Inverse proportion

Topic summary

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.