Proportional Reasoning-2 — Class 8 Mathematics (Ganita Prakash Part 2)
What the book actually covers (2026-27) This chapter is Chapter 3 of Ganita Prakash Part 2, pages 55–69. Its six sections are: 3.1 Proportionality — a quick recap · 3.2 Ratios in maps · 3.3 Ratios with more than 2 terms · 3.4 Dividing a whole in a given ratio · 3.5 A slice of the pie (pie charts) · 3.6 Inverse proportions. It contains no compound interest, no mixtures and alligation, and no partnership. Compound and simple interest, profit, loss and discount are all in Fractions in Disguise (Part 2, Chapter 1 — the eighth chapter of the course). Mixtures/alligation and partnership are competitive-exam topics that are not in Ganita Prakash at all. Everything of that kind has been moved to the appendix at the end of this page.
1. Proportionality — a quick recap
Idli batter is rice and urad dal, often mixed 2 : 1. Viswanath mixes 6 cups of rice with 3 of dal; Puneet mixes 4 with 2. Will their idlis taste the same?
They will, because the two ratios are proportional. Two ratios a : b and c : d are proportional when
a × d = b × c (cross-multiplication), equivalently a/b = c/d
Here 6 × 2 = 12 and 3 × 4 = 12, so 6 : 3 :: 4 : 2. Both reduce to 2 : 1 — Viswanath simply makes one and a half times as much batter.
The idea underneath: when two or more related quantities change by the same factor, the relationship is proportional. Scaling everything by the same number never changes the taste; scaling only some of it does.
2. Ratios in maps
Look at the bottom-right corner of almost any map and you will find a ratio such as 1 : 60,00,000. This is the Representative Fraction (RF) — the ratio between a distance measured on the map and the matching distance on the ground.
RF 1 : 60,00,000 means 1 cm on the map = 60,00,000 cm on the ground. Converting:
60,00,000 cm ÷ 100 = 60,000 m 60,000 m ÷ 1000 = 60 km
So 1 cm on this map stands for 60 km, and 5 cm stands for 300 km.
Two warnings that matter.
This is geographical distance, not road distance. A map scale measures straight lines. Roads bend around hills, rivers and towns, so a highway signboard will always read more than the map calculation.
Accuracy shrinks with scale. On a 1 : 60,00,000 map, one millimetre of ruler error is already 6 km on the ground. A map at 1 : 20,00,000 shows the same pair of cities three times further apart on the page, so the same ruler gives a three times more precise answer. Different maps of different scales should all give roughly the same ground distance — that is the point of the exercise — but not identically.
3. Ratios with more than 2 terms
Viswanath grinds a spice mix from 8 spoons of coriander seeds, 4 red chillies, 2 spoons of toor dal and 1 spoon of fenugreek. That is a four-term ratio:
8 : 4 : 2 : 1
Puneet has only 2 red chillies. Since 2 is half of 4, every other ingredient must also be halved — scaling only some of them would change the proportion and therefore the taste. He needs 4 spoons of coriander, 2 chillies, 1 spoon of toor dal and half a spoon of fenugreek:
8 : 4 : 2 : 1 :: 4 : 2 : 1 : 0.5
The general test
Two multi-term ratios a : b : c : d and p : q : r : s are proportional when
a/p = b/q = c/r = d/s
— one common factor throughout. Here 8/4 = 4/2 = 2/1 = 1/0.5 = 2 ✓
Terms do not have to be whole numbers. Concrete for pillars and beams is mixed cement : sand : gravel :: 1 : 1.5 : 3, and Puneet's mix ends in 0.5. You may clear the decimals if you like — 1 : 1.5 : 3 doubled is 2 : 3 : 6, the same ratio — but you must never round a term. Rounding 0.5 up to 1 would double the fenugreek.
Worked example. A shade of purple is Red : Blue : White :: 2 : 3 : 5. Yasmin has 10 litres of white. White is 5 parts, so 1 part = 10 ÷ 5 = 2 litres. Red = 2 × 2 = 4 litres, Blue = 3 × 2 = 6 litres, and the total is 4 + 6 + 10 = 20 litres.
