Episode 8 — Aptitude and Reasoning / 8.1 — Percentage
8.1.b Tips, Tricks, and Shortcuts
1. The Fraction-Percentage Equivalents Table
Memorize this table. It is the single most powerful tool for solving percentage problems quickly.
| Fraction | Percentage | Fraction | Percentage |
|---|---|---|---|
| 1/1 | 100% | 1/9 | 11.11% |
| 1/2 | 50% | 2/9 | 22.22% |
| 1/3 | 33.33% | 4/9 | 44.44% |
| 2/3 | 66.67% | 5/9 | 55.56% |
| 1/4 | 25% | 7/9 | 77.78% |
| 3/4 | 75% | 8/9 | 88.89% |
| 1/5 | 20% | 1/10 | 10% |
| 2/5 | 40% | 3/10 | 30% |
| 3/5 | 60% | 7/10 | 70% |
| 4/5 | 80% | 9/10 | 90% |
| 1/6 | 16.67% | 1/11 | 9.09% |
| 5/6 | 83.33% | 1/12 | 8.33% |
| 1/7 | 14.28% | 1/15 | 6.67% |
| 2/7 | 28.57% | 1/16 | 6.25% |
| 3/7 | 42.86% | 1/20 | 5% |
| 1/8 | 12.5% | 1/25 | 4% |
| 3/8 | 37.5% | 1/50 | 2% |
| 5/8 | 62.5% | 1/100 | 1% |
| 7/8 | 87.5% |
Why This Works
When a problem says "33.33% of 600", instead of computing (33.33/100) x 600, you immediately recognize 33.33% = 1/3 and compute 600/3 = 200. This saves 10-15 seconds per problem -- which adds up to several minutes over a full exam.
2. Quick Mental Math Techniques
2.1 Breaking Down Percentages
Decompose any percentage into easy parts.
Finding 17.5% of 400:
10% of 400 = 40
5% of 400 = 20 (half of 10%)
2.5% of 400 = 10 (half of 5%)
--------------------------
17.5% of 400 = 70
2.2 The 10% Anchor Method
Always start from 10%, then scale up or down.
10% of any number = just move the decimal one place left
10% of 860 = 86
5% of 860 = 43 (half of 10%)
1% of 860 = 8.6 (one-tenth of 10%)
20% of 860 = 172 (double 10%)
25% of 860 = 215 (10% x 2 + 5%)
30% of 860 = 258 (10% x 3)
15% of 860 = 129 (10% + 5%)
2.3 Percentage of Percentage
20% of 50% = (20/100) x (50/100) = 10/100 = 10%
Quick rule: multiply the two percentages and divide by 100.
a% of b% = ab/100 %
3. The Multiplier Method (Speed Technique)
Instead of calculating increase/decrease step by step, use a single multiplication.
| Change | Multiplier | Example (on 600) |
|---|---|---|
| +10% | 1.1 | 660 |
| +20% | 1.2 | 720 |
| +25% | 1.25 | 750 |
| +33.33% | 4/3 | 800 |
| +50% | 1.5 | 900 |
| -10% | 0.9 | 540 |
| -20% | 0.8 | 480 |
| -25% | 0.75 | 450 |
| -33.33% | 2/3 | 400 |
| -50% | 0.5 | 300 |
How to use: Just multiply the original value by the multiplier once.
A salary of Rs 45,000 is increased by 20%.
New salary = 45,000 x 1.2 = Rs 54,000
Done in one step -- no need to calculate the increase separately and add.
4. The Successive Percentage Change Shortcut
4.1 Two Successive Changes
Net effect of a% followed by b% = a + b + (ab/100) %
Use negative values for decreases.
| Successive Changes | Net Effect |
|---|---|
| +10%, +10% | +21% |
| +20%, +20% | +44% |
| +10%, -10% | -1% |
| +20%, -20% | -4% |
| +25%, -20% | 0% |
| +50%, -33.33% | 0% |
| +100%, -50% | 0% |
Why This Works
When you increase by a% and then decrease by a%, the result is always a net decrease (never zero). The net loss is:
Net change = a + (-a) + a(-a)/100 = -a^2/100
This is always negative. For 10% up then 10% down: net = -(10)^2/100 = -1%.
4.2 Cancelling Pairs (Very Useful)
These pairs cancel each other out -- memorize them:
+100% and -50% cancel out (net 0%)
+50% and -33.33% cancel out (net 0%)
+25% and -20% cancel out (net 0%)
+20% and -16.67% cancel out (net 0%)
+10% and -9.09% cancel out (net 0%)
Pattern: If an increase of r% is to be cancelled, the required decrease is:
Required decrease = [r / (100 + r)] x 100 %
5. The Fraction Trick for Increase/Decrease
When the percentage is a "nice" fraction, convert and use fraction arithmetic.
