Direct Proportion

Understanding how quantities increase or decrease together, with formulas like \( \frac{x}{y} = k \).

Inverse Proportion

Exploring when one quantity increases as the other decreases, using \( xy = k \).

Real-Life Applications

Applying proportions to scenarios like cost, speed, and work.

Problem Solving

Solving examples involving scales, shadows, and mixtures.

Definition and Examples

Direct proportion occurs when \( \frac{x}{y} = k \) (constant). Examples: Cost of sugar (\( \frac{36}{1} = 36 \)), petrol and distance (\( \frac{4}{60} = \frac{1}{15} \)). Formula: \( \frac{x_1}{y_1} = \frac{x_2}{y_2} \).

Activities and Observations

Clock hand activity shows angle proportional to time (\( \frac{T}{A} \) constant). Age table shows not always proportional. More formulas: Simple interest \( I = P \times \frac{r}{100} \times t \), compound interest \( A = P(1 + \frac{r}{100})^t \).

Solved Examples

Example 1: Cloth cost. Example 2: Pole shadow (\( x = 21 \) m). Example 3: Paper weight (\( x = 750 \)). Example 4: Train speed (\( x = 25 \) km, \( y = 200 \) min). Example 5: Map scale (\( y = 1200 \) km).

Direct Proportion

When \( \frac{x}{y} = k \), quantities vary directly: \( x \propto y \).

Inverse Proportion

When \( xy = k \), quantities vary inversely: \( x \propto \frac{1}{y} \).

Constant of Proportion

The value \( k \) in equations.

Unitary Method

Finding value for one unit then scaling.

Scale in Maps

Ratio of map to actual distance.

Variation

Change in one quantity affecting another.

Direct Proportion

• \( x \propto y \)
• \( \frac{x}{y} = k \)
• Examples: Cost, Distance

Inverse Proportion

• \( x \propto \frac{1}{y} \)
• \( xy = k \)
• Examples: Speed-Time, Workers-Time

Formulas

• Direct: \( x_1 y_2 = x_2 y_1 \)
• Inverse: \( x_1 y_1 = x_2 y_2 \)
• Scale: Map ratio

Activities

• Clock angle
• Counters rows
• Tables check