Class 8 › Mathematics › Chapter
Proportional Reasoning-1
Similarity through multiplicative change, ratios and their simplest form, proportion, the Rule of Three (Trairasika), dividing a quantity in a ratio and unit conversions.
1. Observing Similarity in Change
Images (width × height in mm). A, C, D look alike; B and E do not.
- Multiplicative change: images look similar when width and height change by the same factor (C is $\tfrac12$ of A in both; D is $1.5\times$ A in both).
- Additive change fails: B has $20$ mm less width and $20$ mm less height than A, yet it looks distorted (height halves, width does not).
- When widths and heights change by the same factor, the changes are proportional.
2. Ratios and Their Simplest Form
Ratio $a : b$: for every $a$ units of the first quantity there are $b$ units of the second. $a$ and $b$ are the terms. Width : height of image A is $60 : 40$.
Simplest form
$$\text{Divide both terms by their HCF.}\qquad 60 : 40 = 3 : 2\ (\text{HCF } 20)$$
- Proportional ratios: two ratios are proportional (written $a : b :: c : d$) when they have the same simplest form, i.e. their terms change by the same factor.
- $60 : 40 :: 30 : 20$ and $60 : 40 :: 90 : 60$ (all reduce to $3 : 2$). Image B reduces to $2 : 1$ and E to $1 : 1$, so they are not proportional to A.
- Example: $3 : 4$ and $72 : 96$: HCF of $72, 96$ is $24$, giving $3 : 4$, so they are proportional.
- Same units first: convert both terms to one unit before forming a ratio ($600$ mL : $900$ mL $= 2 : 3$).
3. Problem Solving with Proportional Reasoning
- Find the factor of change: Lemonade: $6$ glasses need $10$ spoons of sugar. For $18$ glasses: factor $18 \div 6 = 3$, so $6 : 10 :: 18 : 30$, i.e. $30$ spoons.
- Compare ratios: a $60$ ft wall with $3$ bags of cement and a $40$ ft wall with $2$ bags both give $20 : 1$ (length per bag), so they are equally strong.
- Adding or subtracting the same number changes the ratio. Neelima ($3$) and mother ($30$): $1 : 10$. Nine years later ($12$ and $39$): $4 : 13$. Age ratios are not proportional over time.
- Missing terms for $14 : 21$: $? : 42 \to 28 : 42$ (factor $2$); $6 : ? \to 14y = 6$, $y = \tfrac37$, so $6 : 9$; $2 : ? \to$ divide by HCF $7$: $2 : 3$.
- Strength of coffee: compare decoction to milk. Regular $15 : 35 = 3 : 7$. Strong $20 : 30 = 2 : 3$ (more decoction per milk). Light $10 : 40 = 1 : 4$. To classify a mix, compare its ratio with $3 : 7$ (or compare decoction/total with $0.3$): $300 : 600$ is strong, $150 : 500$ is light, $200 : 400$ is strong, $24 : 56$ is regular, $100 : 300$ is light.
- Counting: a brick pattern with $9$ grey and $6$ coloured bricks gives grey : coloured $= 9 : 6 = 3 : 2$. Count one full block of the pattern.
4. Trairasika: The Rule of Three
Cross multiplication
$$a : b :: c : d \;\Longrightarrow\; ad = bc \;\Longrightarrow\; d = \frac{bc}{a}$$
Why $ad = bc$
If $c = fa$ and $d = fb$ for the same factor $f$, then $f = \tfrac ca = \tfrac db$. Multiply both sides by $ab$: $bc = ad$.
Āryabhaṭa (199 CE): three known numbers: pramāṇa ($a$, measure), phala ($b$, fruit) and ichchhā ($c$, requisition). The unknown ichchhāphala is $d = \dfrac{\text{phala} \times \text{ichchhā}}{\text{pramāṇa}}$.
- Mid-day meal: $120$ students need $15$ kg rice; for $80$ students: $120 : 15 :: 80 : ?$. Factor $\tfrac{80}{120} = \tfrac23$, so $15 \times \tfrac23 = 10$ kg.
- Units must match: car goes $90$ km in $150$ min; for $4$ h write $150 : 90 :: 240 : x$ (not $4$). $x = \tfrac{240 \times 90}{150} = 144$ km.
- Compare prices for the same weight: Himachal tea $200$ g for $\text{₹}200$ ($\text{₹}1000$/kg) vs Meghalaya $1$ kg for $\text{₹}800$. Weight : price is $1000 : 800 = 5 : 4$ for Meghalaya and $1 : 1$ for Himachal, so they are not proportional; Himachal tea is dearer per kg.
- Worked: Earth travels about $940$ million km a year, so a week $\approx \tfrac{940{,}000{,}000}{52} \approx 1.8$ crore km. Buses: $162$ people in $3$ full buses means $54$ per bus; $204$ people need $4$ buses ($12$ empty seats).
