Class 8 › Mathematics › Chapter
Fractions in Disguise
Percentages as fractions with denominator 100, finding a percentage of a quantity, percentages above 100, increase and decrease, profit, loss, discount, tax, simple and compound interest, depreciation, and tricky percentage situations.
1. Fractions as Percentages
Per cent (Latin per centum) means out of 100. A percentage is a fraction with denominator $100$: $x\% = \dfrac{x}{100}$.
Bar model: $\tfrac34 = 75\%$.
- Fraction $\to$ percentage: find an equivalent fraction with denominator $100$, or multiply the fraction by $100$. $\tfrac34 = \tfrac{75}{100} = 75\%$; $\tfrac25 = 40\%$.
- Percentage $\to$ fraction: $z\% = \tfrac{z}{100}$ (then simplify): $24\% = \tfrac{24}{100} = \tfrac{6}{25}$.
- Decimals: $31\% = \tfrac{31}{100} = 0.31$; divide by $100$ to convert a percentage to a decimal, multiply by $100$ for the reverse.
- Why $100$? Our number system is base $10$, so $100$ fits decimals; it is large enough for detail and small enough to grasp. Other options: per decem ($10$), per mille ($1000$).
- Why percentages when we have fractions? A common denominator makes comparison immediate ($26.47\%$ vs $28.88\%$ is easier than $\tfrac{9}{34}$ vs $\tfrac{13}{45}$).
- History: 'per hundred' appears in Kautilya's Arthashastra (4th century BCE) for interest; Romans used taxes of $\tfrac1{20}$, $\tfrac1{100}$.
| Percent | Fraction | Decimal |
|---|---|---|
| $50\%$ | $\tfrac12$ | $0.5$ |
| $25\%$ | $\tfrac14$ | $0.25$ |
| $75\%$ | $\tfrac34$ | $0.75$ |
| $10\%$ | $\tfrac1{10}$ | $0.1$ |
| $5\%$ | $\tfrac1{20}$ | $0.05$ |
| $1\%$ | $\tfrac1{100}$ | $0.01$ |
2. Percentage of a Quantity
$y\%$ of $x$
$$y\% \text{ of } x = \frac{y}{100} \times x$$
- Compare percentages only of the same whole. $25\%$ of $120$ g $= 30$ g of sugar, $35\%$ of $95$ g $= 33.25$ g; the larger percentage is not always the larger amount.
- Three equivalent routes: fraction multiplication ($\tfrac{80}{100} \times 75$), decimal multiplication ($0.8 \times 75 = 60$), and proportion ($80 : 100 :: x : 75$).
- Free-hand tricks: $50\% = \tfrac12$, $25\% = \tfrac14$, $10\%$ = shift the decimal point, $5\% = \tfrac12$ of $10\%$, $20\% = 2 \times 10\%$, $15\% = 10\% + 5\%$, $75\% = 100\% - 25\%$, $90\% = 100\% - 10\%$, $40\% = 2 \times 20\%$.
- Ratio to percentage: millet : water $= 2 : 7$ means millet is $\tfrac29 \times 100 = 22.22\%$ and water $77.78\%$; in $500$ ml, millet $= 111.1$ ml.
- Finding the whole from a part: if $40\%$ of a journey is $92$ km, total $= \tfrac{92 \times 100}{40} = 230$ km, remaining $= 138$ km (or $20\% = 46$ km, so $60\% = 138$ km).
- Estimate first (is it more or less than $50\%$, $25\%$?) and draw a rough bar model; it reduces errors.
3. Percentages Greater than 100
- A percentage above $100$ compares a quantity with a reference that it exceeds. Target $\text{₹}5000$: sales $\text{₹}2000 = 40\%$, $\text{₹}5000 = 100\%$, $\text{₹}6000 = 120\%$ ($20\%$ more than target), $\text{₹}10000 = 200\%$ (double).
- Meaning: $250\% = 2.5$ times the original; $120\% = 1.2$ times; $100\% = 1$.
- Example: harvest $650$ kg vs $260$ kg last year: $\tfrac{650}{260} \times 100 = 250\%$.
- Shortfall: $40\%$ of target achieved means $60\%$ short of target.
- $150\%$ of $\text{₹}5000 = \text{₹}7500$; $210\%$ of $\text{₹}5000 = \text{₹}10500$.
