Number - Direct and inverse proportion

Given that $y$ is directly proportional to $x$, and $y = 20$ when $x = 4$, find the value of $y$ when $x = 9$.

$y = kx$ Substituting the known values: $20 = k \times 4$ $k = \frac{20}{4} = 5$ Now find $y$ for $x = 9$: $y = 5 \times 9$ $y = 45$

First, establish the general equation for direct proportion. Use the given values to solve for the constant $k$. Then, use $k$ to solve for the unknown variable.

The variable $z$ is inversely proportional to $w$. If $z = 10$ when $w = 2$, find $z$ when $w = 5$.

$z = \frac{k}{w}$ Substituting the known values: $10 = \frac{k}{2}$ $k = 10 \times 2 = 20$ Now find $z$ for $w = 5$: $z = \frac{20}{5}$ $z = 4$

Identify the inverse relationship and set up the equation $z = \frac{k}{w}$. Calculate $k$ by multiplying the given $z$ and $w$ values, then divide $k$ by the new $w$ to find the result.

A quantity $A$ is directly proportional to the square of $r$. When $r = 3$, $A = 18$. Find $A$ when $r = 5$.

$A = kr^2$ Substituting the known values: $18 = k \times 3^2$ $18 = 9k$ $k = 2$ Now find $A$ for $r = 5$: $A = 2 \times 5^2$ $A = 2 \times 25$ $A = 50$

Since $A$ is proportional to the square of $r$, the equation is $A = kr^2$. Square the value of $r$ before solving for $k$ or $A$.