OCR GCSE Maths: Inverse Proportion and Applications

Inverse Proportion: Unlocking the Secrets of Relationships

Inverse proportion is a fundamental concept in mathematics that describes how two quantities change in relation to each other. Unlike direct proportion, where an increase in one quantity leads to a proportional increase in the other, in inverse proportion, an increase in one quantity results in a proportional decrease in the other.

Understanding Inverse Proportion

Imagine you have a fixed amount of work to complete. The more people you have working on the task, the less time it will take to finish. This scenario demonstrates inverse proportion. The number of workers and the time taken to complete the work are inversely proportional.

Key Features of Inverse Proportion:

• Product is constant: If two quantities, x and y, are inversely proportional, their product (x * y) will always remain constant.
• Graph is a hyperbola: The graph of an inverse proportion relationship is a hyperbola, showing the curve as one variable increases, the other decreases proportionally.

Solving Inverse Proportion Problems

1. Identify the Inverse Relationship: Determine whether the given situation involves inverse proportion. Look for keywords like "inversely proportional," "decreases as the other increases," or "varies inversely."

2. Set up the Proportion: Express the relationship between the two quantities using a proportion. For example: * If x is inversely proportional to y, then x * y = k, where k* is a constant.

3. Find the Constant: Use the given information to find the value of the constant k. Substitute the given values of x and y into the equation from step 2.

4. Solve for the Unknown: Use the constant k and the remaining known value to solve for the unknown quantity in the problem.

Applications of Inverse Proportion

1. Speed and Time:

• Scenario: A car travels at a constant speed. The time taken to complete a journey is inversely proportional to the speed of the car.
• Explanation: If the car travels faster, it will take less time to cover the same distance.

2. Work and Time:

• Scenario: The number of workers needed to complete a task is inversely proportional to the time taken to complete it.
• Explanation: If you have more workers, it will take less time to finish the job.

3. Gear Ratios:

• Scenario: In a bicycle, the gear ratio (the number of teeth on the front chainring divided by the number of teeth on the rear cassette) is inversely proportional to the speed of the bike.
• Explanation: A higher gear ratio (more teeth on the front chainring) results in a lower speed.

4. Volume and Pressure:

• Scenario: For a fixed mass of gas at a constant temperature, the volume of the gas is inversely proportional to the pressure.
• Explanation: If you increase the pressure on a gas, its volume will decrease.

Remember: Inverse proportion is a crucial concept for understanding how quantities relate to each other. By applying the steps outlined above, you can solve a variety of problems involving inverse proportion and gain a deeper understanding of its practical applications.