Seven topics that show up in almost every aptitude test. Each one gets a plain-English idea, the formula you actually need, and worked examples you can copy the method from.
Chapter 01
LCM & HCF
HCF (Highest Common Factor) is the biggest number that divides two or more numbers exactly. LCM (Lowest Common Multiple) is the smallest number that all of them divide into exactly. Think of HCF as "the biggest shared piece" and LCM as "the smallest number they all fit into."
HCF → break each number into prime factors, multiply the common primes (smallest power)
LCM → multiply all primes that appear, each raised to its highest power
The shared middle is the HCF. Everything in both circles combined (highest powers) is the LCM.
Example 1
Find the HCF and LCM of 12 and 18.
12 = 2 × 2 × 3 → 2² × 3
18 = 2 × 3 × 3 → 2 × 3²
HCF = common primes at lowest power = 2 × 3 = 6
LCM = all primes at highest power = 2² × 3² = 36
HCF = 6, LCM = 36
Example 2
Three bells ring every 4, 6, and 8 minutes. If they ring together now, after how many minutes will they ring together again?
This is asking for LCM(4, 6, 8)
4 = 2², 6 = 2×3, 8 = 2³
LCM = 2³ × 3 = 24
They ring together again after 24 minutes
Rule of thumb: "together again / repeats" questions → LCM. "Split into equal, largest groups" questions → HCF.
Shortcut: LCM × HCF = product of the two numbers (only works for exactly two numbers). So for 12 and 18: 6 × 36 = 216 = 12 × 18 ✓ — handy for checking your answer fast.
Practice
1. Find the LCM of 8 and 12.
2. Find the HCF of 24 and 36.
Chapter 02
Time and Work
The whole trick is to stop thinking in days and start thinking in "how much of the job gets done in one day." If someone finishes a job in 10 days, they complete 1/10th of it every single day.
One day's work = 1 ÷ (days taken to finish alone)
Combined one day's work = add each person's one-day work
Days to finish together = 1 ÷ (combined one day's work)
Stack each person's daily share end to end — the combined bar shows how much of the job disappears in one day when they work together.
Example 1
A can finish a job in 10 days, B can finish it in 15 days. How long will they take working together?
A's one-day work = 1/10, B's one-day work = 1/15
Together per day = 1/10 + 1/15 = 3/30 + 2/30 = 5/30 = 1/6
Days to finish = 1 ÷ (1/6) = 6 days
6 days together
Example 2
A and B together finish a job in 12 days. A alone can do it in 20 days. How long will B alone take?
Same logic extends to pipes filling/emptying a tank — just swap "work" for "tank filled."
Practice
1. A can do a job in 8 days, B in 12 days. How long will they take together?
2. 6 workers finish a job in 10 days. How many days will 4 workers take (same work rate each)?
Chapter 03
Speed, Time & Distance
One relationship drives everything here: distance is speed multiplied by time. Almost every question is this formula rearranged, plus a unit conversion.
Distance = Speed × Time | Speed = Distance ÷ Time | Time = Distance ÷ Speed
km/hr → m/s: multiply by 5/18 | m/s → km/hr: multiply by 18/5
Cover the one you want to find. D on top of S and T means D = S × T; hide S and it's D ÷ T; hide T and it's D ÷ S.
Example 1
A train 150 m long crosses a pole in 10 seconds. Find its speed in km/hr.
Crossing a pole means the train covers a distance equal to its own length
Speed = distance/time = 150/10 = 15 m/s
Convert to km/hr: 15 × 18/5 = 54 km/hr
54 km/hr
Example 2
A man travels from home to office at 40 km/hr and returns the same route at 60 km/hr. Find his average speed for the whole trip.
For a round trip with the same distance both ways, do NOT just average the two speeds
Use: Average speed = 2 × S₁ × S₂ ÷ (S₁ + S₂)
= 2 × 40 × 60 ÷ (40 + 60) = 4800 ÷ 100 = 48 km/hr
48 km/hr
Common trap: the answer is NOT (40+60)/2 = 50. That only works if time, not distance, is equal on both legs.
