Inverse Proportion Quizzes

Introduction

Inverse proportion is an important concept in GCSE Maths that describes a relationship between two quantities where an increase in one quantity causes a proportional decrease in the other, and vice versa. This is the opposite of direct proportion. Understanding inverse proportion is essential for solving real-life problems such as speed and time, work problems, and resource allocation. Mastery of this topic ensures students can recognize and correctly apply inverse relationships in both exams and practical situations.

Core Concepts

What is Inverse Proportion?

Two quantities, $x$ and $y$, are in inverse proportion if increasing one quantity results in a proportional decrease in the other. Mathematically, this is expressed as:

$$ y \propto \frac{1}{x} $$>

Or using a constant of proportionality $k$:

$$ y = \frac{k}{x} $$>

Here, $k$ is a positive constant that represents the product of $x$ and $y$.

Identifying Inverse Proportion

To check if two quantities are inversely proportional, multiply the known pairs:

$$ x_1 \times y_1 = x_2 \times y_2 = k $$>

Example: A task takes 6 workers 10 hours to complete. If 12 workers work at the same rate, the time taken is:

• $6 × 10 = 60$ → constant $k$
• For 12 workers, $y = k ÷ x = 60 ÷ 12 = 5$ hours

Graphical Representation

In inverse proportion, the graph of $y$ against $x$ is a curve called a hyperbola, not a straight line. As $x$ increases, $y$ decreases, and as $x$ decreases, $y$ increases.

Rules & Steps

To solve inverse proportion problems, follow these steps:

1. Identify the two quantities: Determine which quantity decreases as the other increases.
2. Calculate the constant of proportionality: Multiply the known values of $x$ and $y$ to find $k$.
3. Form the equation: Use $y = k/x$ with the calculated $k$.
4. Find the unknown value: Substitute the unknown $x$ or $y$ into the equation.
5. Check consistency: Multiply the calculated $x$ and $y$ to ensure it equals $k$.

Worked Examples

1. Example 1: 5 machines complete a job in 12 hours. How long would it take 10 machines to complete the same job?
   Step 1: Identify $x$ = number of machines, $y$ = time
   Step 2: Calculate $k = x × y = 5 × 12 = 60$
   Step 3: Equation: $y = 60 ÷ x$
   Step 4: $x = 10 → y = 60 ÷ 10 = 6$ hours
2. Example 2: A car can travel 300 km on 20 litres of fuel. How much fuel would it need to travel 450 km at the same efficiency?
   Step 1: $x$ = distance, $y$ = fuel
   Step 2: $k = x × y = 300 × 20 = 6000$
   Step 3: Equation: $y = 6000 ÷ x$
   Step 4: $x = 450 → y = 6000 ÷ 450 = 13.33$ litres
3. Example 3 (Higher Level): 8 workers complete a project in 15 days. How many days would 12 workers need to finish the same project?
   Step 1: $k = x × y = 8 × 15 = 120$
   Step 2: Equation: $y = 120 ÷ x$
   Step 3: $x = 12 → y = 120 ÷ 12 = 10$ days
4. Example 4: A tap can fill a tank in 4 hours. How many hours will 2 taps take to fill the same tank?
   Step 1: $k = x × y = 1 × 4 = 4$ (for 1 tap)
   Step 2: Equation: $y = k ÷ x$
   Step 3: $x = 2 → y = 4 ÷ 2 = 2$ hours
5. Example 5: A printer prints 120 pages in 6 minutes. How long will it take to print 200 pages at the same speed?
   Step 1: $x$ = number of pages, $y$ = time
   Step 2: $k = x × y = 120 × 6 = 720$
   Step 3: Equation: $y = 720 ÷ x$
   Step 4: $x = 200 → y = 720 ÷ 200 = 3.6$ minutes

Common Mistakes

• Assuming direct proportion instead of inverse proportion.
• Incorrectly calculating the constant $k$.
• Swapping $x$ and $y$ when multiplying to find $k$.
• Not checking that the product $x × y$ remains constant after calculation.
• Confusing units in practical problems (e.g., hours vs minutes, litres vs millilitres).

Applications

Inverse proportion is used in many real-life scenarios:

• Work problems: More workers → less time, fewer workers → more time.
• Speed and time: Faster speed → less travel time, slower speed → more travel time.
• Resource allocation: More resources → less time needed to complete a task.
• Manufacturing and logistics: Machines and output often have inverse relationships with production time.

Strategies & Tips

• Always check if the relationship is direct or inverse before solving.
• Calculate the constant of proportionality carefully and keep it consistent.
• Draw a table of values to visualize the relationship.
• Use LaTeX or fractions to avoid rounding errors until the final step.
• Practice problems with multi-step inverse relationships for higher-level preparation.

Summary

Inverse proportion is essential for understanding how quantities interact in an opposite manner. Key steps: identify the quantities, calculate the constant $k$ using $x × y$, write the equation $y = k ÷ x$, and solve for unknowns. Always verify that $x × y = k$ for all values. Mastery of inverse proportion equips students to solve practical problems in exams and real life with confidence. Attempt the quizzes in this subcategory to reinforce your skills and test your understanding in a variety of scenarios!