SSC CHSL maths revision sheet

Read this sheet before every mock and every day in the final week. Don't just read it: cover the formula, say it aloud, then check. Anything that makes you hesitate needs five practice questions that same day.

Each formula has a one-line reason. The reason is what lets you rebuild a formula when memory fails in the exam hall, and it stops you applying a formula to the wrong situation.

Number system

• Divisibility: by 3 and 9, use the digit sum; by 4, the last two digits; by 8, the last three; by 11, the difference of alternate-digit sums is 0 or a multiple of 11.
• HCF × LCM = product of the two numbers. Reason: each prime's lower power goes into the HCF and its higher power into the LCM, so between them they use every factor exactly once.
• Sum of the first n natural numbers = n(n + 1)/2; sum of the first n odd numbers = n².

Percentages

• Successive change: a + b + ab/100. Reason: the second change also acts on the first change.
• A is r% more than B → B is r/(100 + r) × 100% less than A. Reason: the base switches from B to A.

Profit, loss and discount

• Profit % is on CP; discount % is on MP.
• SP = CP × (100 + P%)/100.
• MP/CP = (100 + P%)/(100 − D%). Reason: both sides equal SP when you multiply out.
• Successive discounts: a + b − ab/100.
• CP of x articles = SP of y articles → profit % = (x − y)/y × 100.

Ratio, average and mixture

• Divide in a : b → shares a/(a + b) and b/(a + b) of the total.
• Total = average × count. Reason: an average is the total shared equally.
• Alligation: cheaper : dearer = (dearer − mean) : (mean − cheaper). Reason: what the cheap part gains must equal what the dear part loses.
• The difference between two people's ages never changes.

Interest

• SI = P × R × T/100.
• CI amount = P × (1 + R/100)ⁿ for n years.
• CI − SI for 2 years = P(R/100)². Reason: the only extra in year two is interest on the first year's interest.
• Half-yearly compounding: halve the rate and double the periods.

Time, work and speed

• Total work = LCM of the times; add rates, not times.
• M₁D₁H₁ = M₂D₂H₂ for the same job.
• km/h × 5/18 = m/s. Reason: 1,000 m ÷ 3,600 s = 5/18.
• Average speed for equal distances = 2xy/(x + y). Reason: you spend longer at the slower speed, so the average is below the simple mean.
• Trains: add lengths to cross each other; relative speed adds when opposite and subtracts when same direction.
• Boats: downstream = b + s; upstream = b − s.

Algebra

• (a + b)² = a² + 2ab + b²; a² − b² = (a + b)(a − b).
• x + 1/x = k → x² + 1/x² = k² − 2 and x³ + 1/x³ = k³ − 3k.
• a + b + c = 0 → a³ + b³ + c³ = 3abc.

Geometry

• Triangle angle sum 180°; exterior angle = sum of opposite interior angles.
• Triplets: 3-4-5, 5-12-13, 8-15-17, 7-24-25.
• Similar triangles: area ratio = (side ratio)².
• Centroid 2 : 1; ∠BIC = 90° + ∠A/2; ∠BOC = 2∠A.
• Angle in a semicircle = 90°; cyclic quadrilateral opposite angles add to 180°.
• Tangent ⟂ radius; tangent length = √(d² − r²).
• Polygon interior sum = (n − 2) × 180°; each exterior angle = 360°/n.

Mensuration

• Circle 2πr and πr²; sector θ/360 of each.
• Cylinder πr²h; cone ⅓πr²h; sphere (4/3)πr³ and 4πr².
• 1 m³ = 1,000 litres. Recasting conserves volume, not surface area.

Trigonometry

• sin 30° = cos 60° = 1/2; sin 60° = cos 30° = √3/2; tan 45° = 1; tan 30° = 1/√3; tan 60° = √3.
• sin²θ + cos²θ = 1; sec²θ − tan²θ = 1; cosec²θ − cot²θ = 1.
• sin(90° − θ) = cos θ; tan θ × tan(90° − θ) = 1.

Six worked checks

Q1. A price rises 20% and then falls 20%. Find the net change.
20 − 20 − 400/100 = 4% decrease.

Q2. Find the difference between CI and SI on ₹8,000 at 5% for 2 years.
8,000 × (5/100)² = 8,000 × 1/400 = ₹20.

Q3. A shopkeeper wants a 20% profit after giving a 10% discount. How much above cost should the item be marked?
MP/CP = 120/90 = 4/3, so mark it 33⅓% above cost.

Q4. A person travels to a place at 30 km/h and returns at 20 km/h. Find the average speed.
2 × 30 × 20/50 = 24 km/h, not 25.

Q5. If x + 1/x = 3, find x³ + 1/x³.
27 − 9 = 18.

Q6. Find the length of a tangent from a point 10 cm from the centre of a circle of radius 6 cm.
√(100 − 36) = 8 cm.

Self-test

1. 37½% of 240
2. The single discount equal to successive discounts of 20% and 15%
3. SI on ₹5,000 at 6% a year for 3 years
4. A's salary is 25% more than B's. By what percentage is B's salary less than A's?
5. Time for a 150 m train at 54 km/h to pass a pole
6. Each exterior angle of a regular 12-sided polygon
7. sin 60° ÷ cos 60°
8. Rice at ₹40/kg and ₹55/kg is mixed to make a mixture costing ₹45/kg. Find the mixing ratio.

Answers:

1. 90. 3/8 × 240.
2. 32%. 20 + 15 − 300/100.
3. ₹900. 5,000 × 6 × 3/100.
4. 20%. 25/125 × 100.
5. 10 seconds. 54 km/h = 15 m/s; 150 ÷ 15.
6. 30°. 360 ÷ 12.
7. √3. That is tan 60°.
8. 2 : 1. (55 − 45) : (45 − 40) = 10 : 5.

A 20-minute routine for the final week

What to do next

• Run the 20-minute routine before every mock in the last month.
• Turn every formula you hesitated on into five practice questions today.
• Keep the option elimination techniques in mind to check answers fast.
• Follow the last-month plan for the full revision schedule.