Direct and inverse proportions are important ideas in Year 10 Mathematics. They are especially helpful when learning about ratios and how things relate to each other.

Direct Proportions:

• In direct proportions, when one amount goes up, the other amount goes up too. For example, if you double one quantity, the other quantity also doubles.
• We write direct proportion as (y = kx). Here, (k) is a number that helps us understand the relationship. For example, if (y) is the total cost and (x) is the number of items you buy, then for a price of £5 each, the relationship would be (y = 5x).
• Some important points to remember:
  • On a graph, direct proportions look like a straight line that starts at the origin (the point where both axes meet).
  • The ratio between the two amounts stays the same. If you say (x : y), it can be simplified down to a constant (k).

Inverse Proportions:

• In inverse proportions, when one amount goes up, the other amount goes down. So, if one quantity is doubled, the other one is cut in half.
• We express this relationship with the equation (y = \frac{k}{x}). Again, (k) is the constant. A good everyday example is speed and time; if you go faster for the same distance, you take less time. This can be shown by (d = vt) or rearranged to (t = \frac{d}{v}).
• Key points include:
  • The graph of an inverse proportion is curved and looks like a hyperbola.
  • As one amount gets larger, the product of the two amounts stays the same. This means that when you multiply them together, you get a constant value ((xy = k)).

Differences Between Direct and Inverse Proportions:

1. Relationship:

   • Direct: Both amounts change in the same direction.
   • Inverse: Amounts change in opposite directions.

2. Graphs:

   • Direct: Looks like a straight line starting at the origin.
   • Inverse: Looks like a curved line (hyperbola).

3. Math Formulas:

   • Direct: (y = kx).
   • Inverse: (y = \frac{k}{x}).

Understanding these ideas helps students solve more complex problems with ratios and proportions. This prepares them for more advanced math topics in the future.