Welcome to Ratio, Proportion, and Rates of Change!

Hi there! Welcome to one of the most practical parts of your GCSE Maths course. Whether you are figuring out a sale price, doubling a recipe for cookies, or checking how fast a car is moving, you are using the skills in this chapter. Ratio, proportion, and rates of change make up about 25% of the Foundation tier and 20% of the Higher tier exams, so getting these right is a huge step toward your target grade!

Don't worry if some of this feels a bit fast at first. We will break everything down into simple steps, use real-world examples, and point out the "traps" students often fall into. Let’s dive in!


1. Units and Compound Measures

Before we compare things, we need to make sure we are speaking the same "maths language." This means using the right units.

Changing Units

You need to move freely between units like time, length, area, and mass.
Quick Tip: Always convert to the unit the question asks for before you start your big calculations!

  • Length: \( 1 \text{ km} = 1000 \text{ m} \), \( 1 \text{ m} = 100 \text{ cm} \), \( 1 \text{ cm} = 10 \text{ mm} \).
  • Time: Remember, there are 60 minutes in an hour, not 100! To change 1.5 hours into minutes, do \( 1.5 \times 60 = 90 \text{ minutes} \).

Compound Units (Speed, Density, and Pressure)

A compound unit is just two units joined together. The most common one is speed (\( \text{distance} \div \text{time} \)).

The Formula Triangle Trick:
If you struggle to remember whether to multiply or divide, use a triangle! For speed: D (Distance) goes at the top, with S (Speed) and T (Time) at the bottom.
\( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \)
\( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \)
\( \text{Pressure} = \frac{\text{Force}}{\text{Area}} \)

Example: If a car travels 150 miles in 3 hours, its speed is \( 150 \div 3 = 50 \text{ mph} \).

Key Takeaway: Check your units! If the distance is in km and the time is in minutes, but the answer asks for km/h, you must change those minutes into hours first.


2. Working with Ratios

A ratio is a way of comparing parts of a whole. We use the : symbol for it.

Simplifying Ratios

Just like fractions, you can simplify ratios by dividing both sides by the same number.
Example: Simplify \( 10:15 \). Both divide by 5, so it becomes \( 2:3 \).

Sharing in a Ratio (The "Box" Method)

This is a classic exam question. Follow these steps:
1. Add the parts of the ratio together to find the total number of "shares."
2. Divide the total amount by the number of shares to find the value of "one share."
3. Multiply each part of the original ratio by the value of one share.

Example: Share £40 in the ratio \( 2:3 \).
1. \( 2 + 3 = 5 \) shares total.
2. \( £40 \div 5 = £8 \) per share.
3. Person A gets \( 2 \times £8 = £16 \). Person B gets \( 3 \times £8 = £24 \).
Quick Check: Does \( £16 + £24 = £40 \)? Yes! You’ve got it right.

Common Mistake: Students sometimes divide the total by the larger number in the ratio instead of the sum of the numbers. Always add them first!

Key Takeaway: Ratio is about sharing fairly. Find out what "one part" is worth, and the rest is easy.


3. Scale Factors and Maps

Ratios are also used for scale drawings and maps. A scale of \( 1:100 \) means that 1 cm on the paper represents 100 cm (or 1 metre) in real life.

  • Scale Factor: This is the number you multiply by to change the size of something.
  • Area and Volume: This is a "Higher Tier" favorite! If the length scale factor is \( k \), the area scale factor is \( k^2 \) and the volume scale factor is \( k^3 \).

Example: If a model car is \( \frac{1}{10} \) the length of the real car, its surface area will be \( (\frac{1}{10})^2 = \frac{1}{100} \) of the real area.


4. Fractions and Percentages

Ratios, fractions, and percentages are all cousins—they all describe parts of a whole.

Percentages as Multipliers

To find a percentage of an amount quickly, use a multiplier (the decimal version).
- To find 15%, multiply by 0.15.
- To increase by 20%, multiply by 1.20.
- To decrease by 20%, multiply by 0.80 (because \( 100\% - 20\% = 80\% \)).

Compound Interest

Compound interest is when you earn interest on your interest.
The Formula: \( \text{Total} = \text{Initial Amount} \times (\text{Multiplier})^{\text{number of years}} \)

Example: £500 in a bank at 3% interest for 4 years: \( 500 \times 1.03^4 \).

Did you know? This is also how growth and decay work in science, like how bacteria grow or how a car loses value (depreciation) over time!


5. Direct and Inverse Proportion

Proportion tells us how two things are linked.

Direct Proportion

As one thing goes up, the other goes up at the same rate.
Analogy: The more hours you work, the more money you get paid.
The graph of direct proportion is always a straight line through the origin (0,0).
We use the symbol \( \propto \). If \( y \) is proportional to \( x \), we write \( y = kx \), where \( k \) is a constant number.

Inverse Proportion

As one thing goes up, the other goes down.
Analogy: The more people you have helping to paint a fence, the less time it takes.
The formula is \( y = \frac{k}{x} \) or \( y \propto \frac{1}{x} \). The graph is a curve that never touches the axes.

Quick Review:
- Direct: \( y = kx \) (Multiply)
- Inverse: \( y = \frac{k}{x} \) (Divide)


6. Rates of Change (Gradients)

A "rate of change" is just a fancy way of saying "how fast is this changing?" In a graph, the gradient (steepness) tells you the rate of change.

  • On a Distance-Time graph, the gradient is the speed.
  • On a Velocity-Time graph, the gradient is the acceleration.

Average vs. Instantaneous (Higher Content)

If the graph is a curve, the rate of change is always changing!
- Average Rate: Draw a straight line (a chord) between two points and find its gradient.
- Instantaneous Rate: Draw a tangent (a straight line just touching the curve at one point) and find its gradient.

Don't worry if this seems tricky! Drawing a tangent takes practice. Just remember: the steeper the line, the faster the rate of change.


Final Summary Checklist

  • Can you convert between minutes and hours?
  • Do you remember to add the ratio parts before dividing?
  • Do you know that a 10% increase means a multiplier of 1.1?
  • Can you tell the difference between a direct proportion graph (straight line) and an inverse one (curve)?

You’ve got this! Keep practicing these steps, and you'll find that ratio and proportion are some of the most logical parts of the whole GCSE syllabus.