Warning: not every problem is a Rule of Three
Travel time at $50$ km/h is $2$ h. At $75$ km/h the time decreases, so $50 : 2 :: 75 : ?$ is wrong. The Rule of Three needs quantities that increase or decrease together (direct proportion). Speed and time vary inversely (taken up in Part-II).
Prices are not always proportional to size: shampoo sachet $6$ mL for $\text{₹}2$, small bottle $180$ mL for $\text{₹}154$: volume ratio $6 : 180$ vs price ratio $2 : 154$. Bulk packs are often cheaper per mL and use less packaging.
5. Sharing a Quantity in a Ratio
Dividing $x$ in the ratio $m : n$
$$\text{First part} = m \times \frac{x}{m+n}, \qquad \text{Second part} = n \times \frac{x}{m+n}$$
- Idea: split $x$ into $m + n$ equal groups; the first person gets $m$ groups, the second gets $n$. Each group has $\tfrac{x}{m+n}$.
- $12$ in $3 : 1$: groups $= 4$, size $3$, so $9$ and $3$. $42$ in $4 : 3$: groups $= 7$, size $6$, so $24$ and $18$.
- Profit sharing: investments $\text{₹}75{,}000 : \text{₹}25{,}000 = 3 : 1$; profit $\text{₹}4{,}000$ gives $\text{₹}3{,}000$ and $\text{₹}1{,}000$.
- Changing a mixture: $40$ kg of sand : cement $= 3 : 1$ gives $30$ kg sand and $10$ kg cement. For $5 : 2$ keep $30$ kg sand, so cement $= \tfrac25 \times 30 = 12$ kg. Add $12 - 10 = 2$ kg.
- Adding to one part changes the ratio: red : yellow $= 3 : 5$ in one bucket; add one more bucket of yellow: red $\tfrac38$, yellow $\tfrac58 + 1 = \tfrac{13}{8}$, new ratio $3 : 13$.
- Examples: $\text{₹}4500$ in $2 : 3$ gives $\text{₹}1800$ and $\text{₹}2700$; $240$ mL of acid : water $= 1 : 5$ gives $40$ mL and $200$ mL.
6. Unit Conversions
| Type | Conversion |
|---|---|
| Length | $1\ \text{m} = 3.281\ \text{ft}$ |
| Area | $1\ \text{m}^2 = 10.764\ \text{ft}^2$; $1\ \text{acre} = 43{,}560\ \text{ft}^2$; $1\ \text{hectare} = 10{,}000\ \text{m}^2 = 2.471\ \text{acres}$ |
| Volume | $1\ \text{mL} = 1\ \text{cc}$; $1\ \text{L} = 1000\ \text{mL} = 1000\ \text{cc}$ |
| Temperature | $F = \tfrac95 C + 32$; $C = \tfrac59(F - 32)$; $0^\circ\text{C} = 32^\circ\text{F}$, $25^\circ\text{C} = 77^\circ\text{F}$ |
- Use proportion with conversions: $1$ acre costs $\text{₹}15{,}00{,}000$, so $2400\ \text{ft}^2 = \tfrac{2400}{43560}$ acre.
- Crowdedness compares density: people per sq km (Delhi $\tfrac{30\text{ million}}{1484} \approx 20{,}000$, Mumbai $\tfrac{20\text{ million}}{550} \approx 36{,}000$, so Mumbai is more crowded).
- Body ratios: a crane of height $155$ cm has neck : body $= 4 : 6$, so neck $= \tfrac{4}{10} \times 155 = 62$ cm; apply the same ratio to any height.
7. Key Results and Quick Revision
| Idea | Rule |
|---|---|
| Ratio $a : b$ | For every $a$ of one quantity, $b$ of the other |
| Simplest form | Divide both terms by the HCF |
| Proportional ratios | $a : b :: c : d \iff ad = bc$ |
| Fourth quantity | $d = \tfrac{bc}{a}$ (Rule of Three) |
| Share $x$ in $m : n$ | $m\cdot\tfrac{x}{m+n}$ and $n\cdot\tfrac{x}{m+n}$ |
Common mistakes:
• Adding or subtracting the same number to both terms and expecting the same ratio.
• Mixing units (hours vs minutes, kg vs g) inside a proportion.
• Using the Rule of Three when one quantity goes down as the other goes up.
• Dividing the whole by one term instead of by $m + n$.
• Putting terms in the wrong order (keep the same quantity in the same position in both ratios).
• Assuming price is always proportional to quantity.
• Adding or subtracting the same number to both terms and expecting the same ratio.
• Mixing units (hours vs minutes, kg vs g) inside a proportion.
• Using the Rule of Three when one quantity goes down as the other goes up.
• Dividing the whole by one term instead of by $m + n$.
• Putting terms in the wrong order (keep the same quantity in the same position in both ratios).
• Assuming price is always proportional to quantity.