Direct Proportion with Powers and Roots

The Four-Step Process for Calculating $k$

Knowing the exact mathematical relationship between two variables allows you to predict outcomes without needing to run an experiment again. OCR exam questions often ask you to "formulate an equation" or "find a formula linking $y$ and $x$".

To do this, you must always calculate the value of $k$. You find $k$ by dividing the $y$-value by the $x^n$ (or root) value using the initial pair of numbers provided in the question. This follows a strict four-step process:

1. Define: Write the general algebraic equation (e.g., $y = kx^2$).
2. Find $k$: Substitute the initial known values for both variables and solve for $k$.
3. Formulate: Rewrite the equation with your newly calculated value of $k$.
4. Solve: Substitute the new value given in the question into this specific equation to find the final unknown.

Worked Examples (Powers and Roots)

How do we apply these four steps perfectly in an exam to secure all the method marks? The examples below demonstrate exactly how to structure your answers. Note that when solving equations that result in a negative root (e.g., $x^2 = 16$), OCR expects you to ignore the negative answer if the variables represent real-world physical quantities like length or time.

Example 1: Square ($y \propto x^2$) $m$ is directly proportional to the square of $h$. When $h = 5$, $m = 75$. Find $m$ when $h = 8$.

Step 1: Define the relationship as an algebraic equation.

• $m = kh^2$

Step 2: Substitute known values to find the constant of proportionality ($k$).

• $75 = k(5^2)$
• $75 = 25k$
• $k = 3$

Step 3: Formulate the specific equation.

• $m = 3h^2$

Step 4: Solve the equation for the new value.

• $m = 3(8^2)$
• $m = 3 \times 64 = 192$

Example 2: Cube Root ($y = k\sqrt[3]{x}$) $y \propto \sqrt[3]{x}$. When $x = 64, y = 12$. Find $x$ when $y = 15$.

Step 1: Define the relationship.

• $y = k\sqrt[3]{x}$

Step 2: Find $k$.

• $12 = k\sqrt[3]{64}$
• $12 = 4k$
• $k = 3$

Step 3: Formulate the specific equation.

• $y = 3\sqrt[3]{x}$

Step 4: Solve for the new value of $x$.

• $15 = 3\sqrt[3]{x}$
• $5 = \sqrt[3]{x}$
• $x = 5^3 = 125$

Graphs and Tables

Plotting a non-linear proportional relationship directly on a graph will never give you a straight line, but there is a clever trick to fix that. A graph of $y$ against $x$ for a relationship like $y \propto x^2$ creates a curved parabola segment. However, if you plot $y$ against $x^2$ on the axes, the result is a straight line passing perfectly through the origin $(0, 0)$. The gradient of this straight line is exactly equal to $k$.

If you are given a table of values instead of a graph, you can test for proportion by checking if the ratio $\frac{y}{x^n}$ remains constant for every single pair of values. OCR questions often provide a table with just one complete pair of $x$ and $y$ values. You must use that specific pair to find $k$ before you can calculate the missing numbers to fill in the rest of the table.

Higher Tier Contexts

Every time an engineer designs a heavy machine, they must calculate how mass scales with volume using proportional formulas. In Higher Tier OCR papers, proportion questions are often disguised within physics contexts. For example, the mass ($m$) of a sphere is proportional to the cube of its diameter ($d$), written as $m = kd^3$.

You may also face percentage increase problems in Higher Tier exams. If $y \propto x^2$ and $x$ increases by $20\%$, the new $x$-value is represented by the multiplier $1.2$. Because the relationship is squared, the new $y$-value is found using $1.2^2 = 1.44$. This means $y$ has actually increased by $44\%$.