- The "fractions in disguise" of this chapter are percentages — a percentage is just a fraction whose denominator is fixed at 100. So $25\% = \dfrac{25}{100}$.
- Fraction → percentage: multiply by $100$. Percentage → fraction: put it over $100$ and simplify. Percentage → decimal: divide by $100$. These three forms — Fraction, Decimal, Percentage — are the FDP trio.
- $y\%$ of a quantity $Q$ means $\dfrac{y}{100}\times Q$. Percentages let us compare proportions fairly even when the wholes are different.
- Used everywhere: profit/loss, discount, GST, interest, growth and decline. Profit% / Loss% are always on the cost price; increase/decrease% are always on the original (base) value.
- Percentages can be more than 100% (achieving 120% of a target = 1.2 times it). Board weightage: ~4–5 marks — a percentage-of-quantity sum, and a profit/discount or increase/decrease word problem.
1. What "per cent" really means
You hear it daily: "Mega Sale — up to 50% off!", "Hiya scored 83% in her exams". The symbol % is read per cent, from the Latin per centum = "out of a hundred". So $25\%$ means 25 out of every 100 — 25 marks out of 100, 25 rupees out of 100, and so on.
This gives the single most important idea of the chapter:
Examples: $20\% = \dfrac{20}{100}=\dfrac15,\qquad 33\% = \dfrac{33}{100}.$
"50% of a quantity $s$" therefore means $50\times\dfrac{1}{100}\times s = \dfrac{50}{100}s = \dfrac12 s$ — exactly half. Percentages are fractions in disguise: that is the title of the chapter.
2. Turning a fraction into a percentage
To write a fraction as a percentage, find the equivalent fraction with 100 as the denominator — or, more simply, just multiply the fraction by 100.
- Method 1 (equivalent fraction): $\dfrac34 = \dfrac{3\times25}{4\times25}=\dfrac{75}{100}=75\%.$
- Method 2 (multiply by 100): set $\dfrac34=\dfrac{x}{100}\Rightarrow x=\dfrac34\times100 = 75$, so $75\%.$
Surya's orange mix is $\dfrac34$ red. As a percentage: $\dfrac34\times100=75\%$ red, so the yellow makes up $100-75=25\%$.
$\dfrac25 = \dfrac{2\times20}{5\times20}=\dfrac{40}{100}=40\%.$ Or directly $\dfrac25\times100 = 40\%.$ Both methods agree.
3. Turning a percentage into a fraction
Reverse the process: put the number over $100$ and simplify. A percentage $z\%$ equals any fraction equivalent to $\dfrac{z}{100}$.
$24\% = \dfrac{24}{100} = \dfrac{12}{50}=\dfrac{6}{25}.$ All of $\dfrac{24}{100},\dfrac{12}{50},\dfrac{6}{25},\dfrac{48}{200}$ are correct equivalent forms.
Why 100 and not 10 or 1000? A biscuit with $\dfrac{9}{34}$ sugar vs $\dfrac{13}{45}$ sugar is hard to compare — but as $26.47\%$ vs $28.88\%$ the second is clearly sugary. We pick $100$ because our number system is base-10 (so $10,100,1000$ fit decimals neatly), and $100$ is the "sweet spot": large enough for detail, small enough to grasp mentally.
4. The FDP trio — Fraction, Decimal, Percentage
Every proportion has three faces. Converting between them is the workhorse skill of this chapter.
Decimal → Percentage: multiply by $100$. $\;0.7 = 70\%.$
Fraction ↔ Decimal: divide numerator by denominator. $\;\dfrac12=0.5.$
| Per cent | 50% | 25% | 75% | 10% | 1% | 5% |
|---|---|---|---|---|---|---|
| Fraction | $\tfrac12$ | $\tfrac14$ | $\tfrac34$ | $\tfrac{1}{10}$ | $\tfrac{1}{100}$ | $\tfrac{1}{20}$ |
| Decimal | 0.5 | 0.25 | 0.75 | 0.1 | 0.01 | 0.05 |
Yes: $50\% = \dfrac{50}{100}=\dfrac12 = 0.5.$ So $50\%$ of $24 = \dfrac12\times24 = 0.5\times24 = 12$ — the fraction $\tfrac12$ and the decimal $0.5$ are the same tool wearing different clothes.
