Proportional reasoning extends our understanding of ratios and rates. In this chapter we explore direct and inverse proportion, and apply them to real-life problems like speed-time, workers-days, and pipes-cisterns.
Direct Proportion
Two quantities x and y are in direct proportion if as x increases, y increases at the same rate, i.e., x/y = k (a constant). We write x ∝ y.
Formula: x1/y1 = x2/y2 (or k = x/y is constant).
Inverse Proportion
Two quantities x and y are in inverse proportion if as x increases, y decreases such that x × y = k (a constant). We write x ∝ 1/y.
Formula: x1 × y1 = x2 × y2.
5 pens cost Rs 30. How much do 8 pens cost?
This is direct proportion. Cost/pens = 30/5 = 6.
Cost of 8 pens = 8 × 6 = Rs 48.
Alternate: 5/30 = 8/? → ? = (8 × 30)/5 = Rs 48.
A car travels 150 km in 3 hours. How far does it travel in 5 hours at the same speed?
Direct proportion. Distance = speed × time.
150/3 = d/5 → d = (150 × 5)/3 = 250 km.
6 workers complete a task in 12 days. How many days will 9 workers take?
More workers → fewer days: inverse proportion.
6 × 12 = 9 × d → d = 72/9 = 8 days.
A pipe fills a tank in 4 hours. Another pipe empties it in 6 hours. If both are open, how long to fill the tank?
Fraction filled per hour: 1/4 - 1/6 = 3/12 - 2/12 = 1/12.
Time = 12 hours.
If 15 m of cloth costs Rs 540, find the cost of 24 m.
Direct proportion: 15/540 = 24/? → cost = (24 × 540)/15 = Rs 864.
A photograph 4 cm × 6 cm is enlarged so the longer side becomes 9 cm. What is the new width?
Direct proportion: 4/6 = w/9 → w = (4 × 9)/6 = 6 cm.
20 men can complete a wall in 15 days. Working 6 hours a day, how many men are needed to finish it in 10 days working 9 hours a day?
Using compound variation (men × days × hours = constant):
20 × 15 × 6 = m × 10 × 9 → m = 1800/90 = 20 men.
- Key formulas:
- Direct: x/y = k, or x1/y1 = x2/y2
- Inverse: xy = k, or x1y1 = x2y2
- Work rate: work = rate × time; combined rate = sum of individual rates
- Unitary method: find value of 1 unit, then scale.
Common mistakes
- Applying direct proportion when the relationship is inverse (e.g., workers vs time).
- In combined work problems, forgetting that an emptying pipe reduces the rate.
- Not checking whether quantities are consistent in units.
Summary
Direct proportion means same-rate change; inverse proportion means opposite change. The unitary method is a powerful tool for both. Always identify the type of proportion before setting up the equation.