Chapter 7 · Grade 8
Proportional Reasoning-1
Ganita Prakash · NCERT · Reprint 2026–27
⚖️ 📐 🔢📐 1. Proportional Similarity
When we resize an image, the result looks similar only if both dimensions change by the same factor. If they change by different factors, the image looks distorted.
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Key Idea: Two quantities change proportionally when they are multiplied (or divided) by the same factor. Adding or subtracting the same amount does NOT make quantities proportional.
@edugrown
Multiplying by the same factor → similar shape. Subtracting the same number → distorted shape.
⚠️
DON'T CONFUSE: Adding or subtracting the same number from both dimensions does NOT make images proportional. Only multiplication/division by the same factor works!
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⚖️ 2. Ratios
A ratio of the form a : b means: for every a units of the first quantity, there are b units of the second. The numbers a and b are called the terms of the ratio.
⭐ RATIO NOTATION
a : b
Read as "a is to b" — a and b are the terms
💡 MINI EXAMPLE
A rectangle has width 6 cm and height 4 cm.
Width : Height = 6 : 4
This means: for every 6 units of width, there are 4 units of height.
🔑
Same factor = Proportional ratios. Two ratios like 60:40 and 30:20 are proportional because you multiply both terms of the first by ½ to get the second.
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🔢 3. Ratios in Simplest Form
A ratio is in its simplest form when both terms share no common factor other than 1. We simplify by dividing both terms by their HCF (Highest Common Factor).
🪜 How to Reduce a Ratio to Simplest Form
1
Write the ratio a : b
2
Find the HCF of a and b
3
Divide both terms by the HCF
4
The result is the simplest form
💡 MINI EXAMPLE
60 : 40 → HCF = 20 → 60÷20 : 40÷20 = 3 : 2
90 : 60 → HCF = 30 → 90÷30 : 60÷30 = 3 : 2
Both give 3:2 → they are proportional! ✓
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REMEMBER: Two ratios are proportional if and only if their simplest forms are identical.
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🟰 4. Proportion ( :: )
When two ratios are equal in their simplest forms, we say they are in proportion. The symbol :: means "is proportional to".
⭐ PROPORTION NOTATION
a : b :: c : d
Reads: "a is to b as c is to d" — meaning ratios a:b and c:d are equal
@edugrown
When a:b :: c:d, then cross-multiplying gives ad = bc
⭐ CROSS MULTIPLICATION RULE
If a : b :: c : d then ad = bc
Also: the unknown fourth term d = bc / a
⚠️
COMMON MISTAKE: Adding the same number to both terms of a ratio changes the ratio! Example: ratio of ages changes as years pass, because addition ≠ multiplication.
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🧮 5. Solving Proportions
Given three of the four quantities in a : b :: c : d, we can find the fourth using the factor method or cross multiplication.
⚙️ Method 1 — Factor Method
🪜 Steps
1
Find the factor of change in the known term: divide the new value by the original value
2
Multiply the corresponding original value by that same factor
3
That product is the missing term
💡 MINI EXAMPLE
Find x if 4 : 10 :: 12 : x
Factor = 12 ÷ 4 = 3 → x = 10 × 3 = 30
So 4 : 10 :: 12 : 30 ✓
⚙️ Method 2 — Cross Multiplication
🪜 Steps
1
Write the proportion: a : b :: c : d
2
Cross-multiply: a × d = b × c
3
Solve for the unknown: d = (b × c) / a
💡 MINI EXAMPLE
Find x if 5 : 8 :: 15 : x
5 × x = 8 × 15 → 5x = 120 → x = 24
⭐
EXAM TIP: Always convert units to be the same before setting up a proportion. For example, convert hours to minutes if the other ratio uses minutes.
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🏺 6. Trairasika — The Rule of Three
📜 ANCIENT INDIA
The Trairasika (Rule of Three) was described by the mathematician Āryabhaṭa (199 CE). It solves problems where 4 quantities are linked proportionally — 3 are known and 1 must be found.
ichchhāphala = (phala × ichchhā) ÷ pramāṇa
pramāṇa = given measure · phala = given result · ichchhā = desired quantity
In modern terms: given a : b :: c : d, use cross multiplication to find the unknown: d = (b × c) / a
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NOT ALWAYS APPLICABLE: The Rule of Three only works for directly proportional quantities. If one quantity increases and the other decreases (like speed and time), you cannot use this method directly!
