CLASS 8 MATHS · PART 1 · CHAPTER 7 · STUDY NOTES
Proportional Reasoning – 1
Understand proportional reasoning – 1 through short explanations, labelled models and original practice. Read one topic, explain it in your own words, then check.
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Resize both directions by the same factor
A picture keeps its shape when its width and height grow in the same ratio.Compare outlines using equal units
y (units)
x (units)
- Original 8 by 6 · Original 8 by 6
- Correct 12 by 9 · Both dimensions multiplied by 1.5.
- Stretched 12 by 6 · Only the width changed.
- 18 by 6 · (8, 6)
- 212 by 9 · (12, 9)
- 312 by 6 · (12, 6)
All outlines start at the origin so their proportions can be compared. One horizontal unit equals one vertical unit. Equal units on both axes.
Read the plotted values
| x | y |
|---|---|
| 0 | 0 |
| 8 | 0 |
| 8 | 6 |
| 0 | 6 |
| 0 | 0 |
| x | y |
|---|---|
| 0 | 0 |
| 12 | 0 |
| 12 | 9 |
| 0 | 9 |
| 0 | 0 |
| x | y |
|---|---|
| 0 | 0 |
| 12 | 0 |
| 12 | 6 |
| 0 | 6 |
| 0 | 0 |
- Scale factor
- The multiplier applied to each corresponding length.
- Aspect ratio
- The ratio of width to height.
An 8 cm by 6 cm picture has ratio 4:3. Multiplying both lengths by 1.5 gives 12 cm by 9 cm, still 4:3.
Changing only the width to 12 cm gives ratio 2:1 and stretches the picture. Adding the same amount to both lengths usually changes the ratio too.
Take a moment. What do you think?
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Question
A 10 by 8 picture is enlarged to width 15. What height keeps its shape?
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A ratio tells us how two amounts compare
Write the amounts in the stated order and use the same units for like quantities.Part-to-part and part-to-whole
- Ratio
- a:b compares an amount a with an amount b; the order matters.
- Equivalent ratios
- Ratios obtained by multiplying or dividing both terms by the same nonzero number.
12 red beads to 18 blue beads is 12:18 = 2:3. Divide both terms by their HCF, 6.
2:3 means 2 red for every 3 blue. It does not mean that 2/3 of all the beads are red: the red share is 2/(2 + 3) = 2/5.
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Question
Simplify 45 minutes to 2 hours.
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Keep the recipe constant by multiplying both parts
If a batch becomes three times as large, each ingredient becomes three times as large.A constant recipe passes through zero
Concentrate (spoons)
Water (glasses)
- Same recipe · 1.5 spoons per glass.
- 1Original batch · (4, 6)
- 2New batch · (10, 15)
Equal measures and unchanged recipe are assumed. No water means no concentrate in this model.
Read the plotted values
| Water | Concentrate |
|---|---|
| 0 | 0 |
| 4 | 6 |
| 10 | 15 |
| 12 | 18 |
- Direct proportion
- Two quantities are directly proportional when their ratio stays constant, so y = kx.
A recipe uses 4 glasses of water and 6 spoons of concentrate. A new batch of 10 glasses needs 6 × (10/4) = 15 spoons.
This assumes equal-sized glasses and spoons and the same recipe. A ratio alone does not guarantee the strength or quality of a real building material.
- 1
Find the scale factor: new amount ÷ old amount.
- 2
Multiply the matching amount by that same factor.
- 3
Check that the ratio remains unchanged.
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Question
A recipe uses 3 cups of rice for 5 people. How much for 20 people at the same serving size?
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Equal additions do not usually keep a ratio
Ask whether one amount is always a fixed multiple of the other.A fixed fee changes the relationship
Total bill (rupees)
Items (count)
- With ₹20 fee · Cost = 3n + 20.
- No fixed fee · Cost = 3n.
- 1Fixed fee · (0, 20)
- 2No fee · (0, 0)
Whole-number item counts are intended. Lines help compare the rules between those counts.
Read the plotted values
| Items | Total bill |
|---|---|
| 0 | 20 |
| 5 | 35 |
| 10 | 50 |
| Items | Total bill |
|---|---|
| 0 | 0 |
| 5 | 15 |
| 10 | 30 |
- Constant difference
- A fixed amount obtained by subtracting two quantities; this differs from a constant ratio.
A child aged 4 and an adult aged 28 have ratio 1:7. Four years later their ages are 8 and 32, ratio 1:4. Their age difference stays 24.
A price of ₹3 per item plus ₹20 delivery gives cost 3n + 20. Doubling the item count does not double the total bill.
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Question
Is a bill of ₹5 per item with no fixed fee directly proportional to item count?
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Three known values give the fourth
Write the matching quantities in corresponding positions before calculating.Use a unit rate or a scale factor
- Proportion
- An equality of ratios, a:b = c:d, where the required denominators are nonzero.
- Rule of three
- A way of finding a fourth proportional value from three known values.