4. Dividing a whole in a given ratio
To divide a quantity x in the ratio a : b : c : …
- Add the terms — this is how many equal parts the whole is cut into.
- Divide x by that sum — this is one part.
- Multiply each term by one part.
- Add the answers back and check they give x.
As a formula, the parts are
x × a ÷ (a + b + c + …), x × b ÷ (a + b + c + …), and so on.
Example. 110 units of concrete at 1 : 1.5 : 3. The terms add to 5.5, and 110 ÷ 5.5 = 20, so the mix needs 20 units of cement, 30 of sand and 60 of gravel. Check: 20 + 30 + 60 = 110 ✓
Example. A triangle with angles in the ratio 1 : 3 : 5. Angles of a triangle add to 180°, and 1 + 3 + 5 = 9, so one part is 20°. The angles are 20°, 60° and 100°.
Read what the ratio is counting. In the coins question, 100 coins are shared as ₹10 : ₹5 : ₹2 : ₹1 coins in the ratio 4 : 3 : 2 : 1. That gives 40, 30, 20 and 10 coins — worth ₹400, ₹150, ₹40 and ₹10, a total of ₹600. Dividing ₹100 in the ratio 4 : 3 : 2 : 1 would be completely wrong: the ratio of values is 40 : 15 : 4 : 1, nothing like the ratio of counts.
Ratios of sides versus ratios of angles
These behave differently, and the chapter puts them side by side deliberately.
Sides in the ratio 3 : 4 : 5 — a triangle exists (3, 4, 5 works, and so does 6, 8, 10 and 30, 40, 50). All of them are right-angled, since 3² + 4² = 5². But they are not congruent to each other: a ratio fixes shape and angles, not size. They are similar — the same shape at different scales, like an enlarged photograph.
Sides in the ratio 1 : 3 : 5 — impossible. The sides would be k, 3k and 5k, and
k + 3k = 4k, which is less than 5k for every k > 0
so the two shorter sides can never reach across the longest one. This is the triangle inequality: any two sides must together exceed the third. Scaling does not rescue it — the shortfall grows with k.
But angles in the ratio 1 : 3 : 5 are perfectly fine — 20°, 60°, 100°, as computed above. Angles only need to sum to 180°; sides carry the extra inequality condition. The same three numbers behave completely differently depending on what they measure.
5. A slice of the pie
A pie chart shows proportions of a whole. Each slice's angle must be proportional to its value, and the circle is 360°, so
angle = (value ÷ total) × 360°
or, when the data is given as percentages, angle = percentage × 3.6°.
Worked example — grades of 40 students
| Grade | A | B | C | D | E |
|---|---|---|---|---|---|
| Students | 12 | 10 | 8 | 6 | 4 |
Divide 360° in the ratio 12 : 10 : 8 : 6 : 4. Simplify first — the HCF is 2, giving 6 : 5 : 4 : 3 : 2. These add to 20 parts, so one part is 360 ÷ 20 = 18°.
| Grade | Parts | Angle |
|---|---|---|
| A | 6 | 108° |
| B | 5 | 90° |
| C | 4 | 72° |
| D | 3 | 54° |
| E | 2 | 36° |
Check: 108 + 90 + 72 + 54 + 36 = 360° ✓
Drawing it
- Draw a circle and mark a radius AB.
- Measure 108° from AB and draw radius AC — that slice is grade A.
- From AC measure 90° for grade B, then 72°, then 54°, then 36°.
- The last radius should close exactly onto AB. If it does not, your angles did not total 360°.
- Colour and label the slices.
Always add the angles before drawing. Rounding each slice separately loses a degree here and there, and the chart will not close. If the total misses 360, adjust the largest slice.
Reading a pie chart backwards
The same relationship works in reverse. If a slice is 30° out of 360°, it is 30/360 = 1/12 of the whole. So if that slice represents 18 children, the survey covered 18 × 12 = 216 children.
