Simple Direct Proportion and Currency

Step 2: Scale up to the required amount using multiplication.

• $17 \text{ kg} \times 3 = 51 \text{ kg}$

Worked Example: Scaling Method

A stack of 8 identical tiles is 11.2 cm high. What is the height of a stack of 12 tiles?

Step 1: Calculate the scale factor between the quantities.

• $12 \div 8 = 1.5$

Step 2: Apply the scale factor to the height.

• $11.2 \text{ cm} \times 1.5 = 16.8 \text{ cm}$

Algebraic Proportion ($y = kx$)

Plotting a direct proportion relationship on a graph always creates a perfectly straight line passing exactly through the origin $(0,0)$. If a straight line does not pass through the origin, the variables are definitely not in direct proportion.

In algebra, we write this relationship using the proportionality symbol as $y \propto x$. To perform calculations, we change this into an equation:

$$y = kx$$

The letter $k$ represents the constant of proportionality, which is the fixed multiplier that connects the two variables.

Worked Example: Finding Unknowns algebraically

$y$ is directly proportional to $x$. When $x = 12$, $y = 30$. Calculate the value of $x$ when .

Step 1: Write the general equation and substitute the known pair of values to find $k$.

• $y = kx$
• $30 = k \times 12$
• $k = 30 \div 12 = 2.5$

Step 2: Write the specific equation for this relationship.

• $y = 2.5x$

Step 3: Substitute the new value into the equation and solve for the unknown.

• $45 = 2.5x$
• $x = 45 \div 2.5 = 18$

Ratio Equality and Solving for Unknowns

Mixing a specific shade of purple paint requires the exact same mathematical logic whether you are filling a tiny cup or a giant bucket. A ratio compares the size of one part to another. Proportion (Ratio Equality) occurs when two different ratios represent the identical relationship, written mathematically as $a : b = c : d$.

You can solve equivalent ratios by finding the multiplier between corresponding parts. OCR exams also frequently require you to simplify ratios into the form $1 : n$ by dividing both sides by the left-hand number.

Worked Example: Ratio Equality

The ratio of sand to cement in a concrete mix is $4 : 9$. A builder uses 27 kg of cement. Calculate how much sand is used.

Step 1: Set up the equivalent ratios.

• $4 : 9 = x : 27$

Step 2: Find the scale factor for the known parts (cement).

• $27 \div 9 = 3$

Step 3: Apply the scale factor to the unknown part (sand).

• $4 \times 3 = 12 \text{ kg}$

Worked Example: The $1 : n$ Form

Write the ratio $8\text{ m} : 360\text{ cm}$ in the form $1 : n$.

Step 1: Convert both quantities to identical units.

• $800\text{ cm} : 360\text{ cm}$

Step 2: Divide both sides by the left-hand number (800).

• $1 : 0.45$

Currency Conversion

Buying a souvenir abroad might feel like spending monopoly money until you calculate exactly what it costs in your home currency. An exchange rate is simply a real-world application of direct proportion.

The base currency is the currency given as a unit of "1" (for example, the £1 in £1 = $1.28).

• To convert from the base currency to the foreign currency:

Multiply by the rate.

• To convert from the foreign currency back to the base currency: Divide by the rate.

Worked Example: Multi-step Currency Conversion

A laptop costs $850 in the USA. The exchange rates are \pounds1 = $1.25 and \pounds1 = €1.16. Calculate the cost of the laptop in Euros (€).

Step 1: Convert the foreign currency (USD) back to the base currency (GBP) by dividing.

• $\$850 \div 1.25 = \pounds680.00$

Step 2: Convert the base currency (GBP) into the new foreign currency (EUR) by multiplying.

• $\pounds680 \times 1.16 = \u20ac788.80$