5. Percentage of a quantity
The central formula. $y\%$ of a quantity $Q$ means take $\dfrac{y}{100}$ of it:
Madhu ate $120$ g of biscuits with $25\%$ sugar. Sugar $= \dfrac{25}{100}\times120 = 30$ g. Madhav ate $95$ g with $35\%$ sugar $= \dfrac{35}{100}\times95 = 33.25$ g. So Madhav ate more sugar.
Mental-maths shortcuts (used constantly):
- $10\%$ = divide by $10$. $\;\;20\% = $ double of $10\%$. $\;\;5\% = $ half of $10\%$.
- $25\% = \dfrac14$ (a quarter), $50\% = \dfrac12$, $75\% = \dfrac34$.
- Handy identity: $20\% + 5\% = 25\%$, i.e. $\left(\dfrac{20}{100}Q\right)+\left(\dfrac{5}{100}Q\right)=\dfrac{25}{100}Q.$
Max marks $=75$; an A needs $80\%$. Minimum $= \dfrac{80}{100}\times75 = \dfrac45\times75 = 60$ marks (also $0.8\times75 = 60$).
6. Ratios → percentages, and "find the whole"
A ratio can be turned into a fraction, then a percentage.
Millet : water $=2:7$, so millet is $\dfrac{2}{2+7}=\dfrac29$ of the mixture. As a percentage $\dfrac29\times100 = 22.22\%$ (water $=77.78\%$). In $500$ ml of mixture, millet $=22.22\%$ of $500 = 5\times22.22 = 111.1$ ml.
Sometimes the part is known and we want the whole. Use the proportional relationship $\dfrac{\text{percent}}{100}=\dfrac{\text{part}}{\text{whole}}$.
$40\%$ of the journey is $92$ km, so $\dfrac{40}{100}=\dfrac{92}{d}\Rightarrow d = 92\times\dfrac{100}{40}=230$ km. Remaining distance $= 230-92 = 138$ km. (Mentally: $40\%$ is $92$, so $20\%$ is $46$, and $60\%$ is $138$.)
7. Percentages greater than 100%
A percentage can exceed $100$ — it just means "more than the whole". $200\%$ of a value is $2$ times it; $150\% = 1.5$ times; $250\% = 2.5$ times.
Day 4 sales $= ₹6000$. Percentage of target $= \dfrac{6000}{5000}\times100 = \dfrac65\times100 = 120\%$. He achieved $120\%$ — i.e. $20\%$ more than the target.
Last year $260$ kg, this year $650$ kg. This year $= \dfrac{650}{260}\times100 = 250\%$ of last year — that is $2.5$ times the earlier harvest.
8. Comparing proportions fairly
You cannot compare two raw amounts when the wholes differ — convert both to percentages first. This is the whole point of percentages.
English $42/50$, Science $70/80$. As percentages: English $= \dfrac{42}{50}\times100 = 84\%$; Science $= \dfrac{70}{80}\times100 = 87.5\%$. Although she lost fewer marks in English, her Science score is the higher proportion — so she did better in Science.
Caution from the chapter: when comparing percentages, you are comparing proportions, not absolute values. A club that "grew $100\%$" may be tiny; a club that grew "$80\%$" may be huge — the base matters.
9. Percentage increase and decrease
For change over time, the percentage is always measured against the original (base) value.
Price went ₹30 → ₹42, an increase of ₹12. $\text{% increase} = \dfrac{12}{30}\times100 = 40\%.$
Footfall $160\to100$, a decrease of $60$. $\text{% decrease} = \dfrac{60}{160}\times100 = 37.5\%.$
Two ways of saying the same thing: "population in 1991 is $165\%$ of 1961" $\equiv$ "population increased by $65\%$ from 1961 to 1991", because $q = p + 65\%\text{ of }p = 1.65p$.
