Episode 8 — Aptitude and Reasoning / 8.1 — Percentage
8.1 Quick Revision -- Percentage Cheat Sheet
Use this for last-minute revision before exams. Everything important on one page.
Core Formulas
Percentage = (Part / Whole) x 100
Part = (Percentage / 100) x Whole
Whole = (Part x 100) / Percentage
% Increase = [(New - Old) / Old] x 100
% Decrease = [(Old - New) / Old] x 100
New after r% increase = Old x (1 + r/100)
New after r% decrease = Old x (1 - r/100)
Old from New (increase) = New / (1 + r/100)
Old from New (decrease) = New / (1 - r/100)
Successive Percentage Changes
Two changes a% and b%:
Net change = a + b + (ab / 100)
(Use negative sign for decrease)
Three changes a%, b%, c%:
Net multiplier = (1 + a/100)(1 + b/100)(1 + c/100)
Common results (memorize):
+10%, +10% --> +21% +10%, -10% --> -1%
+20%, +20% --> +44% +20%, -20% --> -4%
+25%, +25% --> +56.25% +25%, -25% --> -6.25%
+50%, +50% --> +125% +50%, -50% --> -25%
Base Switching Formulas
A is r% MORE than B --> B is less than A by: [r / (100 + r)] x 100 %
A is r% LESS than B --> B is more than A by: [r / (100 - r)] x 100 %
Quick reference:
25% more <--> 20% less
33.33% more <--> 25% less
50% more <--> 33.33% less
100% more <--> 50% less
20% more <--> 16.67% less
Fraction-Percentage Table
1/2 = 50% 1/3 = 33.33% 2/3 = 66.67%
1/4 = 25% 3/4 = 75% 1/5 = 20%
2/5 = 40% 3/5 = 60% 4/5 = 80%
1/6 = 16.67% 5/6 = 83.33% 1/7 = 14.28%
2/7 = 28.57% 3/7 = 42.86% 1/8 = 12.5%
3/8 = 37.5% 5/8 = 62.5% 7/8 = 87.5%
1/9 = 11.11% 1/10 = 10% 1/11 = 9.09%
1/12 = 8.33% 1/15 = 6.67% 1/16 = 6.25%
1/20 = 5% 1/25 = 4% 1/50 = 2%
Multiplier Table
% Change Multiplier Fraction Form
------- ---------- -------------
+10% 1.1 11/10
+12.5% 1.125 9/8
+14.28% ~1.1428 8/7
+16.67% ~1.1667 7/6
+20% 1.2 6/5
+25% 1.25 5/4
+33.33% ~1.3333 4/3
+50% 1.5 3/2
+66.67% ~1.6667 5/3
+100% 2.0 2/1
-10% 0.9 9/10
-12.5% 0.875 7/8
-14.28% ~0.8571 6/7
-16.67% ~0.8333 5/6
-20% 0.8 4/5
-25% 0.75 3/4
-33.33% ~0.6667 2/3
-50% 0.5 1/2
Growth and Decay
Population after n years = P x (1 + r/100)^n
Depreciation after n yrs = V x (1 - r/100)^n
Population n years ago = Current / (1 + r/100)^n
Price-Consumption Relationship
Expenditure = Price x Consumption
Price up r%, keep expenditure same:
Reduce consumption by [r / (100 + r)] x 100 %
Price down r%, keep expenditure same:
Increase consumption by [r / (100 - r)] x 100 %
Election Problems Template
Total votes
- Did not vote = x%
= Votes cast
- Invalid votes
= Valid votes
Winner gets P% of valid votes
Loser gets (100 - P)% of valid votes
Margin = (2P - 100)% of valid votes
Product Change Rule
When two quantities multiply to give a third:
Result = Quantity1 x Quantity2
If Q1 changes by a% and Q2 changes by b%:
Net % change in Result = a + b + ab/100
Applications:
- Area = Length x Breadth
- Revenue = Price x Quantity
- Distance = Speed x Time
- Volume of cube: side^3 changes by (1 + r/100)^3
Shortcut Methods Summary
| Technique | When to Use |
|---|---|
| Fraction conversion | % is a recognizable fraction (33.33%, 12.5%, etc.) |
| 10% anchor | Break any % into multiples of 10%, 5%, 1% |
| Multiplier method | Any increase/decrease calculation |
| Assume 100 | No actual values given in the problem |
| Back-substitution | MCQ -- plug options into the problem |
| Approximation | Options are far apart |
| Units digit check | Eliminate wrong MCQ options |
Common Traps -- Do NOT Fall For These
TRAP: "A is 25% more than B" does NOT mean "B is 25% less than A"
FIX: B is less than A by 25/125 x 100 = 20%
TRAP: Successive changes of +20% and -20% do NOT cancel out
FIX: Net = 20 - 20 - 400/100 = -4%
TRAP: "A is 150% of B" does NOT mean "A is 150% more than B"
FIX: 150% of B means A = 1.5B (50% more, not 150% more)
TRAP: Confusing percentage points with percentage change
FIX: "5% to 8%" = 3 percentage points = 60% increase
TRAP: Using wrong base for percentage change
FIX: Always use the ORIGINAL value as the base
Powers to Memorize (For Growth/Depreciation)
(1.05)^2 = 1.1025 (0.95)^2 = 0.9025
(1.05)^3 = 1.157625 (0.95)^3 = 0.857375
(1.1)^2 = 1.21 (0.9)^2 = 0.81
(1.1)^3 = 1.331 (0.9)^3 = 0.729
(1.2)^2 = 1.44 (0.8)^2 = 0.64
(1.2)^3 = 1.728 (0.8)^3 = 0.512
(1.25)^2 = 1.5625 (0.75)^2 = 0.5625
Exam Day Tips
- Convert percentages to fractions immediately -- avoid decimal multiplication.
- Use multipliers -- one multiplication instead of two steps.
- Check with assumed values -- if stuck, let the unknown = 100.
- Read "more than" vs "of" very carefully -- they are different.
- When in doubt, back-solve from the MCQ options.
- Verify your answer by plugging it back into the original condition.
- Watch the base -- increase/decrease is always relative to the original.
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