SSC CGL Quantitative Aptitude: Quick Revision Formulas & Shortcuts
Master high-yield formulas, short tricks, and quick-solving methods for both Arithmetic and Advance Mathematics. Boost your speed and accuracy in SSC CGL Tier-1 and Tier-2.
🎯 Table of Contents
Arithmetic forms the core of the SSC CGL Quantitative Aptitude section. Focus on the core relationships and percentage conversions to solve these quickly.
Percentage conversions allow you to transform complex multiplication into simple division. Learn the fraction-to-percentage conversions below:
| Fraction | Percentage Value | Fraction | Percentage Value |
|---|---|---|---|
| 1/2 | 50% | 1/9 | 11.11% (11 1/9%) |
| 1/3 | 33.33% (33 1/3%) | 1/11 | 9.09% (9 1/11%) |
| 1/4 | 25% | 1/12 | 8.33% (8 1/3%) |
| 1/5 | 20% | 1/15 | 6.67% (6 2/3%) |
| 1/6 | 16.67% (16 2/3%) | 1/16 | 6.25% (6 1/4%) |
| 1/7 | 14.28% (14 2/7%) | 1/24 | 4.16% (4 1/6%) |
| 1/8 | 12.5% (12 1/2%) | 1/25 | 4% |
(Note: Use + for increase/gain and – for decrease/loss/discount)
Decrease (%) = [ P / (100 + P) ] * 100
Profit and loss calculations are simplified when using the direct relation between Cost Price (CP), Marked Price (MP), and Discount.
If there is a loss of L%, replace Profit% with -L%
Alternative: Profit (%) = [ (True Weight – False Weight) / False Weight ] * 100
Compound interest calculations can be simplified using effective rates or the difference formulas for SI and CI.
CI = A – P
For 3 Years: Diff (D3) = P * (R / 100)^2 * [ (300 + R) / 100 ]
Efficiency is inversely proportional to time taken when the total work is constant.
If A does work in x days, and B in y days: Together they take [ (x * y) / (x + y) ] days
Where M=Men, D=Days, H=Hours, E=Efficiency, W=Work
Relative speed depends on the direction of travel, while boat problems require defining the stream’s influence.
1 m/s = 18/5 km/h
If different distances: Avg Speed = Total Distance / Total Time
Downstream Speed (D) = u + v
Upstream Speed (U) = u – v
Boat Speed in still water (u) = (D + U) / 2
Stream Speed (v) = (D – U) / 2
Advance Maths accounts for a significant portion of SSC CGL, especially in Tier-2. Focus on memorizing trigonometric identities and geometric theorems.
Symmetric algebraic terms of the form x + 1/x are highly recurring in SSC exams.
• (a + b)^2 – (a – b)^2 = 4ab
• a^3 + b^3 + c^3 – 3abc = (a + b + c)(a^2 + b^2 + c^2 – ab – bc – ca)
• a^3 + b^3 + c^3 – 3abc = 0.5 * (a + b + c)[(a – b)^2 + (b – c)^2 + (c – a)^2]
• IF a + b + c = 0, then: a^3 + b^3 + c^3 = 3abc
• x^3 + 1/x^3 = k^3 – 3k
• x^4 + 1/x^4 = (k^2 – 2)^2 – 2
• x^5 + 1/x^5 = (x^2 + 1/x^2)(x^3 + 1/x^3) – (x + 1/x) = (k^2 – 2)(k^3 – 3k) – k
• x^6 + 1/x^6 = (k^3 – 3k)^2 – 2
• x^3 – 1/x^3 = k^3 + 3k
Standard values and reciprocal relationships are the most efficient ways to solve trigonometry questions.
• sec^2 θ – tan^2 θ = 1 ⇒ (sec θ – tan θ) = 1 / (sec θ + tan θ)
• cosec^2 θ – cot^2 θ = 1 ⇒ (cosec θ – cot θ) = 1 / (cosec θ + cot θ)
| Ratio | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin θ | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| cos θ | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| tan θ | 0 | 1/√3 | 1 | √3 | ∞ |
• sin^2 A + sin^2 B = 1 | cos^2 A + cos^2 B = 1
Understand the properties of centers of triangles and properties of circles involving chords and tangents.
• Orthocenter: ∠BOC = 180° – ∠A
• Circumcenter: ∠BSC = 2∠A
• Equilateral Triangle: r = a / 2√3 | R = a / √3
• Transverse Common Tangent (TCT) = √[ d^2 – (R + r)^2 ]
• Tangent Secant Property: PT^2 = PA * PB (Where PT is tangent, PAB is secant)
Formulas for 2D areas and 3D volumes/surfaces must be memorized thoroughly.
| Shape (3D) | Volume | Curved Surface Area (CSA) | Total Surface Area (TSA) |
|---|---|---|---|
| Cuboid | l * b * h | 2h(l + b) | 2(lb + bh + hl) |
| Cube | a^3 | 4a^2 | 6a^2 |
| Cylinder | π * r^2 * h | 2 * π * r * h | 2 * π * r * (r + h) |
| Cone | (1/3) * π * r^2 * h | π * r * l (where l = √[r^2+h^2]) | π * r * (r + l) |
| Sphere | (4/3) * π * r^3 | 4 * π * r^2 | 4 * π * r^2 |
| Hemisphere | (2/3) * π * r^3 | 2 * π * r^2 | 3 * π * r^2 |
Using shortcuts is key to saving time. Apply these mental math hacks to speed up your calculation by 2x.