4. Using Percentages to Compare Proportions
- Marks out of different maximums cannot be compared directly. Convert to percentages: English $\tfrac{42}{50} = 84\%$, Science $\tfrac{70}{80} = 87.5\%$, so Science is better (even though more marks were lost there).
- Know Your Contents (KYC): convert each ingredient to a percentage of total weight (e.g. $\tfrac{99}{150} \times 100 = 66\%$ sugar). Check that all percentages add up to $100$.
5. Percentage Increase and Decrease
Change
$$\text{Percentage change} = \frac{\text{amount of increase (or decrease)}}{\text{original amount (base)}} \times 100$$
- Increase: tomatoes $\text{₹}30 \to \text{₹}42$: $\tfrac{12}{30} \times 100 = 40\%$.
- Decrease: footfall $160 \to 100$: $\tfrac{60}{160} \times 100 = 37.5\%$.
- Multiplier method: increase by $p\%$ $\Rightarrow$ new value $= (1 + \tfrac{p}{100}) \times$ old; decrease by $p\%$ $\Rightarrow (1 - \tfrac{p}{100}) \times$ old.
- Same statement: '$165\%$ of $1961$ population' $=$ 'increased by $65\%$' $= 1.65p$.
- 'Demand fell by $85\%$' means demand now is $15\%$ of before.
- Inflation is the percentage increase in prices (e.g. rice $\text{₹}38 \to \text{₹}42$: $\tfrac{4}{38} \times 100 \approx 10.5\%$).
- A number increased by $20\%$ becomes $90$: number $= \tfrac{90}{1.2} = 75$.
6. Profit, Loss, Discount and Tax
CP
Cost price
Price paid to buy the item.
MP
Marked price
Price quoted (MRP). Discounts are on MP.
SP
Selling price
Price actually paid by the customer after any discount.
| Quantity | Formula |
|---|---|
| Profit / loss | $\text{Profit} = SP - CP$ ($SP \gt CP$); $\text{Loss} = CP - SP$ ($SP \lt CP$) |
| Profit % | $\dfrac{\text{Profit}}{CP} \times 100$ |
| Loss % | $\dfrac{\text{Loss}}{CP} \times 100$ |
| SP with profit $p\%$ | $SP = CP \times (1 + \tfrac{p}{100})$ |
| SP with loss $p\%$ | $SP = CP \times (1 - \tfrac{p}{100})$ |
| Discount | $\text{Discount} = MP - SP$; $SP = MP \times (1 - \tfrac{d}{100})$ |
| Tax (GST $g\%$) | Final price $= \text{price} \times (1 + \tfrac{g}{100})$ |
- Examples: CP $300$, SP $430$: profit $130$, profit $\% = \tfrac{130}{300} \times 100 = 43.3\%$. CP $350$, SP $300$: loss $50$, loss $\% = 14.28\%$.
- Loss of $18\%$ on $\text{₹}2650$: $SP = 0.82 \times 2650 = \text{₹}2173$.
- Finding CP: a $25\%$ profit and $SP = 50$: $CP = \tfrac{50}{1.25} = 40$. A $12\%$ loss and $SP = 80$: $CP = \tfrac{80}{0.88}$.
- Discount example: $35\%$ off $\text{₹}1800$ gives $\text{₹}1170$; with $CP = 900$, profit $= 270$, i.e. $30\%$.
- Profit base: normally profit % is on cost ($\text{₹}500 \to \text{₹}750$: $50\%$); on revenue it would be $\tfrac{250}{750} = 33.33\%$.
- Gross profit $=$ sales $-$ cost of goods; net profit $=$ gross profit $-$ other expenses (transport, salaries, electricity). In this chapter, 'profit' means gross profit.
- GST is a percentage tax added to the price: $\text{₹}8250 \times 1.18$.
7. Growth, Compounding and Decline
Interest: extra money paid on a deposit (or by a borrower). Principal ($p$): the amount deposited. Rate ($r$): interest per year as a fraction (p.a. = per annum). Term ($t$): number of years.
No compounding
Simple interest
Interest is paid out each term; principal stays $p$. Interest $= prt$. Amount $= p(1 + rt)$.
Compounding
Interest added back
Each term's interest is added to the principal. Amount $= p(1 + r)^t$.