A neat special case: if the total happens to be 360, then one item equals exactly one degree and no arithmetic is needed at all.
6. Inverse proportion
This is the part of the chapter that costs students marks.
Puneeth's father rides Lucknow to Kanpur in 3 hours at 30 km/h. At 60 km/h, how long?
Writing the usual proportion 30 : 60 :: 3 : x gives x = 6 hours — the faster trip taking longer. That is absurd, and the absurdity is the point: the rule of three assumes a direct proportion, and this is not one.
Direct and inverse, side by side
| Direct proportion | Inverse proportion | |
|---|---|---|
| Behaviour | double one, the other doubles | double one, the other halves |
| What is constant | the quotient x/y | the product xy |
| Test | x₁/y₁ = x₂/y₂ = … = k | x₁y₁ = x₂y₂ = … = k |
| Typical situation | a fixed rate — km per litre, ₹ per metre | a fixed job or quantity being shared — one tank, one route, one wall |
The Lucknow–Kanpur table
| Mode | Walk | Bicycle | Motorcycle | Car |
|---|---|---|---|---|
| Speed (km/h) | 5 | 15 | 30 | 60 |
| Time (hours) | 18 | 6 | 3 | 1.5 |
Walking to bicycle: speed ×3, time ÷3. Bicycle to motorcycle: ×2 and ÷2. Motorcycle to car: ×2 and ÷2. The two quantities change by the same factor in opposite directions.
And the product is always the same:
5 × 18 = 15 × 6 = 30 × 3 = 60 × 1.5 = 90
That 90 is not an accident — it is the distance in kilometres, which of course does not depend on how you travel. In every inverse proportion the constant is some real fixed thing, and being able to name it is the best check that you have chosen the right relationship.
Deciding which one you have
Ask: if I double the first quantity, does the second double or halve?
| Situation | Which | The constant is |
|---|---|---|
| Taps filling a tank and the time taken | inverse | the tank's volume |
| Painters and days to paint a fixed wall | inverse | worker-days of labour |
| Speed of a cyclist and time on a fixed route | inverse | the length of the route |
| Petrol in the tank and distance travelled | direct | mileage, km per litre |
| Metres of cloth and price at a fixed rate | direct | the rate per metre |
| Pages in a book and reading time at fixed speed | direct | reading speed, pages per hour |
A trap worth naming. More pumps filling one tank is inverse — the job is fixed and gets shared. More tanks for one pump is direct — the job itself is growing while the rate stays put. The words look almost identical; ask which quantity is being held constant before writing anything.
Working together — add the rates, never the times
Ram cuts a quantity of vegetables in 1 hour; Shyam takes 1.5 hours. Together?
The answer is not 2.5 hours (adding) and not 1.25 hours (averaging). Two people together must be faster than the quicker of them alone, so the answer must be under 1 hour.
Convert each to work per hour, taking the whole job as 1 unit:
- Ram: 1 unit per hour
- Shyam: 1 ÷ 1.5 = 2/3 unit per hour
- Together: 1 + 2/3 = 5/3 units per hour
Then invert once at the end: one unit takes 1 ÷ (5/3) = 3/5 hour = 36 minutes ✓
The same method handles pumps, taps and painters. A small pump filling a tank in 3 hours and a large one in 2 hours together fill 1/3 + 1/2 = 5/6 per hour, so the tank takes 6/5 = 1.2 hours = 1 hour 12 minutes.
Why times cannot be added. Hours-per-tank is an inverted measure — a bigger number means slower work. Inverted measures do not combine by addition. Tanks-per-hour does, because the two pumps really are pouring into the same tank at the same moment.
State the assumptions
Every one of these models rests on assumptions, and the chapter asks for them explicitly.
For "3 workers paint a fence in 4 days; how long will 4 workers take?" the answer 3 days assumes:
- all workers work at the same rate;
- they work the same hours each day;
- they do not obstruct one another;
- the work divides freely, with no stage waiting on another.
