductive (certain), inductive (probable), abductive (the best explanation) or analogical.
- Series: find the rule (differences, ratios, squares, alternation), then apply it once more.
- Codes and relationships: a letter-shift or a pattern; a family tree drawn on paper; directions drawn as a sketch.
- Aptitude: a small set of formulas, each worked below, and the habit of checking the base of every percentage.
1. Types of Reasoning
| Type | Moves from | Conclusion | Example |
|---|---|---|---|
| Deductive | General premises to a particular conclusion | Certain, if the premises are true and the form is valid | All metals conduct electricity; copper is a metal; so copper conducts electricity |
| Inductive | Particular observations to a general conclusion | Probable; more cases make it stronger, never certain | Every crow seen so far is black; so all crows are black |
| Abductive | An observation to its most likely explanation | A plausible hypothesis, to be tested (C. S. Peirce's term) | The grass is wet this morning; the likeliest explanation is that it rained overnight |
| Analogical | A likeness between two cases | What holds of one probably holds of the other | A drug that lowers blood pressure in rats may do so in humans |
Two more terms the paper uses: mathematical reasoning proves results by deduction from axioms (and by mathematical induction, which, despite its name, is a deductive proof); statistical reasoning is inductive, drawing conclusions about a population from a sample, with a stated uncertainty.
EXAMPLE 1
"The sum of the first n odd numbers is n²: it holds for n = 1, and if it holds for n it holds for n + 1." This is proof by mathematical induction, and the conclusion is certain: deductive reasoning.
EXAMPLE 2
"In a survey of 1,000 voters, 54% favoured the proposal, so about 54% of all voters do." This is inductive (statistical) reasoning; its conclusion is probable, with a margin of error.
2. Number Series and Letter Series
Find the rule, test it on every term given, then apply it once more. Try these in order:
- Differences between terms: constant (arithmetic), or themselves in a pattern (2, 4, 6, …).
- Ratios: constant (geometric), or ×2, ×3, ….
- Squares, cubes and primes: 1, 4, 9, 16; 1, 8, 27, 64; 2, 3, 5, 7, 11.
- Mixed operations: ×2 + 1; ×3 − 1.
- Two series interleaved: the odd-placed and even-placed terms follow different rules.
EXAMPLE 1
2, 6, 12, 20, 30, ? — the differences are 4, 6, 8, 10, so the next difference is 12 and the next term 42. (The terms are also 1×2, 2×3, 3×4, 4×5, 5×6, and 6×7 = 42.)
EXAMPLE 2
3, 7, 15, 31, 63, ? — each term is twice the last plus one: 63×2 + 1 = 127.
Letter series
Number the alphabet A = 1 to Z = 26 and work with the numbers. Useful anchors: E = 5, J = 10, O = 15, T = 20, Y = 25 ("EJOTY"). The reverse position of a letter is 27 minus its position (A ↔ Z, B ↔ Y, M ↔ N).
EXAMPLE 3
B, E, H, K, ? — positions 2, 5, 8, 11: add 3 each time, so 14 = N.
EXAMPLE 4
AZ, BY, CX, ? — the first letters go forward from A, the second back from Z, each pair adding to 27: the next pair is DW.
3. Codes and Relationships
Coding and decoding
A word is coded by a rule: each letter moved forward or back by a fixed number, letters replaced by their reverse, the word written backwards, or letters replaced by numbers. Find the rule from the given pair, then apply it.
EXAMPLE 1
If CAT is coded DBU, each letter moves forward by 1. So DOG is coded EPH.
EXAMPLE 2
If each letter is replaced by its position, BAD = 2 + 1 + 4 = 7 when the code is the sum. Then CAB = 3 + 1 + 2 = 6.
Blood relations
Draw the family tree: one generation per row, men and women marked, marriages with a double line. Read the relationship from the speaker's point of view.
EXAMPLE 3
Pointing to a man, Meena says, "He is the son of my grandfather's only son." Her grandfather's only son is her father, and his son is her brother.
Directions
Sketch every move. For distance from the start, use Pythagoras on the net north–south and east–west movement. Turning right from north faces east; turning left from north faces west.
EXAMPLE 4
A man walks 3 km north, then 4 km east. He is √(3² + 4²) = 5 km from the start, to the north-east.
4. Mathematical Aptitude
Fractions
To compare fractions, put them over a common denominator, or turn them into decimals. 3/5 = 0.6 and 5/8 = 0.625, so 5/8 is larger. The sum 1/2 + 1/3 = 3/6 + 2/6 = 5/6.
