Direct and Inverse Proportions
Overview
Direct and Inverse Proportions is a Class 8 chapter that explores how two quantities relate to each other when one changes. In direct proportion, both quantities increase or decrease together at the same rate. In inverse proportion, when one quantity increases, the other decreases proportionally. These concepts are widely used in everyday life such as calculating speed, time, cost, and work done by workers. Understanding these relationships helps solve real-world problems efficiently using simple ratios and equations.
Typed Notes
Direct Proportion
Two quantities are said to be in direct proportion when an increase in one quantity causes a proportional increase in the other, and a decrease in one causes a proportional decrease in the other. For example, if you buy more items, you pay more money. The ratio between the two quantities always remains constant. If x and y are in direct proportion, then x/y = k, where k is the constant of proportion. We also write x₁/y₁ = x₂/y₂ to compare two situations. Example: If 5 pens cost ₹30, then 10 pens cost ₹60.
Inverse Proportion
Two quantities are in inverse proportion when an increase in one quantity causes a proportional decrease in the other quantity, and vice versa. For example, more workers complete a job in fewer days. The product of the two quantities always remains constant. If x and y are in inverse proportion, then x × y = k (constant). We write x₁ × y₁ = x₂ × y₂ to solve problems. Example: If 4 workers finish work in 12 days, then 6 workers will finish the same work in 8 days because 4×12 = 6×8 = 48.
Identifying Direct vs Inverse Proportion
To identify whether two quantities are in direct or inverse proportion, observe how they change together. If both increase or both decrease together, it is direct proportion. If one increases while the other decreases, it is inverse proportion. For example, speed and time for a fixed distance are inversely proportional — higher speed means less time. But distance and time at constant speed are directly proportional — more time means more distance covered. Always check whether the product or the ratio remains constant to confirm the type of proportion.
Solving Problems Using Proportion
To solve proportion problems, first identify whether the relationship is direct or inverse. For direct proportion, use x₁/y₁ = x₂/y₂ and cross-multiply to find the unknown. For inverse proportion, use x₁ × y₁ = x₂ × y₂ and divide to find the unknown. Always write down the known values clearly, set up the equation, and solve step by step. These methods are useful in problems involving cost and quantity, speed and time, number of workers and days, pipes filling tanks, and many other practical situations students encounter daily.
Key Concepts
Revision ke liye ek nazar — exam se pehle yahi padhna hai.
- 01In direct proportion, x/y = k (constant), meaning x and y increase or decrease together at the same rate.
- 02In inverse proportion, x × y = k (constant), meaning when x increases, y decreases proportionally.
- 03Two quantities x and y are in direct proportion if x₁/y₁ = x₂/y₂.
- 04Two quantities x and y are in inverse proportion if x₁ × y₁ = x₂ × y₂.
- 05Unitary method is the base of both direct and inverse proportion — find the value for one unit first.
Important Formulas
Questions & Solutions
Pehle khud try karo — phir solution kholo. Yahi tariqa topper banata hai.
If 8 kg of rice costs ₹320, find the cost of 15 kg of rice.
A car travels 150 km in 3 hours. How far will it travel in 5 hours at the same speed?
12 workers can complete a construction job in 20 days. How many days will 15 workers take to complete the same job?
A tap fills a tank in 6 hours. How long will 4 such taps take to fill the same tank?
If 35 students can sit in 5 rows, how many students can sit in 8 rows with the same arrangement?
Direct proportion means two quantities change in the same direction — their ratio is constant (x/y = k). Inverse proportion means two quantities change in opposite directions — their product is constant (x × y = k). Use x₁/y₁ = x₂/y₂ for direct proportion and x₁y₁ = x₂y₂ for inverse proportion. These concepts apply to real-life problems involving cost, speed, time, and workers.
Chapter-end quiz
4 MCQsOption tap karo — turant pata chalega sahi ya galat, explanation ke saath.
Q1.If x and y are in direct proportion and x = 4 when y = 20, what is y when x = 7?
Q2.Which of the following is an example of inverse proportion?
Q3.If 6 pipes fill a tank in 12 hours, how many hours will 9 pipes take?
Q4.In direct proportion, if x₁ = 3, y₁ = 9, x₂ = 5, what is y₂?
Mark karo — dashboard par progress dikhega.