Assumption 3 is where the model breaks. It predicts that 12 workers finish in 1 day and 24 workers in half a day, which cannot be true of one fence. Inverse proportion describes the arithmetic faithfully; it does not know about crowding.
7. Common mistakes
- Assuming every proportion is direct. Ask the doubling question first. An answer saying a faster bus takes longer is the signal you chose wrongly.
- Adding or averaging the times for two workers or two pipes. Add the rates, invert at the end.
- Using the ratio on the wrong quantity. The coin ratio counts coins, not rupees.
- Forgetting to add the newcomers. "10 more families move in" means 30 families, not 10.
- Confusing similar with congruent. A ratio of sides fixes shape, not size.
- Treating a ratio of sides like a ratio of angles. 1 : 3 : 5 works for angles and fails for sides.
- Checking only the first two columns of a proportion table. One mismatch anywhere breaks it.
- Pie-chart angles that miss 360°. Add them before you draw.
8. Chapter summary
- a : b :: c : d exactly when a × d = b × c.
- Multi-term ratios are proportional when a/p = b/q = c/r = … — every term scaled by the same factor.
- Dividing x in a ratio: add the terms, divide x by the sum, multiply out, and check the parts add back to x.
- RF 1 : n means 1 cm on the map is n cm on the ground; 1 : 60,00,000 gives 1 cm = 60 km, and this is straight-line, not road, distance.
- Pie chart: angle = (value ÷ total) × 360°, or percentage × 3.6°. The angles must total exactly 360°.
- Direct proportion: the quotient x/y is constant.
- Inverse proportion: the product xy is constant, and that constant is always some real fixed thing.
- Working together: add rates, never times.
- A ratio of sides fixes shape and angles but not size; sides also have to satisfy the triangle inequality, which angles do not.
Appendix — beyond the current syllabus
Everything below is genuinely useful, but none of it is in Chapter 3 of Ganita Prakash Part 2. Do not present it as this chapter's content in a Class 8 exam.
Simple and compound interest — this is Chapter 8, not Chapter 10
Interest, profit, loss and discount belong to Fractions in Disguise (Part 2, Chapter 1 — the eighth chapter of the course), where they are developed as applications of percentage. Study them there. For reference:
Simple interest: SI = (P × R × T) ÷ 100 Compound interest: A = P(1 + R/100)ⁿ, and CI = A − P
For half-yearly compounding, halve the rate and double the number of periods; for quarterly, quarter the rate and quadruple the periods.
Mixtures and alligation
Not in Ganita Prakash at any point. The alligation rule — for mixing two grades at prices c and d to reach a mean price m, the quantities are in the ratio (d − m) : (m − c) — is a competitive-exam shortcut. The multi-term ratio work in §3.3 of this chapter is the school-syllabus version of the same territory, and it is enough for Class 8.
Partnership
Also not in Ganita Prakash. The idea — profits shared in the ratio of capital × time invested — is a straightforward application of §3.4's dividing-in-a-ratio, so if you meet it in an aptitude test you already have the tool. It is not examinable at Class 8.
Compound ratios and chained ratios
If a : b = 3 : 4 and b : c = 5 : 7, you can find a : b : c by making the two b-terms match: multiply the first ratio by 5 and the second by 4, giving 15 : 20 and 20 : 28, so a : b : c = 15 : 20 : 28. A useful olympiad technique, not part of the chapter.
Where this chapter goes next
- Class 9–10 Statistics — pie charts become one of several data displays, alongside histograms and frequency polygons.
- Class 9–10 Coordinate Geometry — direct proportion y = kx is the straight line through the origin; inverse proportion xy = k is the hyperbola.
- Class 11–12 Physics — Boyle's law (PV = constant) is inverse proportion; Ohm's law (V = IR) is direct.
- Competitive aptitude tests — time and work, pipes and cisterns, and time-speed-distance are all built directly on §3.6.