Example: A value increases by 16.67%.
16.67% = 1/6
So the multiplier = 1 + 1/6 = 7/6
If original = 42:
New value = 42 x 7/6 = 49
Common multipliers in fraction form:
| % Change | Fraction Multiplier |
|---|---|
| +12.5% | 9/8 |
| +14.28% | 8/7 |
| +16.67% | 7/6 |
| +20% | 6/5 |
| +25% | 5/4 |
| +33.33% | 4/3 |
| +50% | 3/2 |
| +66.67% | 5/3 |
| +100% | 2/1 |
| -12.5% | 7/8 |
| -14.28% | 6/7 |
| -16.67% | 5/6 |
| -20% | 4/5 |
| -25% | 3/4 |
| -33.33% | 2/3 |
| -50% | 1/2 |
Why This Works
Fraction multipliers let you avoid decimal arithmetic entirely. When the original value is divisible by the denominator, the answer comes out clean -- no calculator needed.
6. Common Exam Traps
Trap 1: Confusing the Base
"A is 25% more than B. What percent is B less than A?"
Wrong answer: 25% Correct answer: 20%
The base changes when you flip the comparison. Always use:
[r / (100 + r)] x 100 for "more" to "less" conversion
[r / (100 - r)] x 100 for "less" to "more" conversion
Trap 2: Adding Successive Percentages
"Price goes up 20%, then up 10%. Total increase?"
Wrong answer: 30% Correct answer: 32%
Use the formula: 20 + 10 + (20 x 10)/100 = 32%
Trap 3: Percentage OF vs Percentage MORE/LESS
"A is 150% of B" means A = 1.5B (A is 50% MORE than B, not 150% more) "A is 150% more than B" means A = 2.5B
Trap 4: Percentage Change When Going Back
"A value drops by 50% and then increases by 50%. Is it back to original?"
No. It is at 75% of original. Net change = -25%.
Trap 5: Percentage Points vs Percentage Change
"Rate changed from 5% to 8%."
Change in percentage points = 3 Percentage change = (3/5) x 100 = 60% increase
7. Time-Saving Tricks for MCQs
7.1 Back-Solving
When stuck, plug each answer option into the problem. Start with option (b) or (c) since MCQ options are usually in ascending order.
Example: "A number when increased by 20% gives 360. Find the number."
Options: (a) 280 (b) 300 (c) 320 (d) 340
Try (b): 300 x 1.2 = 360. Correct. Done in 3 seconds.
7.2 Approximation
When options are far apart, approximate aggressively.
Example: "Find 17.6% of 493."
Approximate: 17.6% of 500 = 88
Actual will be slightly less than 88.
If options are 76, 82, 87, 93 -- pick 87 (closest).
7.3 Elimination Using Digit Sums or Units Digits
Check the units digit of your answer against the options.
23% of 400 = 92
Units digit must be 2. Eliminate any option that doesn't end in 2.
7.4 The "Ratio" Shortcut
When comparing two quantities with percentage differences, use ratios.
"A earns 20% more than B. B earns what % less than A?"
A : B = 120 : 100 = 6 : 5
B is less than A by 1 part out of 6 = 1/6 = 16.67%
7.5 Using Assumed Values
When no actual values are given, assume a convenient number (usually 100 or LCM of denominators).
"A price increases by 25% then decreases by 20%. Net effect?"
Assume original = 100
After 25% increase = 125
After 20% decrease = 125 x 0.8 = 100
Net change = 0%
8. Pattern Recognition for Speed
8.1 Revenue / Expenditure Changes
Revenue = Price x Quantity
If Price changes by a% and Quantity changes by b%:
Net % change in Revenue = a + b + ab/100
This is the successive percentage formula applied to products.
8.2 Area Changes
Area of rectangle = Length x Breadth
If Length changes by a% and Breadth changes by b%:
Net % change in Area = a + b + ab/100
Example: Length +10%, Breadth +20%:
Net change in area = 10 + 20 + (10 x 20)/100 = 32%
8.3 The "Two Variables Product" Rule
Whenever two quantities multiply to give a third, and both change by percentages, use the successive percentage formula to find the net change in the product.
9. Summary of Shortcuts
| Situation | Shortcut |
|---|---|
| Finding X% of a number | Convert to fraction, multiply |
| Percentage increase/decrease | Single multiplier |
| Two successive changes | a + b + ab/100 |
| "More than" to "Less than" | r/(100+r) x 100 |
| "Less than" to "More than" | r/(100-r) x 100 |
| Product of two changing quantities | a + b + ab/100 |
| Cancelling an increase of r% | Decrease by r/(100+r) x 100 |
| Quick check | Plug in answer options |
| No values given | Assume 100 |
Previous: 8.1.a Concepts and Formulas | Next: 8.1.c Solved Examples