1. Percentage-to-Fraction Shortcuts
"Find 37.5% of 240" is slow as a decimal multiplication and fast as a fraction — 37.5% is exactly 3/8, and 240 ÷ 8 × 3 is mental math. Memorizing this table pays off on nearly every question in the syllabus that touches a percentage.
50% = 1/2 12.5% = 1/8
33.33% = 1/3 6.25% = 1/16
25% = 1/4 11.11% = 1/9
20% = 1/5 16.66% = 1/6
16.66% = 1/6 9.09% = 1/11
14.28% = 1/7 8.33% = 1/12
10% = 1/10 37.5% = 3/8
66.66% = 2/3 62.5% = 5/8
75% = 3/4 83.33% = 5/637.5% = 3/8
240 / 8 = 30
30 x 3 = 90
37.5% of 240 = 90
(Compare: 0.375 x 240 by hand is slower and more error-prone
than dividing by 8 then multiplying by 3.)The general recipe for any percentage not in the table: find a nearby "nice" percentage (a multiple of 10, 25, or 50), calculate that, then adjust — e.g. 47% of a number is 50% minus 3%, both of which are fast.
2. Successive Percentage Change
A price rises 20%, then falls 20%. The final price is not the original — this is the single most common percentage trap on any aptitude test, and the fix is a formula worth memorizing exactly.
Net % change = a + b + (a x b)/100
where a and b are the two percentage changes (use NEGATIVE
values for a decrease).
Example: +20% then -20%
Net = 20 + (-20) + (20 x -20)/100
= 0 + (-400)/100
= 0 - 4
= -4%
The price ends up 4% LOWER than the original -- not unchanged.
The intuition behind why it's never zero: the 20% decrease is calculated on the
already-increased price, which is a bigger number than the original — so
the decrease removes more in absolute terms than the increase added. This asymmetry
is exactly what the (a×b)/100 correction term captures.
A price rises 10%, then rises another 10%, then falls 10%.
Step 1: combine the first two: 10 + 10 + (10x10)/100 = 21%
Step 2: combine that with the third: 21 + (-10) + (21x-10)/100
= 21 - 10 - 2.1
= 8.9%
Net change: +8.9%Combining +20% then -20% gives the same net result as combining -20% then +20% — the formula is symmetric in a and b. What DOES matter is that each step's percentage is always calculated on the result of the previous step, never on the original value, which is the actual source of the "traps you if you just add them" behavior.
3. The Base-Value Trap
"A's salary is 25% more than B's. By what percentage is B's salary less than A's?" The answer is not 25% — this trap appears constantly because the two questions use different base values.
Let B's salary = 100
A's salary = 100 + 25% of 100 = 125
"A is 25% more than B" -- correctly compares A to a BASE OF B
(125 - 100)/100 = 25% -- correct, uses B as the base
"B is what % less than A?" -- must compare B to a BASE OF A
(125 - 100)/125 = 25/125 = 20%
B's salary is 20% less than A's, NOT 25% less.
The general shortcut, worth deriving once and keeping: if X is p% more
than Y, then Y is (100p)/(100+p)% less than X. Plugging p=25 gives
2500/125 = 20, matching the worked example. This formula is a compressed version of
the same "which number is the base" reasoning, not a separate rule to memorize
blindly.
4. Percentage Points vs. Percentage Change
"Interest rates rose from 5% to 8%." Did they rise by 3% or by 60%? Both numbers are real, they just answer different questions — and test questions deliberately exploit this ambiguity.
From 5% to 8%:
Percentage POINT change: 8 - 5 = 3 percentage points
(simple subtraction of the two percentage VALUES)
Percentage CHANGE: (8 - 5)/5 x 100 = 60%
(the RELATIVE change, treating 5% as the base value)
"Rose by 3 percentage points" and "rose by 60%" are both
TRUE, correct descriptions of the exact same change.A test question asking "by what percentage did the rate increase" wants the 60% relative-change answer; a question stating "the rate increased by 3 percentage points" is giving you the point-difference directly, not something to convert. Data interpretation questions (Weeks 17-18) rely on this distinction constantly, since a chart's raw percentage values invite exactly this ambiguity.
5. Practice Problems
30 problems, no calculator — Easy, Medium & Tough
Use Section 1's fraction shortcuts wherever possible instead of decimal arithmetic. Easy problems should take under 30 seconds each; Medium under 60 seconds; Tough under 90 seconds.
Easy (1–10)
- Find 20% of 350.
- Find 45% of 200.
- What percentage is 45 of 180?
- Convert 3/8 to a percentage.
- Find 12.5% of 640.
- If 25% of a number is 50, find the number.
- Increase 200 by 10%.
- Decrease 500 by 20%.
- What is 150% of 80?
- Express 0.75 as a percentage.
1) 70. 2) 90. 3) 45/180×100=25%. 4) 3/8=37.5%. 5) 12.5%=1/8; 640/8=80. 6) 50/0.25=200. 7) 200×1.1=220. 8) 500×0.8=400. 9) 150% of 80=120. 10) 75%.