Example: Higher speed → less travel time. Speed and time are inversely proportional, not directly.
Example: Higher speed → less travel time. Speed and time are inversely proportional, not directly.
📊 Directly vs Inversely Proportional
| Type | Behaviour | Example |
|---|---|---|
| Directly proportional | One increases → other also increases | More students → more rice needed |
| Inversely proportional | One increases → other decreases | Higher speed → less travel time |
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EXAM TIP: Before applying the Rule of Three, always ask: "If A goes up, does B also go up?" If yes → directly proportional → Rule of Three applies. If B goes down → inversely proportional → do NOT set up proportion as usual.
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🤝 7. Sharing a Quantity in a Ratio
To divide a total quantity x in the ratio m : n:
⭐ RATIO DIVISION FORMULA
First part = m × x / (m + n)
Second part = n × x / (m + n)
Second part = n × x / (m + n)
Total number of parts = m + n · Size of each part = x ÷ (m + n)
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Dividing x in ratio m:n — find size of each part = x ÷ (m+n), then multiply
🪜 Method: Divide x in ratio m : n
1
Add the terms of the ratio: m + n
2
Find the size of each unit: x ÷ (m + n)
3
First share = m × [x ÷ (m+n)]
4
Second share = n × [x ÷ (m+n)]
💡 MINI EXAMPLE
Divide ₹600 in ratio 2 : 3
Total parts = 2 + 3 = 5 | Each part = 600 ÷ 5 = 120
First share = 2 × 120 = ₹240 | Second share = 3 × 120 = ₹360
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VERIFY: Always check that your two shares add up to the original total. 240 + 360 = 600 ✓
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📏 8. Unit Conversions
When setting up proportions, all units in the same ratio position must be the same. Always convert before calculating!
📐 Length
- 1 metre = 3.281 feet
- 1 foot = 12 inches
🌳 Area
- 1 sq. m = 10.764 sq. ft
- 1 acre = 43,560 sq. ft
- 1 hectare = 10,000 sq. m
- 1 hectare = 2.471 acres
💧 Volume
- 1 mL = 1 cc (cubic cm)
- 1 litre = 1,000 mL
🌡️ Temperature
- F = (9/5 × C) + 32
- C = (5/9) × (F − 32)
- 0°C = 32°F
⭐
EXAM TIP: In proportion problems, units matter! Check that corresponding terms use the same unit — convert everything to grams, centimetres, seconds etc. before setting up the proportion.
💡 MINI EXAMPLE
A car travels 90 km in 150 minutes. How far in 4 hours?
Convert 4 hours → 240 minutes (same unit as 150 min)
150 : 90 :: 240 : x → x = (240 × 90) ÷ 150 = 144 km
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⚡ Quick Revision
📌 Key Formulas
Ratio: a : b → for every a units, there are b units of the second quantity
Proportion: a : b :: c : d ⟺ ad = bc (cross multiplication)
Unknown term: d = (b × c) / a
Simplest form: divide both terms of a:b by their HCF
Divide x in ratio m:n → First = m·x/(m+n), Second = n·x/(m+n)
Temperature: F = (9/5 × C) + 32 | C = (5/9)(F − 32)
🔑 Key Rules
- Ratios are proportional iff they have the same simplest form
- Multiplying/dividing both terms by the same number keeps the ratio equivalent
- Adding/subtracting the same number from both terms changes the ratio
- Rule of Three only works for directly proportional quantities
- Always use the same units when comparing ratios
- Two ratios a:b and c:d are proportional if ad = bc
⚠️ Common Mistakes to Avoid
- Using different units in the same position of a proportion
- Applying Rule of Three to inversely proportional quantities
- Confusing adding the same value with multiplying by the same factor
- Forgetting to check that the two shares add back to the original total
- Not reducing ratio to simplest form before checking proportionality
🏺 Historical Note
- Trairasika = "Rule of Three" — used by Āryabhaṭa (199 CE)
- pramāṇa (given), phala (result), ichchhā (desired) → find ichchhāphala
- Formula: ichchhāphala = (phala × ichchhā) ÷ pramāṇa
@edugrown · Mathematics Notes · Grade 8