Ratio, scale and sharing

Q: What are the three steps for sharing an amount in a ratio?

The three steps are: Add the parts, Divide the total by that sum, and Multiply the result by each original ratio number.

Q: How do area and volume change when a shape is enlarged by scale factor \(k\)?

If the lengths are multiplied by \(k\), the area is multiplied by \(k^{2}\) and the volume is multiplied by \(k^{3}\).

Q: How do you use multipliers to calculate percentage increases and decreases?

To increase , add the percentage to 100% (e.g., 1.15 for +15%). To decrease , subtract it from 100% (e.g., 0.85 for -15%).

Q: What is the main difference between simple interest and compound interest?

Simple interest is calculated only on the original sum, while compound interest is calculated on the original sum plus any interest already earned.

Q: What defines a direct proportion relationship?

In direct proportion , as one quantity increases, the other increases at the same rate; the graph is a straight line through the origin (0,0).

Practice

Mathematics (1MA1) · Pearson Edexcel GCSE (9-1) · 6 min read · Updated: 6 Sept 2026

Welcome to Ratio, Proportion, and Rates of Change!

In this chapter, we are going to explore how different quantities relate to each other. Whether you are scaling up a recipe for a party, working out which "Special Offer" at the supermarket is actually a bargain, or reading a map for a hike, you are using ratio and proportion. These skills are not just for exams; they are "maths for life." Don't worry if some parts seem a bit confusing at first—we will break everything down into simple, manageable steps.

1. Understanding Ratios

A ratio is a way of comparing two or more amounts. It shows how much of one thing there is compared to another.

We use a colon (:) to separate the numbers. For example, if a bag has 2 red marbles and 3 blue marbles, the ratio of red to blue is 2 : 3.

Simplifying Ratios

Just like fractions, ratios can be simplified to make them easier to work with. To simplify a ratio, divide all the numbers by the highest common factor.

Example: Simplify the ratio 10 : 15.
Both 10 and 15 can be divided by 5.
\(10 \div 5 = 2\)
\(15 \div 5 = 3\)
The simplified ratio is 2 : 3.

Ratios and Fractions

It is important to see the link between ratios and fractions.
In the ratio 2 : 3, there are 5 parts in total (\(2 + 3 = 5\)).
The first part is \(\frac{2}{5}\) of the total.
The second part is \(\frac{3}{5}\) of the total.

Quick Review:
• Always check if you can simplify your final answer.
• Order matters! "Red to Blue" (2:3) is different from "Blue to Red" (3:2).

One of the most common exam questions asks you to share a total amount into a specific ratio. You can use the "Add, Divide, Multiply" method.

The 3-Step Dance for Sharing:

Step 1: Add the parts of the ratio to find the total number of shares.
Step 2: Divide the total amount by the number of shares to find the value of one share.
Step 3: Multiply that value by each number in the original ratio.

Example: Share £40 in the ratio 1 : 3.
1. Add: \(1 + 3 = 4\) shares.
2. Divide: \(£40 \div 4 = £10\) (this is the value of one share).
3. Multiply:
\(1 \times £10 = £10\)
\(3 \times £10 = £30\)
Answer: £10 : £30 (Notice that £10 + £30 = £40, so we know we are right!)

Common Mistake to Avoid:
Don't divide the total by just one of the numbers in the ratio. You must divide by the sum of the parts!

3. Scale Factors and Maps

Scale is used to represent real-life objects in a smaller (or larger) size, like on a map or a model car. The scale factor tells us how many times bigger or smaller the representation is.

Maps and Ratios

Map scales are often written as ratios like 1 : 25,000. This means 1 cm on the map represents 25,000 cm in real life.

Tip: To convert 25,000 cm to metres, divide by 100 (250 m). To convert to kilometres, divide by another 1000 (0.25 km).

Similarity

Two shapes are similar if they are the same shape but different sizes. One is an enlargement of the other.
• Lengths: To get from the small shape to the big shape, multiply by the scale factor (\(k\)).
• Areas: To compare areas, we use the scale factor squared (\(k^{2}\)).
• Volumes: To compare volumes, we use the scale factor cubed (\(k^{3}\)).

Did you know?
If you double the length of a cube's sides, you don't double the volume. The volume actually becomes 8 times larger (\(2 \times 2 \times 2 = 8\))!

4. Percentages

A percentage is simply a ratio out of 100. "Percent" means "per hundred."

