Direct and Inverse Proportion for the TMUA

Direct Proportion

Direct proportion occurs when one variable increases in proportion to the increase in another. For instance, if one chocolate bar costs $£6$, then $x$ bars will cost $£6x$. In this case, the number of bars and the total price are in direct proportion because there is no reduction for bulk buying.

In general, if $y$ is directly proportional to $x$, we write the relationship as $y \propto x$ or as the equation $y = kx$, where $k$ is a non-zero constant known as the constant of proportionality. You may use any letter for the constant, though $k$ is standard.

Graphs of Direct Proportion

If a graph is plotted of $y$ against $x$ and they are in direct proportion, the graph must be a straight line that passes through the origin $(0, 0)$. If the line does not pass through the origin, the variables are not in direct proportion.

Example: Identifying Direct Proportion Graphically

Consider the diagram below showing a graph of $y$ against $x$. Is $y$ directly proportional to $x$?

Method 1: By inspection, the graph is a straight line, but it does not pass through the origin. Therefore, $y$ is not directly proportional to $x$.

Method 2: Using the equation $y = kx$. From the graph, when $x = 4$, $y = 18$. Substituting these into the formula gives $18 = 4k$, so $k = 4.5$. However, if we check another point, such as $x = 6, y = 24$, we find that $24 \neq 6 \times 4.5$. Since $k$ is not constant, the relationship is not direct proportion.

Example: Solving Direct Proportion Problems

On days when the temperature is above $20^{\circ}C$, the number of ice creams sold in a shop, $n$, is proportional to $(t - 15)$, where $t$ is the temperature in $^{\circ}C$. When the temperature is $20^{\circ}C$, the shop sells 30 ice creams. How many will it sell when the temperature is $28^{\circ}C$?

1. Set up the equation: $n = k(t - 15)$.
2. Find $k$: Substitute $t = 20$ and $n = 30$. This gives $30 = k(20 - 15)$, so $30 = 5k$ and $k = 6$.
3. Solve for the new value: Substitute $t = 28$ and $k = 6$ into the equation. $n = 6(28 - 15) = 6(13) = 78$ ice creams.

Inverse Proportion

If two variables are in inverse proportion, one variable increases as the other decreases. For example, the thickness of ice on a pond might be inversely proportional to the air temperature: as the temperature drops, the ice thickness increases.

We say $x$ is inversely proportional to $y$ is equivalent to $x$ being proportional to $\frac{1}{y}$. This is written as $x \propto \frac{1}{y}$ or $x = \frac{k}{y}$.

Graphs of Inverse Proportion

A graph of $y$ against $x$ for an inverse proportion relationship takes the form of a reciprocal curve, as shown below:

Example: Verifying Inverse Proportion

Is it reasonable to say $y$ is inversely proportional to $x$ in the following graph?

If $y = \frac{k}{x}$, then $xy$ must equal a constant $k$. We check several points:

• When $x = 1, y = 12 \implies k = 12$
• When $x = 2, y = 6 \implies k = 12$
• When $x = 3, y = 4 \implies k = 12$
• When $x = 12, y = 1 \implies k = 12$ Since $xy$ is consistently 12, it is reasonable to assume $y \propto \frac{1}{x}$.

Example: Solving Inverse Proportion Problems

The number of pairs of gloves sold, $n$, is inversely proportional to the temperature $t$. When $t = 5$, $n = 15$. Find $n$ when $t = 3$.

1. Set up the equation: $n = \frac{k}{t}$.
2. Find $k$: $15 = \frac{k}{5} \implies k = 75$.
3. Solve: When $t = 3$, $n = \frac{75}{3} = 25$ pairs of gloves.

Proportion with Powers

Proportionality can also involve powers of variables, such as $y \propto x^2$ or $y \propto \sqrt{x}$. The general form is $y = kx^n$ for integer or fractional values of $n$.

Example: Proportionality to a Square

The number of water bottles sold, $n$, is proportional to the square of the temperature $t$. At $12^{\circ}C$, 72 bottles are sold. How many are sold at $24^{\circ}C$?

1. Set up the equation: $n = kt^2$.
2. Find $k$: $72 = k \times 12^2 \implies 72 = 144k \implies k = 0.5$.
3. Solve: When $t = 24$, $n = 0.5 \times 24^2 = 0.5 \times 576 = 288$ bottles.