Medium (11–20)
- A number increased by 20% becomes 240. Find the original number.
- A price rises by 25% and then falls by 20%. Find the net percentage change.
- X is 30% more than Y. By what percentage is Y less than X?
- In an election, a candidate got 60% of votes and won by 4,000 votes. Find the total votes cast.
- A student scored 72 out of 90. Find the percentage.
- A number is decreased by 10% and then increased by 10%. Find the net percentage change.
- If the price of sugar increases by 25%, by what percentage should consumption be reduced to keep expenditure unchanged?
- Find the number whose 15% is 45.
- A's income is 25% less than B's. By what percentage is B's income more than A's?
- The population of a town increases by 10% annually. If it's currently 22,000, find the population after 2 years.
11) 240/1.2=200. 12) 25-20+(25×-20)/100=5-5=0%. 13) X=1.3Y; Y is 30/1.3≈23.08% less than X. 14) Diff of 20% (0.6T-0.4T) = 4000 → T=20,000. 15) 72/90×100=80%. 16) -10+10+(-10×10)/100=0-1=-1% (net decrease). 17) Reduction%=(25/(100+25))×100=20%. 18) 45/0.15=300. 19) A=0.75B; B is (0.25/0.75)×100=33.33% more than A. 20) 22000×1.1×1.1=26,620.
Tough (21–30)
- A rise of 20%, then a fall of 20%, then a rise of 20% again. Find the net percentage change.
- In an exam, 35% of students failed in Math and 45% failed in English, while 20% failed in both. Find the percentage who passed in both subjects.
- A's salary is increased by 20% and then reduced by 20%. If his new salary is Rs. 19,200, find his original salary.
- Two numbers are respectively 20% and 50% more than a third number. Find the ratio of the two numbers.
- If A's income is 20% more than B's, and B's income is 25% more than C's, find by what percentage A's income is more than C's.
- In a college, 60% of students are boys. If 75% of boys and 40% of girls passed an exam, find the overall pass percentage.
- The price of an item is first increased by 30% and then decreased by x% so that the final price equals the original price. Find x.
- A number is first increased by 25%, then the result is decreased by 25%. Find the net percentage change on the original number.
- A reduction of 20% in the price of an item enables a person to buy 5 kg more for Rs. 400. Find the original price per kg.
- In a class, 40% of students play cricket, 30% play football, and 10% play both. Find the percentage playing neither.
21) +20 then -20 gives -4%; combine -4% with another +20%: -4+20+(-4×20)/100=16-0.8=15.2%. 22) Failed at least one=35+45-20=60%; passed both=40%. 23) 0.96×Original=19,200 → Original=20,000. 24) Third number=100 → first=120, second=150 → ratio 4:5. 25) A=1.2×1.25C=1.5C → 50% more than C. 26) 100 students: boys passed=0.75×60=45, girls passed=0.4×40=16 → total=61%. 27) x=30/130×100≈23.08%. 28) 25-25+(25×-25)/100=0-6.25=-6.25%. 29) 500/p-400/p=100/p=5 → p=Rs. 20/kg. 30) At least one=40+30-10=60%; neither=40%.
6. Knowledge Check
Four quick questions. Expand each to check your answer.
Q1
Why is a 20% increase followed by a 20% decrease not a net 0% change?
Why is a 20% increase followed by a 20% decrease not a net 0% change?
The 20% decrease is calculated on the already-increased (larger) value, not the original — so it removes more in absolute terms than the increase added. This asymmetry always produces a net decrease whenever equal-percentage increase and decrease are applied in sequence, captured by the formula's negative (a×b)/100 correction term.
Q2
If A's salary is 25% more than B's, why is B's salary NOT 25% less than A's?
If A's salary is 25% more than B's, why is B's salary NOT 25% less than A's?
"25% more" uses B's salary as the base for comparison, while "what % less" must use A's (larger) salary as the base. Since the base value is different in each direction, the two percentages aren't required to match — the correct answer works out to 20%, not 25%, because 25% of the larger number A is a bigger absolute amount than 25% of the smaller number B.
Q3
What's the difference between "the rate rose by 3 percentage points" and "the rate rose by 60%," when going from 5% to 8%?
What's the difference between "the rate rose by 3 percentage points" and "the rate rose by 60%," when going from 5% to 8%?
"3 percentage points" is the simple arithmetic difference between the two percentage values (8-5=3). "60%" is the relative change, treating the original 5% as the base value ((8-5)/5×100=60%). Both are correct, true statements about the same underlying change — they just answer "how much did the value change by" versus "what fraction did it change by relative to where it started."
Q4
Why is converting 37.5% to the fraction 3/8 faster for mental math than multiplying by the decimal 0.375?
Why is converting 37.5% to the fraction 3/8 faster for mental math than multiplying by the decimal 0.375?
Dividing by 8 and multiplying by 3 are both simple integer operations easy to do in your head, especially when the original number is a multiple of 8. Multiplying by a three-decimal-place number like 0.375 directly is slower and much more error-prone under time pressure, even though both approaches give the identical result.