Digital sum is the sum of digits of a number continued until a single digit remains. In this process, 9 is treated as 0 (or ignored).
Application: In the equation 345 * 12 = 4140, check digital sums:
• DS(345) = 3+4+5 = 12 ⇒ 1+2 = 3.
• DS(12) = 1+2 = 3.
• Product DS = 3 * 3 = 9 (or 0).
• DS(4140) = 4+1+4+0 = 9. The LHS and RHS digital sums match! Use this to eliminate incorrect options in seconds.
Pythagorean triplets are sets of three integers (a, b, c) that satisfy a^2 + b^2 = c^2. Memorizing them saves time in geometry, trigonometry, and mensuration.
| Primary Triplets | Common Multiples (also valid triplets) |
|---|---|
| 3, 4, 5 | 6, 8, 10 | 9, 12, 15 | 12, 16, 20 | 15, 20, 25 |
| 5, 12, 13 | 10, 24, 26 | 15, 36, 39 |
| 8, 15, 17 | 16, 30, 34 |
| 7, 24, 25 | 14, 48, 50 |
| 9, 40, 41 | 18, 80, 82 |
| 20, 21, 29 | 40, 42, 58 |
Instead of using the formula a + b + ab/100, which becomes complex for non-integer rates (like 12.5%, 16.67%), use the ratio method.
• 12.5% increase = 1/8 increase ⇒ Ratio = 8 to 9
• 16.67% increase = 1/6 increase ⇒ Ratio = 6 to 7
• Multiply initial and final ratios: (8 * 6) to (9 * 7) ⇒ 48 to 63 ⇒ 16 to 21
• Net Increase = (5 / 16) * 100 = 31.25%
Practice these actual exam-pattern questions and see how the formulas and shortcuts are applied to find solutions in under 30 seconds.
View Shortcut Solution
Shortcut Approach:
• Profit of 25% is entirely due to water added.
• 25% = 1/4 in fraction.
• This means if Milk (Cost Price part) = 4 units, then Water (Profit part) = 1 unit.
• Total Mixture = Milk + Water = 4 + 1 = 5 units.
• Percentage of water in the mixture = [ Water / Total Mixture ] * 100 = [ 1 / 5 ] * 100 = 20%.
• Time saved: No need to assume prices or weights!
View Shortcut Solution
Shortcut Approach:
Use the symmetric formula: x⁵ + 1/x⁵ = (x² + 1/x²)(x³ + 1/x³) – (x + 1/x)
Given k = 3:
• x² + 1/x² = k² – 2 = 3² – 2 = 7
• x³ + 1/x³ = k³ – 3k = 3³ – 3(3) = 27 – 9 = 18
• Therefore, x⁵ + 1/x⁵ = (7 * 18) – 3 = 126 – 3 = 123.
• Time saved: Solved in 3 simple arithmetic steps!
View Shortcut Solution
Shortcut Approach:
• When two circles of radii R and r touch each other externally, the distance between their centers (d) is exactly R + r.
• Substituting d = R + r in DCT formula:
DCT = √[ (R + r)² – (R – r)² ] = √[ 4Rr ] = 2√(Rr)
• Here, R = 9 cm and r = 4 cm.
• DCT = 2 * √(9 * 4) = 2 * √36 = 2 * 6 = 12 cm.
• Time saved: Use 2√(Rr) directly for externally touching circles.
View Shortcut Solution
Shortcut Approach (Value Putting Method):
• In trigonometric identity expressions, you can substitute a standard angle θ that does not make any denominator zero.
• Let’s put θ = 0°:
sin 0° = 0, cos 0° = 1.
• Substitute these values in the expression:
[ (1³ + 0³) / (1 + 0) ] + [ (1³ – 0³) / (1 – 0) ] = [ 1 / 1 ] + [ 1 / 1 ] = 1 + 1 = 2.
• Mathematical Proof: Using a³±b³ identities:
(cos² θ – sin θ cos θ + sin² θ) + (cos² θ + sin θ cos θ + sin² θ) = (1 – sin θ cos θ) + (1 + sin θ cos θ) = 2.
View Shortcut Solution
Shortcut Approach:
• Volume of a sphere V = (4/3) * π * r³, which means V is directly proportional to r³ (radius cubed).
• A 10% increase means the radius increases from 10 to 11 (ratio = 11/10).
• Volume ratio change = (11/10)³ = 1331 / 1000.
• Percentage increase = [ (1331 – 1000) / 1000 ] * 100 = [ 331 / 1000 ] * 100 = 33.1%.
• Alternative: Successive percentage of 10% three times: 10% & 10% = 21%. Then 21% & 10% = 21 + 10 + 2.1 = 33.1%.
SSC CGL मात्रात्मक अभिरुचि: त्वरित रिवीजन सूत्र और शॉर्टकट्स
अंकगणित (Arithmetic) और अग्रिम गणित (Advance Maths) दोनों के लिए उच्च-प्रतिफल (high-yield) सूत्रों, शॉर्ट ट्रिक्स और त्वरित समाधान विधियों में महारत हासिल करें। SSC CGL Tier-1 और Tier-2 में अपनी गति और सटीकता बढ़ाएं।
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