Practice
1. A car covers 180 km in 3 hours. Find its speed.
2. Convert 72 km/hr into m/s.
Chapter 04
Profit & Loss
CP is what something cost to buy or make. SP is what it sold for. If SP is more, it's profit; if less, it's loss — and the percentage is always measured against the CP.
2. An item bought for ₹500 is sold at a 10% loss. Find the selling price.
Chapter 05
Percentage
"Percent" just means "out of 100." Once you can convert between a percentage, a fraction, and a decimal in your head, most percentage questions become simple multiplication.
x% of a number N = (x ÷ 100) × N
% change = (change in value ÷ original value) × 100
35 out of 100 squares shaded = 35%. That's really all "percent" means — a count out of a hundred.
Example 1
What is 35% of 240?
35% of 240 = (35 ÷ 100) × 240
= 0.35 × 240 = 84
84
Example 2
A student's marks increased from 80 to 92. Find the percentage increase.
Increase = 92 − 80 = 12
% increase = (12 ÷ 80) × 100 = 15%
15% increase
Always divide by the original (starting) value, never the new one.
Practice
1. What is 25% of 160?
2. Price rises from ₹200 to ₹250. Find the percentage increase.
Chapter 06
Simple & Compound Interest
Simple interest is calculated only on the original amount, every year. Compound interest is calculated on the original amount plus whatever interest has already piled up — so it grows faster.
Simple Interest (SI) = (P × R × T) ÷ 100
Compound Interest (CI) = P × (1 + R/100)ᵀ − P
P = Principal, R = Rate % per year, T = Time in years
Same starting amount and rate — SI grows in a straight line, CI curves upward because each year's interest also earns interest.
Example 1 — Simple Interest
Find the simple interest on ₹5,000 at 8% per year for 3 years.
SI = (5000 × 8 × 3) ÷ 100
= 120000 ÷ 100 = ₹1,200
₹1,200
Example 2 — Compound Interest
Find the compound interest on ₹5,000 at 8% per year for 2 years.
Amount = 5000 × (1 + 8/100)² = 5000 × (1.08)²
= 5000 × 1.1664 = ₹5,832
CI = 5832 − 5000 = ₹832
₹832
Notice CI (₹832 for 2 years) is a bit more than double one year's SI (8% of 5000 = ₹400) — that extra bit is "interest on interest."
Practice
1. Find the simple interest on ₹4,000 at 5% per year for 2 years.
2. Find the compound interest on ₹2,000 at 10% per year for 2 years.
Chapter 07
Logical Reasoning
Reasoning questions don't need formulas — they need pattern-spotting. Look for what changes between terms (add, multiply, alternate) or what rule connects letters, numbers, and positions.
Common types: number/letter series, coding-decoding, blood relations, direction sense, seating arrangement
Spot the rule connecting each term to the next, then apply it one more time. Here every step is "×3", so the next term is 54 × 3 = 162.
Example 1 — Number series
Find the next number: 3, 6, 12, 24, ?
Check the pattern between terms: 3→6 (×2), 6→12 (×2), 12→24 (×2)
Each term is double the previous one
Next term = 24 × 2 = 48
48
Example 2 — Coding-decoding
If CAT is coded as DBU, how is DOG coded?
Compare CAT → DBU letter by letter: C→D, A→B, T→U
Each letter has simply shifted forward by 1 in the alphabet
Apply the same shift to DOG: D→E, O→P, G→H
EPH
Example 3 — Direction sense
Riya walks 5 km North, then turns right and walks 3 km. Which direction is she facing now, and how would she head straight back to her starting point?
Facing North, turning right means she now faces East
She walked 5 km North then 3 km East
To go straight back to the start, she'd need to head West and South — i.e. South-West
She's facing East; the start point is to the South-West
Tip: sketch a quick compass and trace the path — direction questions are much easier drawn than imagined.