source: ncert
topic: arithmetic
class_level: class_8
chapter: ch13
difficulty: beginner
Class 8 | Ch13: Direct and Inverse Proportion DIRECT PROPORTION: x and y are in direct proportion if x/y=k (constant) or x1/y1=x2/y2. As x increases, y increases proportionally. Symbol: x∝y. Graph: straight line through origin. Examples: cost∝quantity, distance∝time(constant speed), simple interest∝time. INVERSE PROPORTION: x and y are in inverse proportion if xy=k (constant) or x1×y1=x2×y2. As x increases, y decreases proportionally. Symbol: x∝1/y. Graph: hyperbola (not straight line). Examples: speed∝1/time, workers∝1/days, pipes∝1/fill time. SOLVED EXAMPLES: Example 1 (Direct): If 15 bags of rice cost ₹450, find cost of 24 bags. 15/450=24/x → x=24×450/15=₹720. OR cost per bag=30, 24×30=720. Example 2 (Inverse): 8 workers finish in 12 days. How many days for 6 workers? 8×12=6×d → d=96/6=16 days. (fewer workers → more days → inverse) Example 3 (Inverse): 3 pipes fill tank in 8 hours. How long for 4 pipes? 3×8=4×t → t=24/4=6 hours. Example 4: Car travels 150km in 3 hours. How long for 250km at same speed? 3/150=t/250 → t=500/100=5 hours. (direct proportion) IDENTIFYING TYPE: More→More: Direct proportion. More→Less: Inverse proportion. Workers and days: MORE workers, LESS days → INVERSE. Speed and time: MORE speed, LESS time → INVERSE. COMMON MISTAKES: Workers and days is INVERSE not direct. In inverse proportion: x1×y1=x2×y2 (NOT x1/y1=x2/y2).