Direct and Inverse Proportions — Class 8 Mathematics
1. Direct Proportion
Two quantities x and y are in DIRECT proportion if they increase or decrease TOGETHER at the SAME RATE. y ∝ x → y = kx, where k is the CONSTANT OF PROPORTIONALITY. Equivalent: y/x = k (constant). 'If x DOUBLES, y DOUBLES. If x is HALVED, y is HALVED. The RATIO y/x NEVER changes.'
Identifying Direct Proportion from a Table
| x | 2 | 4 | 6 | 8 | 10 |
|---|---|---|---|---|---|
| y | 6 | 12 | 18 | 24 | 30 |
| Check ratios: y/x = 6/2=3, 12/4=3, 18/6=3, 24/8=3, 30/10=3. All equal → DIRECT proportion. k = 3. Equation: y = 3x. |
Graph of Direct Proportion
The graph of y = kx is a STRAIGHT LINE passing through the ORIGIN (0,0). Slope = k. 'The origin-intercept is the visual signature of direct proportion — when x=0, y MUST be 0.'
Worked Examples — Direct Proportion
Example 1 — Cost: 5 pens cost ₹35. Find cost of 12 pens. y/x = k → 35/5 = k → k = 7. So cost = 7x. For x=12: cost = 7×12 = ₹84. OR use proportion: 5:35 = 12:y → 5/35 = 12/y → y = (12×35)/5 = 84.
Example 2 — Map Scale: On a map, 5 cm represents 200 km. Find distance represented by 8 cm. 5cm:200km → 1cm=40km. 8cm = 8×40 = 320 km. Using proportion: 5/200 = 8/d → d = (8×200)/5 = 320 km.
Example 3 — Shadow: A pole of height 6 m casts a shadow of 8 m. At the same time, a tower casts 40 m shadow. Find the tower height. h/s = k → 6/8 = h/40 → h = (6×40)/8 = 30 m.
2. Inverse Proportion
Two quantities x and y are in INVERSE proportion if as ONE increases, the OTHER decreases such that their PRODUCT stays CONSTANT. y ∝ 1/x → x × y = k (constant). 'If x DOUBLES, y is HALVED. If x is tripled, y becomes ONE-THIRD. The PRODUCT x×y NEVER changes.'
Identifying Inverse Proportion from a Table
| x | 1 | 2 | 3 | 4 | 6 | |---|---|---|---|----|----|---| | y | 12 | 6 | 4 | 3 | 2 | Check products: 1×12=12, 2×6=12, 3×4=12, 4×3=12, 6×2=12. All equal → INVERSE proportion. k = 12. Equation: xy = 12 or y = 12/x.
Graph of Inverse Proportion
The graph of xy = k is a RECTANGULAR HYPERBOLA. It NEVER touches the axes (asymptotes). As x→∞, y→0. As x→0, y→∞.
Worked Examples — Inverse Proportion
Example 1 — Work and Men: 6 men can build a wall in 10 days. How many days will 15 men take? Men×Days = constant (total work units). 6×10 = 15×d → d = 60/15 = 4 days. 'More men → fewer days. Inverse proportion.'
Example 2 — Speed and Time: A car takes 4 hours to travel a distance at 60 km/h. How long at 80 km/h? Speed×Time = Distance (constant). 60×4 = 80×t → t = 240/80 = 3 hours. 'Higher speed → less time. Inverse proportion.'
Example 3 — Food and Persons: A stock of food feeds 120 persons for 30 days. How many days will it feed 200 persons? Persons×Days = k. 120×30 = 200×d → d = 3600/200 = 18 days.
Example 4 — Gears: A gear with 40 teeth meshes with one of 15 teeth. If the smaller gear rotates at 48 rpm, what is the rpm of the larger gear? Teeth×rpm = k (inverse). 40×r = 15×48 → r = 720/40 = 18 rpm.
3. Direct vs Inverse Proportion — How to Tell
| Criterion | Direct Proportion | Inverse Proportion |
|---|---|---|
| As x increases | y INCREASES | y DECREASES |
| Equation | y = kx | xy = k |
| Constant | y/x = k | xy = k |
| Graph | Straight line through origin | Rectangular hyperbola |
| Real-world example | Cost increases with quantity | Time decreases with more workers |
The Unitary Method for Both
Direct: Find value for 1 unit. Then multiply. Inverse: Find total work/cost/constant. Then divide.
