Chapter 7 ยท Grade 8
Proportional Reasoning-1
Ganita Prakash ยท NCERT ยท Reprint 2026โ27
โ๏ธ ๐ ๐ข๐ 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.
๐ก
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.
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Multiplying by the same factor โ similar shape. Subtracting the same number โ distorted shape.
โ ๏ธ
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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โ๏ธ 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.
โญ RATIO NOTATION
a : b
Read as "a is to b" โ a and b are the terms
๐ก 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.
๐
Same factor = Proportional ratios. Two ratios like 60:40 and 30:20 are proportional because you multiply both terms of the first by ยฝ to get the second.
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๐ข 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).
๐ช 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
๐ก MINI EXAMPLE
60 : 40 โ HCF = 20 โ 60รท20 : 40รท20 = 3 : 2
90 : 60 โ HCF = 30 โ 90รท30 : 60รท30 = 3 : 2
Both give 3:2 โ they are proportional! โ
๐ง
REMEMBER: Two ratios are proportional if and only if their simplest forms are identical.
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๐ฐ 4. Proportion ( :: )
When two ratios are equal in their simplest forms, we say they are in proportion. The symbol :: means "is proportional to".
โญ PROPORTION NOTATION
a : b :: c : d
Reads: "a is to b as c is to d" โ meaning ratios a:b and c:d are equal
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When a:b :: c:d, then cross-multiplying gives ad = bc
โญ CROSS MULTIPLICATION RULE
If a : b :: c : d then ad = bc
Also: the unknown fourth term d = bc / a
โ ๏ธ
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 โ multiplication.
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๐งฎ 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.
โ๏ธ Method 1 โ Factor Method
๐ช 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
๐ก MINI EXAMPLE
Find x if 4 : 10 :: 12 : x
Factor = 12 รท 4 = 3 โ x = 10 ร 3 = 30
So 4 : 10 :: 12 : 30 โ
โ๏ธ Method 2 โ Cross Multiplication
๐ช Steps
1
Write the proportion: a : b :: c : d
2
Cross-multiply: a ร d = b ร c
3
Solve for the unknown: d = (b ร c) / a
๐ก MINI EXAMPLE
Find x if 5 : 8 :: 15 : x
5 ร x = 8 ร 15 โ 5x = 120 โ x = 24
โญ
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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๐บ 6. Trairasika โ The Rule of Three
๐ ANCIENT INDIA
The Trairasika (Rule of Three) was described by the mathematician ฤryabhaแนญa (199 CE). It solves problems where 4 quantities are linked proportionally โ 3 are known and 1 must be found.
ichchhฤphala = (phala ร ichchhฤ) รท pramฤแนa
pramฤแนa = given measure ยท phala = given result ยท ichchhฤ = desired quantity
In modern terms: given a : b :: c : d, use cross multiplication to find the unknown: d = (b ร c) / a
โ ๏ธ
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 โ less travel time. Speed and time are inversely proportional, not directly.
Example: Higher speed โ less travel time. Speed and time are inversely proportional, not directly.
๐ Directly vs Inversely Proportional
| Type | Behaviour | Example |
|---|---|---|
| Directly proportional | One increases โ other also increases | More students โ more rice needed |
| Inversely proportional | One increases โ other decreases | Higher speed โ less travel time |
โญ
EXAM TIP: Before applying the Rule of Three, always ask: "If A goes up, does B also go up?" If yes โ directly proportional โ Rule of Three applies. If B goes down โ inversely proportional โ do NOT set up proportion as usual.
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๐ค 7. Sharing a Quantity in a Ratio
To divide a total quantity x in the ratio m : n:
โญ RATIO DIVISION FORMULA
First part = m ร x / (m + n)
Second part = n ร x / (m + n)
Second part = n ร x / (m + n)
Total number of parts = m + n ยท Size of each part = x รท (m + n)
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Dividing x in ratio m:n โ find size of each part = x รท (m+n), then multiply
๐ช Method: Divide x in ratio m : n
1
Add the terms of the ratio: m + n
2
Find the size of each unit: x รท (m + n)
3
First share = m ร [x รท (m+n)]
4
Second share = n ร [x รท (m+n)]
๐ก MINI EXAMPLE
Divide โน600 in ratio 2 : 3
Total parts = 2 + 3 = 5 | Each part = 600 รท 5 = 120
First share = 2 ร 120 = โน240 | Second share = 3 ร 120 = โน360
๐ง
VERIFY: Always check that your two shares add up to the original total. 240 + 360 = 600 โ
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๐ 8. Unit Conversions
When setting up proportions, all units in the same ratio position must be the same. Always convert before calculating!
๐ Length
- 1 metre = 3.281 feet
- 1 foot = 12 inches
๐ณ 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
๐ง Volume
- 1 mL = 1 cc (cubic cm)
- 1 litre = 1,000 mL
๐ก๏ธ Temperature
- F = (9/5 ร C) + 32
- C = (5/9) ร (F โ 32)
- 0ยฐC = 32ยฐF
โญ
EXAM TIP: In proportion problems, units matter! Check that corresponding terms use the same unit โ convert everything to grams, centimetres, seconds etc. before setting up the proportion.
๐ก MINI EXAMPLE
A car travels 90 km in 150 minutes. How far in 4 hours?
Convert 4 hours โ 240 minutes (same unit as 150 min)
150 : 90 :: 240 : x โ x = (240 ร 90) รท 150 = 144 km
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โก Quick Revision
๐ Key Formulas
Ratio: a : b โ for every a units, there are b units of the second quantity
Proportion: a : b :: c : d โบ ad = bc (cross multiplication)
Unknown term: d = (b ร c) / a
Simplest form: divide both terms of a:b by their HCF
Divide x in ratio m:n โ First = mยทx/(m+n), Second = nยทx/(m+n)
Temperature: F = (9/5 ร C) + 32 | C = (5/9)(F โ 32)
๐ 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
โ ๏ธ 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
๐บ Historical Note
- Trairasika = "Rule of Three" โ used by ฤryabhaแนญa (199 CE)
- pramฤแนa (given), phala (result), ichchhฤ (desired) โ find ichchhฤphala
- Formula: ichchhฤphala = (phala ร ichchhฤ) รท pramฤแนa
@edugrown ยท Mathematics Notes ยท Grade 8