Unit ratios

Unit ratios and map scales are practical tools for comparing quantities and interpreting real-world distances. A unit ratio simplifies a ratio so that one term is 1, while map scales allow us to calculate actual distances from a scaled representation.

1. Unit Ratios:

A unit ratio expresses a relationship where one of the terms is reduced to 1. This is often used to compare quantities in a standardised way, such as speeds, costs, or densities.

Steps to Find a Unit Ratio:

1. Divide by the first term: If simplifying to $1:n$, divide both parts of the ratio by the first term.
2. Express the ratio: The simplified form will show how much of the second quantity corresponds to 1 of the first.

Example 1: Simplify $3:15$ to a unit ratio:

• Divide both terms by the first term $3$. $\frac{3}{3}:\frac{15}{3}=1:5$.

The unit ratio is $1:5$, meaning the second quantity is 5 times the first.

Example 2: Convert $12:4$ to a unit ratio where the second term is 1:

• Divide both terms by the second term $4$. $\frac{12}{4}:\frac{4}{4}=3:1$.

The unit ratio is $3:1$, meaning the first quantity is 3 times the second.

2. Map Scales:

A map scale represents the relationship between a distance on a map and the actual distance it represents in real life. Map scales are often written as a ratio, such as $1:50,000$, meaning 1 unit on the map corresponds to 50,000 units in reality.

Steps to Use a Map Scale:

1. Interpret the scale: Understand the ratio and the units involved (e.g., centimetres, kilometres).
2. Measure the map distance: Use a ruler to find the distance on the map.
3. Convert to real distance: Multiply the map distance by the scale factor to find the actual distance.

Example 1: A map has a scale of $1:25,000$. If the distance between two points on the map is $4\text{cm}$, what is the actual distance?

• Multiply the map distance by the scale factor: $4\times 25,000=100,000\text{cm}$.
• Convert to kilometres: $\frac{100,000}{100,000}=1\text{km}$.

The actual distance is $1\text{km}$.

3. Converting Between Units in Map Scales:

Sometimes, you need to convert between units to make calculations easier. For example, if a scale is $1:50,000$ and a map distance is measured in millimetres, convert millimetres to centimetres or metres before applying the scale.

Example 2: A scale is $1:100,000$, and the map distance is $30\text{mm}$. Find the real distance in kilometres:

• Convert $30\text{mm}$ to $3\text{cm}$ (since $1\text{cm}=10\text{mm}$).
• Multiply by the scale factor: $3\times 100,000=300,000\text{cm}$.
• Convert $300,000\text{cm}$ to kilometres: $\frac{300,000}{100,000}=3\text{km}$.

The real distance is $3\text{km}$.

4. Summary:

• Unit ratios express one term of the ratio as 1, simplifying comparisons.
• Map scales use ratios to represent the relationship between map distances and actual distances.
• To use a map scale, measure the map distance, multiply by the scale factor, and convert units if necessary.