4. Common Mistakes
- Misidentifying proportion type: 'More workers → job done faster' = INVERSE. 'More items → higher cost' = DIRECT.
- Setting up the proportion incorrectly: For direct: x₁/x₂ = y₁/y₂. For inverse: x₁×y₁ = x₂×y₂.
- Forgetting the constant: Always find k FIRST. In direct, k = y/x. In inverse, k = xy.
5. AP Exam Focus
| Topic | Marks |
|---|---|
| Identify proportion type | 2-3 |
| Direct proportion word problems | 3-4 |
| Inverse proportion word problems | 3-4 |
| Mixed problems | 4-5 |
Key Exam Tips
- IDENTIFY the proportion type BEFORE solving. Write: 'This is a case of DIRECT/INVERSE proportion because...'
- Show the constant k. Write the equation. Then solve for the unknown.
- For work problems: 'Men×Days' or 'Pipes×Hours' gives total work units. This is ALWAYS inverse.
Quick Self-Test
- 8 oranges cost ₹40. What is the cost of 15 oranges? (Answer: Direct — k=5. Cost=15×5=₹75.)
- 12 workers complete a job in 8 days. How many days for 16 workers? (Answer: Inverse — 12×8=96. 96/16=6 days.)
- Identify: A car travels 240 km in 4 hours at 60 km/h. How far in 7 hours? (Answer: Direct — distance ∝ time. 60×7=420 km.)
- If 20 cows graze a field in 12 days, how many days for 30 cows? (Answer: Inverse — 20×12=240. 240/30=8 days.)
- Is y=5x direct or inverse? (Answer: Direct — y/x = 5 = constant.)
Mixed Proportion Problems — The Real Test
Problem 1 — Pipes Filling a Tank: 6 pipes fill a tank in 1 hour 20 minutes (80 minutes). How long will 5 pipes take? This is INVERSE (more pipes → less time). Total work = 6×80 = 480 pipe-minutes. With 5 pipes: time = 480/5 = 96 minutes = 1 hour 36 minutes.
Problem 2 — Cost and Quantity: If 15 books cost ₹525, what is the cost of 8 books? DIRECT (more books → more cost). Cost per book = 525/15 = ₹35. 8 books = 8×35 = ₹280.
Problem 3 — Workers and Time with Partial Completion: 20 workers complete a wall in 15 days. After 5 days, 5 workers leave. How many total days to complete? Total work = 20×15 = 300 worker-days. Work done in 5 days = 20×5 = 100. Remaining = 200. Workers remaining = 15. Days needed = 200/15 = 13.33 days. Total = 5+13.33 = 18.33 days.
Identifying Proportion Type from Word Problems
| Scenario | Type | Why |
|---|---|---|
| More articles → Higher cost | DIRECT | y/x constant |
| More workers → Less time | INVERSE | xy constant (total work) |
| More distance → More fuel | DIRECT | Proportional increase |
| Faster speed → Less time (fixed distance) | INVERSE | Speed×Time = Distance (constant) |
| More students → Less food per student (fixed stock) | INVERSE | Students×Days = Constant |
The Constant 'k' in Real Terms
In direct proportion: k = y/x = 'rate' or 'unit value' (price per kg, speed in km/h, density in g/cm³). In inverse proportion: k = xy = 'total amount' (total work, total distance, total cost). 'Understanding what k REPRESENTS physically helps you set up the equation correctly.'
Graphical Understanding
- Direct: y = kx. Plot y vs x → STRAIGHT LINE through origin. If the line is straight but doesn't pass through origin, it's LINEAR but NOT directly proportional.
- Inverse: xy = k. Plot y vs x → RECTANGULAR HYPERBOLA. Also, if you plot y vs 1/x, you get a straight line through origin — confirming inverse proportion.
AP Exam — Common Question Types
- Fill in blank table: Given some x-y pairs, find missing values using proportion.
- Word problem: Identify type, set up equation, solve.
- Which is direct/inverse?: Five scenarios given, identify each type.
- Graph identification: Given a graph, identify proportion type.