10. Profit and loss
Three prices: Cost Price (CP) = what the seller paid; Marked Price (MP) = the quoted/MRP price; Selling Price (SP) = what the buyer finally pays.
- If $\text{SP} > \text{CP}$: profit $= \text{SP}-\text{CP}$.
- If $\text{SP} < \text{CP}$: loss $= \text{CP}-\text{SP}$.
$$\text{Profit%} = \dfrac{\text{Profit}}{\text{CP}}\times100,\qquad \text{Loss%} = \dfrac{\text{Loss}}{\text{CP}}\times100.$$
CP $= ₹300$, SP $= ₹430$. Profit $= 430-300 = ₹130$. $\text{Profit%} = \dfrac{130}{300}\times100 = 43.3\%.$
CP of 10 kg $= ₹350$, SP $= ₹300$. Loss $= 350-300 = ₹50$. $\text{Loss%} = \dfrac{50}{350}\times100 = 14.28\%.$
CP $= ₹2650$, sold at $18\%$ loss. SP is $82\%$ of CP $= 0.82\times2650 = ₹2173.$ (Check: loss $= \dfrac{18}{100}\times2650 = 477$, and $2650-477 = 2173.$)
Note: "profit margin" here means gross profit (Sales − cost of goods). After subtracting other expenses you get net profit.
11. Discount and taxes (GST)
A discount of $30\%$ means the price is reduced by $30\%$ — so you pay $70\%$ of the marked price.
MRP $= ₹1800$, discount $35\%$. SP $= 65\%$ of $1800 = 0.65\times1800 = ₹1170.$ If CP was $₹900$, profit $= 1170-900 = ₹270$, so $\text{Profit%} = \dfrac{270}{900}\times100 = 30\%.$
Taxes (GST, income tax) are stated as percentages. On a bill, GST is added on top of the price and goes to the government. E.g. a ₹450 item with $9\%$ CGST + $9\%$ SGST adds $40.50 + 40.50 = ₹81$, total ₹531.
On a ₹200 cake: "$30\%+20\%$" compounds — first $30\%$ off gives ₹140, then $20\%$ off ₹140 gives ₹112. A flat $50\%$ off gives ₹100. So $30\%+20\%\neq50\%$; the single $50\%$ discount is cheaper.
12. Interest — simple vs compound (growth)
Interest is extra money paid on a deposit (or loan). Principal $=$ amount deposited; rate $r$ is per annum (p.a.). After 1 year at rate $r$, the amount becomes principal $+ r\%$ of principal.
Over $t$ years there are two cases:
- No compounding (interest paid out each year): $\text{Amount} = P(1+rt)$, where $r$ is written as a decimal. Interest $=P\times r\times t.$
- Compounding (interest added back, so it earns more next year): $\text{Amount} = P(1+r)^{t}.$
Without compounding: interest each year $= 0.1\times6000 = ₹600$, so total $= 6000 + 3\times600 = ₹7800$ (gain $30\%$).
With compounding: $6000\times1.1\times1.1\times1.1 = 6000\times1.331 = ₹7986$ (gain $33.1\%$). Compounding always gives more.
13. Decline (depreciation)
Many items lose value over time — this is depreciation. A decrease of $r\%$ each period multiplies the value by $\left(1-\dfrac{r}{100}\right)$ each time.
TV bought for ₹21000, drops $5\%$ in a year. New value $= 95\%$ of $21000 = 0.95\times21000 = ₹19950.$
Start $1250$, falls $10\%$ each decade $\Rightarrow$ multiply by $0.9$ each time: $1250\times0.9\times0.9\times0.9 = 911.25 \approx 910$ after 3 decades.
14. NCERT "Figure it Out" — selected solutions
Fractions as percentages: $\dfrac35 = 60\%$, $\dfrac{9}{20}=45\%$, $\dfrac{72}{150}=48\%$, $\dfrac13 = 33.33\%$.