If a/b = c/d, cross multiplication gives ad = bc. To find d when a ≠ 0, use d = bc/a.
At a constant speed, 60 km in 90 minutes gives 120 km in 180 minutes. Converting 3 hours to 180 minutes comes before scaling.
Check the ratios
5/80 = 8/128 = 1/16Keep books above books and cost above cost in the proportion.
Let’s look a little closer.
Go deeper: Trairāśika
The rule of three is also discussed in Indian mathematical traditions using the term Trairāśika. Its mathematics here is the equality of two ratios, not a rule to apply when a ratio changes.
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Question
Five identical notebooks cost ₹80. What do eight cost at the same unit price?
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Compare costs for the same amount
Different pack sizes become comparable after finding one unit price.Compare equal quantities
Unit price (rupees/kg)
Pack (comparison index)
- Unit-price points · 1 is the 250 g pack; 2 is the 1 kg pack.
- 1250 g: ₹600/kg · (1, 600)
- 21 kg: ₹480/kg · (2, 480)
The horizontal positions identify two packs; they are not a continuous quantity. The vertical values are comparable prices per kilogram.
Read the plotted values
| Pack | Unit price |
|---|---|
| 1 | 600 |
| 2 | 480 |
- Unit rate
- An amount per one unit of another quantity, such as rupees per kilogram.
A 250 g pack costing ₹150 has unit price ₹150/0.25 kg = ₹600/kg. A 1 kg pack at ₹480 is cheaper per kilogram.
Pack prices need not be directly proportional to size because packaging, discounts and other costs may differ.
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Question
Which has lower unit price: 300 g for ₹90 or 500 g for ₹160?
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Find the value of one ratio part
Add the ratio parts first. Then divide the total into that many equal parts.Five equal parts make the whole
- Proportional sharing
- Dividing a total so that the shares have a specified ratio.
Share ₹65 in the ratio 2:3. There are 5 parts, so one part is ₹65/5 = ₹13. The shares are ₹26 and ₹39.
Always check both the sum and the ratio. If you are sharing indivisible objects, each share must be a whole number.
- 1
Add the ratio terms: 2 + 3 = 5.
- 2
Divide the total by 5 to find one part.
- 3
Multiply one part by each term.
- 4
Check ₹26 + ₹39 = ₹65 and 26:39 = 2:3.
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Question
Share 84 stickers in the ratio 3:4.
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Change the ingredient that is allowed to change
Find the current amounts before trying to adjust the mixture.Hold blue fixed; increase yellow
- Mixture ratio
- The relative amounts of the named ingredients, in the stated order.
A 42 mL model mixture has blue:yellow = 2:5, giving 12 mL blue and 30 mL yellow. To change to 1:3 by adding only yellow, blue stays 12 mL.
Yellow must become 3 × 12 = 36 mL, so add 6 mL. The new total is 48 mL; it is not still 42 mL. Assume the measured ingredient volumes add in this model.
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Question
A model mixture has 10 units of A and 20 of B. How much B should be added to make A:B = 1:4?
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Area conversions square the length factor
A square metre contains a 100 by 100 grid of square centimetres.Convert each side
- Area conversion
- A conversion in which the length factor is applied in both directions.
- Hectare
- An area of 10,000 square metres.
| Unit | Exact equivalent |
|---|---|
| 1 m² | 10,000 cm² |
| 1 hectare | 10,000 m² |
| 1 L | 1000 mL |
| 1 mL | 1 cm³ |
1 m = 100 cm, so 1 m² = 100 × 100 = 10,000 cm². It is not 100 cm².
1 L = 1000 mL and 1 mL = 1 cm³. Rounded conversions, such as metres to feet, should be marked approximate.
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Question
Convert 2.5 m² to cm².
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An offset or a whole-object constraint changes the method
Some calculations need an added amount or a final rounding decision.Temperature conversion has an offset
Temperature (°F)
Temperature (°C)
- Fahrenheit conversion · F = 1.8C + 32.
- 10°C = 32°F · (0, 32)
- 225°C = 77°F · (25, 77)
This is a conversion graph, not direct proportion. Its vertical intercept is 32°F.
Read the plotted values
| Temperature | Temperature |
|---|---|
| 0 | 32 |
| 25 | 77 |
| 100 | 212 |
- Offset
- A fixed addition in a conversion rule, such as the +32 in °F = 9°C/5 + 32.
The Celsius–Fahrenheit rule is F = 9C/5 + 32. Because of +32, it is not direct proportion. At 25°C, F = 45 + 32 = 77°F.
If a bus seats 40 and 95 people must travel at once, 95/40 = 2.375 tells us that 2 buses are insufficient. We need 3 whole buses.
Go deeper: Fixed journey: faster means less time
For a fixed distance, speed × time stays constant. Doubling speed halves travel time if speed is constant during each journey. This is inverse proportion, studied further in Part 2.
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Question
How many 12-capacity boxes are needed for 50 objects if no box may exceed capacity?
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