Percent = per hundred
\[x\%=\frac{x}{100}\]
Thus \(35\%\) means \(35\) parts out of every \(100\) equal parts.
3 SSC CGL · Mathematics
Percentages
Handwritten-style notes for SSC CGL Tier I and Tier II.
01MAP IT!
The Big Picture — Graphic Mind Map
Every percentage question is a comparison against a base of one hundred.
PERCENT
= PER
HUNDRED
= PER
HUNDRED
1. Convertfraction, decimal and percentage
2. Find a Partrate × base quantity
3. Find the Ratepart ÷ whole × hundred
4. Changeincrease, decrease and succession
5. Reverserecover the original base
6. Applyprice, marks, salary and population
\[\text{percentage}=\frac{\text{part}}{\text{whole}}\times100\%\]
02FOUNDATION
The Foundation (Basics) — Core Concepts
Start with “out of one hundred,” then learn how the changing base controls every advanced problem.
AWhat does percentage mean?
Percentage is a ratio
It has no unit. The compared quantities must be expressed in compatible units.
\[\frac{250\text{ g}}{2\text{ kg}}\times100\%=\frac{250}{2000}\times100\%=12.5\%\]
The base matters
“\(20\%\) of \(A\)” uses \(A\) as the base. A change of base changes the percentage even when the absolute difference is unchanged.
Imagine a hundred-cell board
Colouring \(37\) cells makes the shaded part \(37\%\). The same idea works even when the actual whole is not \(100\).
\[37\%=\frac{37}{100}=0.37\]
BFraction–decimal–percentage conversion
Fraction to percentage
\[\frac ab\times100\%\]
Example: \(\frac38\times100\%=37.5\%\).
Decimal to percentage
\[d\times100\%\]
Move the decimal two places right: \(0.625=62.5\%\).
Percentage to fraction/decimal
\[p\%=\frac p{100}=0.01p\]
Always reduce the resulting fraction.
| Fraction | Percentage | Fraction | Percentage |
|---|---|---|---|
| \(\frac12\) | \(50\%\) | \(\frac13\) | \(33\frac13\%\) |
| \(\frac14\) | \(25\%\) | \(\frac15\) | \(20\%\) |
| \(\frac16\) | \(16\frac23\%\) | \(\frac18\) | \(12.5\%\) |
| \(\frac19\) | \(11\frac19\%\) | \(\frac1{10}\) | \(10\%\) |
| \(\frac1{11}\) | \(9\frac1{11}\%\) | \(\frac1{12}\) | \(8\frac13\%\) |
| \(\frac1{16}\) | \(6.25\%\) | \(\frac1{20}\) | \(5\%\) |
| \(\frac1{25}\) | \(4\%\) | \(\frac1{40}\) | \(2.5\%\) |
The conversion triangle
Percentage and decimal differ by a factor of \(100\). Fraction and percentage connect through multiplication by \(100\%\).
\[\text{decimal}\xleftrightarrow[\div100]{\times100}\text{percentage}\]
CThe three basic question types
ReadIdentify part, whole and rate
Choose baseThe word “of” points to it
TranslateReplace percent by division by \(100\)
SolveCancel before multiplying
CheckEstimate size and unit
Find the part
\[P=\frac r{100}\,W\]
\(18\%\) of \(650\) is \(117\).
Find the rate
\[r=\frac PW\times100\]
\(45\) is \(15\%\) of \(300\).
Find the whole
\[W=\frac{100P}{r}\]
If \(72\) is \(24\%\), the whole is \(300\).
DPercentage increase and decrease
Percentage change
\[\%\text{ change}=\frac{\text{new}-\text{old}}{\text{old}}\times100\%\]
The old value is always the base.
Multiplier method
\[\text{new}=\text{old}\left(1\pm\frac r{100}\right)\]
Increase uses \(+\); decrease uses \(-\).
Reverse the multiplier
\[\text{old}=\frac{\text{new}}{1\pm r/100}\]
Do not simply subtract the same percentage from the new value.
Percentage change is a staircase whose first step is the base
An increase and an equal decrease do not cancel because the second move starts from a changed base.
\[100\xrightarrow{+20\%}120\xrightarrow{-20\%}96\]
ESuccessive percentage changes
Two successive increases
\[a+b+\frac{ab}{100}\%\]
Increase then decrease
\[a-b-\frac{ab}{100}\%\]
Take decrease as negative if using the universal rule.
Universal multiplier
\[\text{net factor}=\prod_i\left(1+\frac{r_i}{100}\right)\]
Net percentage is \((\text{factor}-1)100\%\).
\[+x\%\text{ followed by }-x\%\Rightarrow-\frac{x^2}{100}\%\]
Successive changes multiply, never merely add
Convert every change into a multiplier, multiply the multipliers, then compare the result with \(1\). This method works for any number of stages.
FReverse percentage and original value
After an increase
\[\text{original}=\frac{100}{100+r}\times\text{new}\]
After a decrease
\[\text{original}=\frac{100}{100-r}\times\text{new}\]
Restoring percentage
After a decrease of \(r\%\), required increase to return is
\[\frac{100r}{100-r}\%\]
Reverse percentage is a time machine
The visible value is after a multiplier. Divide by that multiplier to travel back to the original; do not reverse by applying the same rate.
