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.