Introduction
Proportion is a powerful tool that connects two quantities. Whenever a change in one quantity causes a predictable change in another, we can use proportional reasoning to solve problems — from recipes and maps to speeds and prices.
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Key Concepts
Ratio: The comparison of two quantities of the same kind. Written as a:b or a/b.
Proportion: Two ratios are equal. If a:b = c:d, then a, b, c, d are in proportion. Cross-multiplication: a x d = b x c (product of extremes = product of means).
Direct Proportion: When one quantity increases, the other also increases at the same rate. If x and y are in direct proportion: y/x = k (constant) or y1/x1 = y2/x2.
Inverse Proportion: When one quantity increases, the other decreases at the same rate. x x y = k or x1 x y1 = x2 x y2.
Unitary Method: Find the value of one unit first, then scale.
- Applications:
- Speed, Distance, Time: Distance = Speed x Time
- Work problems: more workers, less time (inverse proportion)
- Map scale: actual distance / map distance = scale factor
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Worked Examples
If 8 pens cost Rs. 56, how much do 13 pens cost?
1 pen = 56/8 = Rs. 7. 13 pens = 7 x 13 = Rs. 91.
A car covers 240 km in 4 hours. How far in 7 hours?
Direct proportion: distance / time = constant. 240/4 = 60 km/h. In 7 hours: 60 x 7 = 420 km.
6 workers complete a job in 12 days. How many days for 9 workers?
Inverse proportion: 6 x 12 = 9 x d. d = 72/9 = 8 days.
On a map, 1 cm represents 5 km. Two cities are 13.5 cm apart on the map. Find actual distance.
Actual distance = 13.5 x 5 = 67.5 km.
Are 3, 8, 12, 32 in proportion?
Check: 3 x 32 = 96 and 8 x 12 = 96. Since 96 = 96, yes they are in proportion.
If x and y are in inverse proportion and x = 5 when y = 12, find y when x = 20.
x1 x y1 = x2 x y2 => 5 x 12 = 20 x y2. y2 = 60/20 = 3.
A recipe needs 2 cups of flour for 12 cookies. How many cups for 30 cookies?
Direct proportion: 2/12 = c/30. c = (2 x 30)/12 = 60/12 = 5 cups.
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Common mistakes
- Confusing direct and inverse proportion — first identify: as one increases, does the other increase (direct) or decrease (inverse)?
- In the unitary method, finding the value of MANY units first before finding one unit.
- Forgetting to check whether the units are consistent before setting up ratios.
- In proportion a:b::c:d, mixing up means (b and c) and extremes (a and d).
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Summary
Proportional reasoning — whether direct or inverse — simplifies complex real-world calculations. The key is recognising the type of proportion and setting up the correct equation. Cross-multiplication for proportion, and x x y = k for inverse proportion, are the two most powerful tools. The unitary method provides an intuitive fallback for any proportional problem.