Inverse proportion

Inverse proportion describes a relationship where one quantity increases as another decreases, or vice versa, such that their product remains constant. It is commonly applied in real-life problems, including speed-time calculations, workforce problems, and completing tables of values.

1. Understanding Inverse Proportion:

In inverse proportion, the relationship between two variables $x$ and $y$ is expressed as:

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

Here, $k$ is the constant of proportionality, found by multiplying $x$ and $y$ for any pair of values. Once $k$ is determined, the equation can be used to find unknown values of $x$ or $y$.

2. Completing a Table for Inverse Proportion:

To complete a table of values for inverse proportion:

1. Calculate the constant of proportionality ($k$) using a known pair of values: $$k=x\times y$$
2. Use the formula $y=\frac{k}{x}$ to find missing values in the table.

Example: Complete the table for $x$ and $y$ where $y$ is inversely proportional to $x$:

Step 1: Calculate $k$ using the known values ($x=2$, $y=15$):

$$k=x\times y=2\times 15=30$$

Step 2: Use $y=\frac{k}{x}$ to find the missing values:

• For $x=3$: $y=\frac{30}{3}=10$.
• For $x=5$: $y=\frac{30}{5}=6$.

The completed table is:

3. Worded Problems:

Inverse proportion is often used in worded problems. Read the problem carefully, identify the two inversely related quantities, and use the formula $y=\frac{k}{x}$.

Example: A car travels at an average speed of 60 mph, taking 2 hours to cover a journey. How long would the journey take at 40 mph?

• Step 1: Calculate $k$ (the total distance):
• Step 2: Use $k$ to find the time at 40 mph:

4. Identifying Inverse Proportion:

A relationship is inversely proportional if:

• The product of the two variables is constant ($x\times y=k$).
• One variable increases while the other decreases proportionally.

5. Summary:

• Inverse proportion occurs when two quantities vary such that their product remains constant.
• Use the formula $y=\frac{k}{x}$ to solve problems and complete tables.
• In worded problems, identify the inversely related quantities and use their product to find unknown values.

Inverse proportion is a powerful concept used to solve practical problems involving rates, time, and quantities.