Class 8 › Mathematics › Chapter
Proportional Reasoning-2
Ratios in maps, ratios with many terms, dividing a quantity in a ratio, constructing pie charts, and inverse proportion with time, speed and work problems.
1. Proportionality: A Quick Recap
Proportional relationship: related quantities change by the same factor.
Cross-multiplication test
$$a : b \text{ and } c : d \text{ are proportional} \iff a \times d = b \times c \iff \frac{a}{c} = \frac{b}{d}$$
- Example: $6 : 3$ and $4 : 2$: $6 \times 2 = 12 = 3 \times 4$, so they are proportional (idli batter tastes the same).
- Rule of three: if $a : b :: c : d$, then $d = \dfrac{bc}{a}$ (a direct proportion).
2. Ratios in Maps
Representative Fraction (RF): the ratio of a distance on the map to the corresponding actual ground (geographical) distance, e.g. $1 : 60{,}00{,}000$.
- $1 : 60{,}00{,}000$ means $1$ cm on the map $= 60{,}00{,}000$ cm $= 60$ km on the ground.
- Method: measure the map distance with a ruler, multiply by the RF number, then convert units ($100000$ cm $= 1$ km).
- It gives geographical (straight-line) distance, not road distance.
- Different maps with different scales should give roughly the same ground distance.
- Classroom scale $1 : 50$: $1$ cm on paper $= 50$ cm in the room.
3. Ratios with More than 2 Terms
Multi-term proportion
$$a : b : c : d :: p : q : r : s \;\Longrightarrow\; \frac{a}{p} = \frac{b}{q} = \frac{c}{r} = \frac{d}{s}$$
- A ratio such as $8 : 4 : 2 : 1$ (coriander : chillies : toor dal : methi) says: for every $8$ of the first, there are $4$, $2$, $1$ of the others.
- Scaling a recipe: if only $2$ chillies are available (half of $4$), halve everything: $4 : 2 : 1 : 0.5$, so $8:4:2:1 :: 4:2:1:0.5$.
- One known part: find the value of one part ($\text{known amount} \div \text{its term}$), then multiply every term by it.
- Example (paint $2:3:5$, $10$ L white): $1$ part $= 10 \div 5 = 2$ L, so red $4$ L, blue $6$ L, total $20$ L.
- Example (concrete $1 : 1.5 : 3$, $3$ bags cement): multiply by $3$: $3 : 4.5 : 9$, total $16.5$ bags.
4. Dividing a Whole in a Given Ratio
Share formula
$$\text{Dividing } x \text{ in the ratio } a : b : c : \dots \text{ gives parts } x \cdot \frac{a}{a+b+c+\dots},\; x \cdot \frac{b}{a+b+c+\dots},\; \dots$$
- Steps: add the terms; divide the whole by that sum (value of one part); multiply each term by it.
- Example: $110$ units of concrete in $1 : 1.5 : 3$ (sum $5.5$): $110 \div 5.5 = 20$, giving $20$, $30$, $60$.
- Example: $50$ ml of purple in $2:3:5$ (sum $10$): red $10$, blue $15$, white $25$ ml.
- Angles of a triangle in $1 : 3 : 5$: $180^\circ \times \tfrac19, \tfrac39, \tfrac59 = 20^\circ, 60^\circ, 100^\circ$.
- Triangle sides in a ratio: a $3:4:5$ triangle can be scaled to many sizes (similar, not congruent). A $1:3:5$ triangle is impossible since $1 + 3 \lt 5$ (two sides must sum to more than the third).
- Check: the parts must add up to the whole.
5. Pie Charts
Pie chart: a circle divided into slices whose angles are proportional to the parts of a whole.
Slices are drawn anticlockwise from radius AB.
Slice angle
$$\text{Angle} = \frac{\text{part}}{\text{total}} \times 360^\circ$$
- Steps: (1) reduce the ratio by the HCF (e.g. $12:10:8:6:4 \to 6:5:4:3:2$, sum $20$); (2) each angle $= \tfrac{\text{term}}{20} \times 360^\circ = \text{term} \times 18^\circ$; (3) draw a circle and radius; (4) measure each angle in turn with a protractor from the previous radius; (5) colour and label.
- Check: all angles must add up to $360^\circ$.
- From percentages: $1\% = 3.6^\circ$ (e.g. $50\% = 180^\circ$, $25\% = 90^\circ$).