GComparison language and percentage points
“\(A\) is what percent of \(B\)?”
\[\frac AB\times100\%\]
“\(A\) is how much more than \(B\)?”
\[\frac{A-B}{B}\times100\%\]
The quantity after “than” is the base.
Percentage points
A rate rising from \(20\%\) to \(25\%\) rises by \(5\) percentage points but by
\[\frac{25-20}{20}\times100\%=25\%\]
\[A\text{ is }x\%\text{ more than }B\Rightarrow B\text{ is }\frac{100x}{100+x}\%\text{ less than }A\]
HConstant-product applications
Price and consumption
For constant expenditure, price and consumption vary inversely.
\[\text{price factor}\times\text{consumption factor}=1\]
Price rises by \(r\%\)
Required consumption reduction:
\[\frac{100r}{100+r}\%\]
Price falls by \(r\%\)
Possible consumption increase:
\[\frac{100r}{100-r}\%\]
One side rises, the other compensates
When a product must remain fixed, replace verbal change with multipliers and make their product \(1\).
\[(1+r/100)(1-c/100)=1\]
IPopulation, salary, production, marks and error
| Application | Model | Exam warning |
|---|---|---|
| Population growth | \(P_n=P_0(1+r/100)^n\) | For changing annual rates, multiply distinct yearly factors. |
| Depreciation | \(V_n=V_0(1-r/100)^n\) | Loss is on the reduced value each year. |
| Salary and saving | \(\text{saving}=\text{income}-\text{expenditure}\) | Apply each percentage to its stated base. |
| Marks | \(\text{percentage}=\text{marks obtained}/\text{maximum}\times100\%\) | Pass percentage and obtained percentage may use different statements. |
| Election votes | Convert valid-vote percentages into counts. | Invalid votes are removed before candidate shares if stated so. |
| Percentage error | \(|\text{error}|/|\text{true value}|\times100\%\) | The true/accepted value is the denominator. |
JBase-value algebra and advanced identities
Assume \(100\)
When only percentages matter, let the original be \(100\). This converts rates directly into values.
Product change
If two factors change by \(a\%\) and \(b\%\):
\[\text{net}=a+b+\frac{ab}{100}\%\]
Use signs for decreases.
Square/cube change
\[x\to x(1+r)\Rightarrow x^n\to x^n(1+r)^n\]
Here \(r\) is decimal change; expand only if useful.
03SAVE TIME
Short Tricks & Magic Formulas
Convert percentages into friendly fractions or multipliers before multiplying large values.
1Fraction substitution
\(12.5\%\)
\[12.5\%=\frac18\]
\(16\frac23\%\)
\[16\frac23\%=\frac16\]
\(33\frac13\%\)
\[33\frac13\%=\frac13\]
\(66\frac23\%\)
\[66\frac23\%=\frac23\]
2Swap rule
\(x\%\) of \(y\) equals \(y\%\) of \(x\)
Swap to the easier multiplication. For example, \(4\%\) of \(75\) is easier as \(75\%\) of \(4\).
\[\frac{x}{100}y=\frac{y}{100}x\]
3One-line reverse and restoration tricks
More ↔ less conversion
\[x\%\text{ more}\Rightarrow\frac{100x}{100+x}\%\text{ less in reverse}\]
Loss restoration
\[x\%\text{ decrease needs }\frac{100x}{100-x}\%\text{ increase}\]
Equal opposite changes
\[+x\%,-x\%\Rightarrow-\frac{x^2}{100}\%\]
4Option-first checks
Magnitude
If the rate is below \(100\%\), its part of a positive whole is smaller than that whole.
Multiplier
Replace successive changes with short factors such as \(1.2\), \(0.8\), \(1.125\), or exact fractions.
Base-word scan
Circle “of,” “than,” “original,” “new,” and “valid.” These words reveal the denominator.
Three-dial speed test
Before solving: identify the base, convert change to a multiplier, and estimate the answer range. This eliminates many TCS distractors.
04DON'T RUSH
The SSC / TCS Traps — Red Flags 🚩
Wrong options usually preserve the arithmetic but silently change the base.
Trap 1: equal percentages do not reverse
An increase of \(x\%\) followed by a decrease of \(x\%\) gives a net loss of \(x^2/100\%\), because the bases differ.
Trap 2: “more than” chooses its base
In “\(A\) is what percent more than \(B\),” divide the difference by \(B\). Reversing the sentence changes the answer.
Trap 3: percent vs percentage points
From \(20\%\) to \(25\%\) is \(5\) percentage points but a \(25\%\) relative increase. TCS options often contain both.
05LOCK IT IN
Memory Hooks & Mnemonics
Use a short visual sentence to retrieve the correct base and formula.
“Percent parks at one hundred”
Every percent is a fraction whose denominator is \(100\).
“Change climbs from OLD”
Percentage change always divides the difference by the original value.
“Successive means multiply”
Turn each change into a factor; multiplying factors automatically respects changing bases.
“Reverse? Divide the factor”
The new value already contains the multiplier; division recovers the original.
“Points subtract; percent divides”
Percentage-point change is simple subtraction; relative percent change divides by the old rate.
“Constant product balances”
If price rises while expenditure stays fixed, consumption must fall by the reciprocal factor.
Find partRate × whole
Find ratePart ÷ whole
Find wholePart ÷ rate
SuccessiveMultiply factors
ReverseDivide by factor
ComparisonRead the base word