Example: $\text{₹}6000$ at $10\%$ p.a. for $3$ years
No compounding: $6000 + 3 \times 600 = \text{₹}7800$ ($6000 \times 1.3$, a $30\%$ gain).
Compounding: $6000 \times 1.1 = 6600 \to 7260 \to 7986$, i.e. $6000 \times 1.1^3 = \text{₹}7986$ ($6000 \times 1.331$, a $33.1\%$ gain).
Compounding gives more because interest also earns interest. Without compounding growth is linear; with compounding it is exponential.
- Interest earned: without compounding $prt$; with compounding $p(1 + r)^t - p$.
- Depreciation: reduction in value with age or use. Value after one year at $5\%$ depreciation $= 0.95 \times \text{value}$. TV $\text{₹}21000 \to \text{₹}19950$.
- Repeated decrease: decreasing $10\%$ per period for $t$ periods gives $\text{start} \times 0.9^t$. Village $1250$ after $3$ decades: $1250 \times 0.9^3 = 911.25 \approx 910$.
- General growth: a quantity growing at $r$ per period becomes $p(1 + r)^t$ (population at $3\%$ a year, bacteria at $2.5\%$ an hour).
8. Tricky Percentages
- Percentages compare fractions, not amounts. A gain of $\text{₹}200$ on $\text{₹}100$ ($200\%$) beats $\text{₹}500$ on $\text{₹}1000$ ($50\%$) in percentage but not in rupees.
- Percent discount vs flat discount: $20\%$ vs $\text{₹}50$ off: they are equal at $\text{₹}250$; below that, $\text{₹}50$ off is better (bill $\text{₹}180$); above it, $20\%$ is better (bill $\text{₹}300$).
- Successive discounts do not add. '$30\% + 20\%$' means $0.7 \times 0.8 = 0.56$ of the price, a $44\%$ discount, not $50\%$. For $\text{₹}200$: $200 \to 140 \to 112$, versus $100$ at $50\%$ off.
- Same for successive increases: $+3\%$ then $+4\%$ is $1.03 \times 1.04 = 1.0712$, i.e. $7.12\%$.
- A mishap: a $50\%$ profit margin followed by a $50\%$ discount: $1.5x \times 0.5 = 0.75x$, a $25\%$ loss. To break even, the discount must be $d$ with $1.5x(1 - d) = x$, so $d = \tfrac13 = 33.33\%$.
- Reverse percentages are not symmetric: if Ariba has $120\%$ of Arun's marbles, Arun has $\tfrac{100}{120} = 83.3\%$ of Ariba's.
- Symmetry: $x\%$ of $y = y\%$ of $x$ (both equal $\tfrac{xy}{100}$); e.g. $5\%$ of $40 = 2 = 40\%$ of $5$.
- Nested percentages: $60\%$ of $40\%$ of the students is $24\%$ of them.
9. Key Results and Quick Revision
| Idea | Formula |
|---|---|
| Percentage | $x\% = \tfrac{x}{100}$; fraction $\times 100 = $ percentage |
| $y\%$ of $x$ | $\tfrac{y}{100} \times x$ |
| Percent change | $\tfrac{\text{change}}{\text{original}} \times 100$ |
| Profit % / Loss % | $\tfrac{\text{profit or loss}}{CP} \times 100$ |
| SP after profit / loss $p\%$ | $CP(1 \pm \tfrac{p}{100})$ |
| Simple (no compounding) | $A = p(1 + rt)$ |
| Compound | $A = p(1 + r)^t$ |
| Depreciation / decline | $\text{value} \times (1 - r)^t$ |
Common mistakes:
• Adding successive percentages instead of multiplying the factors.
• Taking profit % on selling price instead of cost price.
• Using the new value, not the original, as the base for a percentage change.
• Forgetting to divide by $100$ when converting $r\%$ to $r$ in $p(1 + r)^t$.
• Comparing percentages of different wholes as if they were amounts.
• Writing $120\%$ increase when the quantity is $120\%$ of the original (that is a $20\%$ increase).
• Adding successive percentages instead of multiplying the factors.
• Taking profit % on selling price instead of cost price.
• Using the new value, not the original, as the base for a percentage change.
• Forgetting to divide by $100$ when converting $r\%$ to $r$ in $p(1 + r)^t$.
• Comparing percentages of different wholes as if they were amounts.
• Writing $120\%$ increase when the quantity is $120\%$ of the original (that is a $20\%$ increase).