Nandini's marbles: 15 white of 25 $\Rightarrow \dfrac{15}{25}\times100 = 60\%$ are white.
Walking to school: 15 of 80 walk $\Rightarrow \dfrac{15}{80}\times100 = 18.75\%.$
Values: $25\%$ of $160 = 40$; $16\%$ of $250 = 40$; $140\%$ of $40 = 56$; $1\%$ of 1 hour $= 1\%$ of $60$ min $= 0.6$ min; $7\%$ of $10$ kg $= 0.7$ kg.
Fill in the blanks: if $30\%$ of $k = 70$, then $60\%$ of $k = 140$, $90\% = 210$, $120\% = 280$ (since $10\%$ of $k = 70/3$). "3 is ___% of 300" $\Rightarrow 1\%$. "___ is $40\%$ of 4" $\Rightarrow 1.6$. "40 is $80\%$ of ___" $\Rightarrow 50$.
Halwa recipe (Rava 40%, Sugar 40%, Ghee 20%): for $2$ kg total, Rava $= 0.4\times2 = 0.8$ kg, Sugar $= 0.8$ kg, Ghee $= 0.2\times2 = 0.4$ kg.
Coffee pickers: $20\%$ takes $18$ days, so $100\%$ takes $5\times18 = 90$ days (assuming the work rate stays constant).
Badminton 10:80:10 of 90 min: warm-up $9$ min, play $72$ min, cool-down $9$ min.
Petrol ₹60 → ₹100: increase $= \dfrac{40}{60}\times100 = 66.66\%$ — option (iv).
A number increased by 20% becomes 90: $1.2\times n = 90\Rightarrow n = 75.$
Samson's car (₹4,40,000 after 15% discount): $85\%$ of original $= 440000\Rightarrow$ original $= 440000\times\dfrac{100}{85} = ₹5,17,647$ (approx).
15. Common mistakes to avoid
- Taking profit/loss % on the selling price instead of the cost price (it's always on CP).
- Treating "$30\% + 20\%$ off" as "$50\%$ off" — successive discounts compound, so they give a smaller total reduction.
- Comparing two percentages as if equal-sized when the wholes (bases) differ.
- Forgetting that a discount of $d\%$ means you pay $(100-d)\%$ of the marked price.
- Confusing simple growth $P(1+rt)$ with compound growth $P(1+r)^t$.
- Computing increase/decrease % on the new value instead of the original.
16. Quick revision checklist
- $y\% = \dfrac{y}{100}$; fraction → % means $\times100$; % → decimal means $\div100$.
- $y\%$ of $Q = \dfrac{y}{100}\times Q$.
- Profit/Loss % on CP; increase/decrease % on the original value.
- Discount of $d\%$ ⟹ pay $(100-d)\%$ of MP.
- Growth: simple $P(1+rt)$, compound $P(1+r)^t$; decline ⟹ multiply by $(1-r)$ each period.
- Percentages can exceed $100\%$ (e.g. $250\%$ = $2.5$ times).
- $34\%$
- $75\%$
- $43\%$
- $80\%$
- $\dfrac{24}{10}$
- $\dfrac{12}{50}$
- $\dfrac{6}{25}$
- $\dfrac{24}{1000}$
- $25$
- $40$
- $64$
- $80$
- $0.5$
- $5.0$
- $0.05$
- $0.005$
- $12\%$
- $28\%$
- $40\%$
- $42\%$
- $13\%$
- $30\%$
- $43.3\%$
- $130\%$
- ₹630
- ₹1170
- ₹1450
- ₹1235
- $60\%$
- $83\%$
- $120\%$
- $160\%$
- $70$
- $72$
- $75$
- $108$
- ₹7800
- ₹7986
- ₹8000
- ₹6600
- ₹100
- ₹110
- ₹112
- ₹120
- English ($84\%$)
- Science ($87.5\%$)
- Equal
- Cannot compare
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