Percentage Change

To find the percentage increase or decrease, use this formula:
\(\text{Percentage Change} = \frac{\text{Change}}{\text{Original Amount}} \times 100\)

Working with Multipliers

Using multipliers is the fastest way to solve percentage problems with a calculator:
• To increase by 15%: Multiply by 1.15 (100% + 15% = 115%).
• To decrease by 15%: Multiply by 0.85 (100% - 15% = 85%).

Simple and Compound Interest

• Simple Interest: You earn interest only on the original amount every year. It stays the same.
• Compound Interest: You earn interest on your interest! The amount grows faster because the interest is added to the total before the next year's interest is calculated.

Key Takeaway:
For original value problems (where you are told the price after a change and asked for the price before), always divide by the multiplier. Do not try to add the percentage back on—it won't work!

5. Direct and Inverse Proportion

Direct Proportion

Two quantities are in direct proportion if, as one goes up, the other goes up at the same rate.
Example: The more apples you buy, the more you pay.
The graph of direct proportion is always a straight line passing through the origin (0,0).

Inverse Proportion

Two quantities are in inverse proportion if, as one goes up, the other goes down.
Example: The more builders you have, the less time it takes to build a wall.
The graph of inverse proportion is a curve that never quite touches the axes.

Memory Aid:
Direct = They go the same way.
Inverse = They go opposite ways.

6. Compound Units (Speed, Density, Pressure)

Compound units involve two or more measures combined. We often use formula triangles to help us remember them.

Speed, Distance, and Time

Formula: \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \)
Units: m/s or mph.

Density, Mass, and Volume

Formula: \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \)
Analogy: Imagine a sponge and a brick of the same size. The brick is "denser" because it has more mass packed into that volume.

Pressure, Force, and Area

Formula: \( \text{Pressure} = \frac{\text{Force}}{\text{Area}} \)
Real-world example: Why do snowshoes stop you from sinking? They increase the area, which reduces the pressure on the snow!

Quick Review Box:
When using these formulas, make sure your units match! If the speed is in km per hour, your time must be in hours, not minutes. To turn minutes into hours, divide by 60 (e.g., 30 mins = 0.5 hours).

7. Rates of Change

A "rate of change" describes how one variable changes compared to another.
On a graph, the gradient (the steepness) represents the rate of change.

• On a Distance-Time graph, the gradient is the speed.
• A flat horizontal line means the object has stopped (speed = 0).
• A steeper line means a faster speed.

Final Tip for Success:
When you see a "Best Buy" question (e.g., Which box of cereal is cheaper?), find the unit price. Work out the cost for 1 gram or 100 grams for both options, then compare. The lowest unit price is the best deal!

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is a ratio and how is it written?

A ratio is a way of comparing two or more amounts using a colon (:) to separate the numbers.

How do you simplify a ratio?

To simplify a ratio, divide all the numbers in the ratio by their highest common factor.

How can you express a ratio as a fraction of the total amount?

To find the fraction for a part, use the part as the numerator and the sum of all parts as the denominator.

What are the three steps for sharing an amount in a ratio?

The three steps are: Add the parts, Divide the total by that sum, and Multiply the result by each original ratio number.

How do you interpret a map scale ratio such as 1 : 25,000?

It means that 1 unit of measurement on the map represents 25,000 of the same units in real life.

How do area and volume change when a shape is enlarged by scale factor \(k\)?

If the lengths are multiplied by \(k\), the area is multiplied by \(k^{2}\) and the volume is multiplied by \(k^{3}\).

What is the formula for calculating percentage change?

\(\text{Percentage Change} = \frac{\text{Change}}{\text{Original Amount}} \times 100\)

How do you use multipliers to calculate percentage increases and decreases?

To increase, add the percentage to 100% (e.g., 1.15 for +15%). To decrease, subtract it from 100% (e.g., 0.85 for -15%).

What is the main difference between simple interest and compound interest?

Simple interest is calculated only on the original sum, while compound interest is calculated on the original sum plus any interest already earned.

What defines a direct proportion relationship?

In direct proportion, as one quantity increases, the other increases at the same rate; the graph is a straight line through the origin (0,0).

What defines an inverse proportion relationship?

In inverse proportion, as one quantity increases, the other decreases; the graph is a curve that never touches the axes.

What is the formula for calculating speed?

\( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \)

What is the formula for density?

\( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \)

What is the formula for pressure?

\( \text{Pressure} = \frac{\text{Force}}{\text{Area}} \)

How is the rate of change represented on a graph?

The gradient (steepness) of the line on a graph represents the rate of change.

How do you solve "Best Buy" or unit price problems?

Calculate the unit price (the cost for 1 unit, like 1g or 100g) for each option and compare; the lowest unit price is the best value.

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