- Reading a pie chart: fraction of a part $= \tfrac{\text{angle}}{360^\circ}$; if one part's count is known, find the total by proportion.
6. Inverse Proportions
Direct
Same direction
Both change by the same factor $n$. The quotient $\tfrac{x}{y} = k$ stays constant (cost vs quantity).
Inverse
Opposite directions
If one changes by factor $n$, the other changes by $\tfrac1n$. The product $xy = k$ stays constant.
Inverse proportion
$$xy = k \quad\Longrightarrow\quad x_1 y_1 = x_2 y_2 \quad\Longrightarrow\quad \frac{x_1}{x_2} = \frac{y_2}{y_1}$$
- Speed and time (fixed distance): walk $5$ km/h, $18$ h; bicycle $15$, $6$; motorcycle $30$, $3$; car $60$, $1.5$. Product $= 90$ km (the distance, the constant $k$). Speed $\times 3$, time $\times \tfrac13$.
- Test for inverse proportion: check that $x \times y$ is the same for every pair (for direct proportion check that $\tfrac{x}{y}$ is constant).
- Typical inverse pairs: number of workers (or taps, pumps, machines) and time; speed and time; people and days food lasts; chairs per row and number of rows.
- Typical direct pairs: length of cloth and price at a fixed rate; pages and reading time at a fixed speed; work done and time (one worker).
7. Time, Work and Inverse Problems
| Problem | Setup | Answer |
|---|---|---|
| 20 workers take 4 days; how long for 10 workers? | $20 \times 4 = 10 \times y_2$ | $8$ days |
| 2 pumps fill a tank in 18 h; with 4 pumps? | $2 \times 18 = 4 \times x$ | $9$ h |
| Food for 80 students lasts 15 days; 20 more join | $80 \times 15 = 100 \times x$ | $12$ days |
| 5 workers move 4500 bricks a day; how many for 18000? | Direct: $4500 : 18000 :: 5 : x$ | $20$ workers |
Working together (rates add)
1. Take the whole job as 1 unit. Rate $= \tfrac{1}{\text{time alone}}$ units per hour.
2. Add the rates. Ram: $1$ h alone $\Rightarrow$ rate $1$. Shyam: $1.5$ h $\Rightarrow$ rate $\tfrac23$. Together: $1 + \tfrac23 = \tfrac53$ units per hour.
3. Work and time are directly proportional: $\tfrac53 : 1 :: 1 : x \Rightarrow x = \tfrac35$ hour.
Shortcut: $\dfrac{1}{t} = \dfrac{1}{t_1} + \dfrac{1}{t_2}$. Pumps taking $3$ h and $2$ h fill a tank together in $\tfrac{1}{1/3 + 1/2} = \tfrac65 = 1.2$ h.
- Assumptions: all workers or pumps work at the same constant rate, and the job (or each family's usage) stays the same.
- One tank, one pump: $2$ tanks in $6$ h is a direct proportion ($5$ tanks take $15$ h).
8. Key Results and Quick Revision
| Idea | Rule |
|---|---|
| Proportional ratios | $a : b :: c : d \iff ad = bc$ |
| Multi-term | $\tfrac ap = \tfrac bq = \tfrac cr = \dots$ |
| Share of $x$ | $x \times \tfrac{\text{term}}{\text{sum of terms}}$ |
| Pie slice | $\tfrac{\text{part}}{\text{total}} \times 360^\circ$ |
| Direct proportion | $\tfrac{x_1}{y_1} = \tfrac{x_2}{y_2} = k$ |
| Inverse proportion | $x_1 y_1 = x_2 y_2 = k$ |
| Map RF $1 : n$ | $1$ cm on map $= n$ cm on ground |
Common mistakes:
• Using a direct proportion for an inverse situation (e.g. $30 : 60 :: 3 : x$ for speed and time).
• Dividing by one term instead of the sum of all terms.
• Forgetting to convert cm to km on maps.
• Taking map distance as road distance.
• Pie chart angles that do not add up to $360^\circ$.
• Adding times instead of adding rates when people work together.
• Using a direct proportion for an inverse situation (e.g. $30 : 60 :: 3 : x$ for speed and time).
• Dividing by one term instead of the sum of all terms.
• Forgetting to convert cm to km on maps.
• Taking map distance as road distance.
• Pie chart angles that do not add up to $360^\circ$.
• Adding times instead of adding rates when people work together.