Year 8: Solving proportionality relationships

What is proportionality?

Proportional relationships mean that two quantities change together in a predictable way. When one quantity doubles, the other doubles. When one triples, the other triples, and so on.

Key concepts:

• Direct proportion:$y=kx$ (where k is the constant of proportionality) As x increases, y increases at the same rate.
• Inverse proportion:$x=k/y$ (where k is the constant of proportionality) As y increases, x decreases at the same rate.

How to solve direct proportion problems:

1. Find the Constant of Proportionality (k): If you have two values for x and y, calculate $k=y/x$.
2. Use the Formula: Once you know k, use the formula $y=kx$ to find y if you know x, or find x if you know y.

How to solve inverse proportion problems:

1. Find the constant of proportionality (k): Calculate $k=x\times y$.
2. Use the Formula: Use the formula $x=k/y$ or $y=k/x$.

Example (Direct): If y is directly proportional to x, and $x=4$ when $y=8$, find y when $x=12$.

$k=8/4=2$. Therefore, $y=2\times 12=24$.

Example (Inverse): If x is inversely proportional to y, and $x=6$ when $y=3$, find x when $y=6$.

$k=6\times 3=18$. Therefore, $x=18/6=3$.