Ratio and proportion form the backbone of quantitative reasoning in primary mathematics and appear consistently in KAR TET Paper I. This topic connects abstract mathematical relationships to everyday situations—sharing sweets among children, mixing ingredients in recipes, calculating distances from maps, and understanding scale models. Mastering this topic is essential because it builds the foundation for percentages, profit-loss, time-work, and speed-distance problems that students will encounter later.
For KAR TET, expect questions that test both conceptual understanding and application skills. You must know how to simplify ratios, identify equivalent ratios, solve proportion problems using the unitary method, and apply these concepts to real-life contexts. The pedagogy component may ask about teaching strategies for making ratio and proportion meaningful to primary learners through concrete materials and familiar situations.
Key Concepts
**Ratio** is a comparison of two quantities of the same kind by division. Written as a:b or a/b, it tells how many times one quantity contains another. A ratio has no unit.
**Equivalent ratios** are ratios that represent the same comparison. Multiplying or dividing both terms by the same non-zero number gives equivalent ratios (2:3 = 4:6 = 6:9).
**Simplest form** of a ratio is obtained by dividing both terms by their HCF. The ratio 12:18 in simplest form is 2:3.
**Proportion** states that two ratios are equal. If a:b = c:d, then a, b, c, d are in proportion, written as a:b :: c:d. Here, a and d are called extremes; b and c are called means.
**Property of proportion**: Product of extremes = Product of means. If a:b :: c:d, then a × d = b × c.
**Unitary method** finds the value of one unit first, then uses it to find the value of the required number of units. It relies on direct or inverse variation.
**Direct proportion**: When one quantity increases, the other increases proportionally (more items cost more money).
**Inverse proportion**: When one quantity increases, the other decreases proportionally (more workers finish work in less time).
Formulas / Key Facts
**Ratio of a to b** = a:b = a/b (both quantities must be in the same unit)
**Simplifying ratio**: Divide both terms by HCF(a, b)
**Proportion condition**: a:b :: c:d means a/b = c/d, which gives a × d = b × c
**Finding fourth proportional**: If a:b :: c:x, then x = (b × c)/a
**Finding third proportional**: If a:b :: b:x, then x = b²/a
**Mean proportional** of a and c is √(a × c)
**Unitary method (direct)**: Value of n units = (Value of m units ÷ m) × n
**Unitary method (inverse)**: If m workers take d days, then 1 worker takes m × d days; n workers take (m × d)/n days
**Dividing quantity Q in ratio a:b**: First part = Q × a/(a+b), Second part = Q × b/(a+b)
Worked Examples
**Example 1: Simplifying and comparing ratios**
*Question*: Express 45 minutes to 2 hours as a ratio in simplest form.
*Solution*:
Convert to same unit: 2 hours = 120 minutes
Ratio = 45:120
HCF of 45 and 120 = 15
Simplest form = 45÷15 : 120÷15 = 3:8
**Example 2: Finding unknown in proportion**
*Question*: If 4:7 :: x:35, find x.
*Solution*:
Using property: 4 × 35 = 7 × x
140 = 7x
x = 140 ÷ 7 = 20
**Example 3: Unitary method (direct proportion)**
*Question*: If 8 notebooks cost ₹120, what is the cost of 13 notebooks?
*Solution*:
Cost of 1 notebook = 120 ÷ 8 = ₹15
Cost of 13 notebooks = 15 × 13 = ₹195
**Example 4: Dividing in a given ratio**
*Question*: Divide ₹560 between A and B in the ratio 3:5.
*Question*: If 6 workers can complete a task in 12 days, how many days will 9 workers take?
*Solution*:
Total work = 6 × 12 = 72 worker-days
Days for 9 workers = 72 ÷ 9 = 8 days
Common Mistakes
**Comparing quantities in different units** → Always convert to the same unit before forming a ratio. "30 cm to 2 m" must become "30 cm to 200 cm" = 3:20, not 30:2.
**Writing ratio with units** → Ratio is a pure number with no unit. Writing "3 kg : 5 kg" is acceptable while setting up, but the final answer is simply 3:5.
**Confusing order in ratio** → Ratio 2:3 is different from 3:2. Always maintain the order as stated in the question (first quantity : second quantity).
**Using direct method when inverse applies** → More workers means less time (inverse), not more time. Identify whether quantities increase together (direct) or one increases while other decreases (inverse).
**Forgetting to simplify** → Always check if the ratio can be reduced. An answer of 8:12 should be written as 2:3 unless specifically asked otherwise.
**Adding ratio terms incorrectly when dividing** → When dividing a quantity in ratio a:b:c, total parts = a+b+c, not a×b×c.
Quick Reference
Ratio a:b means a/b; no units; order matters.
Proportion: a:b :: c:d implies a×d = b×c.
Unitary method: Find value of 1 unit first, then multiply.
Direct proportion: More → More; Inverse proportion: More → Less.
To divide Q in ratio a:b → Parts are Qa/(a+b) and Qb/(a+b).
Always convert quantities to the same unit before forming ratio.
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If the ratio of boys to girls in a class is 3:2 and there are 15 boys, how many girls are there?
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