Time and distance
- Speed = distance ÷ time. To turn km/h into m/s multiply by 5/18; m/s into km/h, by 18/5. So 72 km/h = 20 m/s.
- Average speed is total distance ÷ total time, not the mean of the speeds. For two equal distances at speeds x and y, it is 2xy ÷ (x + y).
- Relative speed: add the speeds when moving towards each other, subtract when moving the same way.
- A train passing a pole covers its own length; passing a platform, its length plus the platform's.
EXAMPLE 1
A car goes from A to B at 40 km/h and returns at 60 km/h. Average speed = 2 × 40 × 60 ÷ (40 + 60) = 4,800 ÷ 100 = 48 km/h, not 50.
EXAMPLE 2
A train 150 m long at 54 km/h (= 15 m/s) passes a platform 300 m long in (150 + 300) ÷ 15 = 30 seconds.
Ratio, proportion and percentage
- To combine A : B = 2 : 3 and B : C = 4 : 5, make B the same in both: A : B = 8 : 12 and B : C = 12 : 15, so A : B : C = 8 : 12 : 15.
- Four numbers are in proportion when a : b = c : d, that is ad = bc.
- Percentage change = (new − old) ÷ old × 100. The base is always the old value.
- Successive changes of a% and b% give a net change of a + b + ab/100 percent. A rise of 20% followed by a fall of 20% is 20 − 20 − 400/100 = −4%: a net fall of 4%.
- If A is r% more than B, then B is r ÷ (100 + r) × 100 percent less than A. If A earns 25% more than B, B earns 25/125 × 100 = 20% less than A.
Profit and loss
- Profit % = profit ÷ cost price × 100 (on cost, unless the question says otherwise).
- Discount is on the marked price: selling price = marked price × (1 − discount/100).
EXAMPLE 3
An article bought for ₹400 is sold for ₹500. Profit = ₹100, and profit % = 100/400 × 100 = 25%.
EXAMPLE 4
A shopkeeper marks a shirt at ₹800 and gives a 10% discount. Selling price = 800 × 0.9 = ₹720. If it cost ₹600, the profit is ₹120, which is 20% of cost.
Interest and discounting
- Simple interest SI = P × R × T ÷ 100.
- Compound interest: amount A = P(1 + R/100)T; CI = A − P.
- For two years, CI − SI = P(R/100)².
- Discounting runs interest backwards. The present value of an amount A due after T years at R% compound interest is A ÷ (1 + R/100)T. The true discount is A minus its present value.
EXAMPLE 5
₹10,000 for 2 years at 10%: SI = 10,000 × 10 × 2 ÷ 100 = ₹2,000. Amount at compound interest = 10,000 × 1.1² = ₹12,100, so CI = ₹2,100. The difference, ₹100, equals 10,000 × (0.1)².
EXAMPLE 6
The present value of ₹11,000 due in one year at 10% is 11,000 ÷ 1.1 = ₹10,000; the true discount is ₹1,000.
Averages
- Average = sum ÷ number of items, so sum = average × number. Most average questions are solved by working with sums.
- A weighted average weights each group's average by its size.
EXAMPLE 7
Five numbers average 20. One is removed and the other four average 18. The removed number is 5 × 20 − 4 × 18 = 100 − 72 = 28.
EXAMPLE 8
A class has 30 boys averaging 60 marks and 20 girls averaging 70. The class average is (30 × 60 + 20 × 70) ÷ 50 = (1,800 + 1,400) ÷ 50 = 64, not 65.
Exam pointers for Unit V.
- Average speed over equal distances: 2xy ÷ (x + y), never (x + y) ÷ 2.
- km/h × 5/18 = m/s.
- Successive changes: a + b + ab/100. Equal rise and fall always leave a net fall.
- Profit % is on cost; discount % is on the marked price.
- For two years, CI − SI = P(R/100)².
- Averages: work with sums. Weighted averages: weight by group size.
- Letters: EJOTY = 5, 10, 15, 20, 25; reverse position = 27 − position.
Final Revision Checklist
- Deductive, inductive, abductive and analogical reasoning, with an example of each.
- The five rules to try on a number series; letter positions and reverse positions.
- Letter-shift and number codes; family trees; direction sketches.
- Fractions; speed, average speed, relative speed, trains.
- Combining ratios; percentage change on the right base; successive changes.
- Profit on cost, discount on marked price; SI, CI, present value and true discount; averages through sums.