Scaling Ratios

A ratio scale expresses the relationship between two quantities or values. It allows us to compare and represent different sizes or magnitudes of objects.

Ratio scales are written in the form 1 : N, where N is a whole number. This ratio indicates that one unit on a scale represents N units in reality. For example, a scale of 1 : 100 means that 1 unit on the scale represents 100 units in reality.

When scaling ratios, the units of measurement must be consistent. If necessary, convert between different units to maintain consistency.

Scaling Ratios in Maps and Models

Scaling ratios is important when creating maps and models. To create a map or model, we often need to shrink or enlarge real-world objects to fit within a certain space or size. Ratio scales help us maintain accurate and consistent proportions between the original object and its representation.

For example, city maps use ratio scales to represent large areas on a smaller sheet of paper. Similarly, architectural models of buildings, vehicles, or other structures use scaling ratios to create accurate, scaled-down representations of the original objects.

Calculating Distances and Sizes Using Ratio Scales

To determine the actual distance or size of an object using scaling ratios, we can use the following method:

Step 1: Identify the scaling ratio (e.g., 1 : N).

Step 2: Measure the distance or size on the map or model.

Step 3: Multiply the measurement by N to find the actual distance or size.

For example, if we have a map with a scale of 1 : 50,000, and we measure a distance of 3 cm between two points on the map, we can calculate the actual distance as follows:

Actual distance = 3\:cm \times 50,000

= 150,000\:cm

= 1.5\:km (since 1\:km = 100,000\:cm)

An example: Mapping London

Imagine creating a map of London, where the city’s layout is scaled down to fit on a single sheet of paper. In this example, the scale used is 1\:cm : 100,000\:cm. This means that for every centimeter on the map, the actual distance represented is 100,000\:cm.

To illustrate how this works, let’s say we want to find the actual distance between Croydon and Forest Hill on this map. If the distance between these two locations on the map measures 9.2\:cm, we can calculate the real-world distance by multiplying the map distance by the scale factor.

9.2\:cm (map distance) \times 100,000\:cm (scale factor) = 920,000\:cm

Since 1\:km equals 100,000\:cm, we can then convert the distance into kilometres:

920,000 cm \div 100,000 cm/km = 9.2\:km

Therefore, the actual distance between Croydon and Forest Hill is 9.2\:km. Understanding and applying ratios in model making and map creation allows for accurate representation and easy interpretation of real-world distances and proportions.

Scaling Ratios in Geometry

In geometry, scale factors are used to describe similar shapes which have the same shape but may differ in size. When two shapes are similar, their corresponding sides are proportional, meaning their scaling ratio remains constant.

The scale factor is the ratio between the lengths of corresponding sides in similar shapes. This factor has a direct relationship with the area and volume of the shapes.

When enlarging or reducing a shape, the scale factor will affect its area and volume. For example, if a shape is enlarged by a scaling factor of 2, its area will increase by a factor of 4 (which is 2^2), and its volume will increase by a factor of 8 (which is 2^3).

Examples

Example 1:

A map is drawn to a scale of 1 : 25000. Two cities are connected by a straight road of length 7.5 km. Calculate, in centimetres, the distance on the map between these two cities.

We are given the scale as 1 : 25,0001 : 25,000, which means that 1\:cm1\:cm on the map corresponds to 25,000\:cm25,000\:cm (or 0.25\:km0.25\:km) in reality. The actual distance between the cities is 7.5 km. To find the distance on the map, we can use the following proportion:

\frac{Map}{Actual} = \frac{1}{25,000}\frac{Map}{Actual} = \frac{1}{25,000}

We know the actual distance is 7.5\:km7.5\:km, so we can set up the equation:

\frac{Map}{7.5\:km} = \frac{1}{25,000\:cm}\frac{Map}{7.5\:km} = \frac{1}{25,000\:cm}

Now, we want to find the value of the map distance. To do so, we can cross-multiply and solve for this:

Map = \frac{7.5\:km}{25,000}Map = \frac{7.5\:km}{25,000}

First, we need to convert the actual distance from km to cm. We know that 1\:km = 100,000\:cm1\:km = 100,000\:cm, so 7.5\:km = 7.5 \times 100,000\:cm = 750,000\:cm7.5\:km = 7.5 \times 100,000\:cm = 750,000\:cm. Now, we can substitute this value into our equation:

Map = \frac{750,000\:cm}{25,000}Map = \frac{750,000\:cm}{25,000}

Map = 30\:cmMap = 30\:cm

So, the distance between the two cities on the map is 30\:cm30\:cm.

Example 2:

A map is drawn to a scale of 1 : 100,000. Find the actual distance apart, in km, of two villages which are represented on the map by the two dots 36 cm apart.

We have the scale as 1 : 100,0001 : 100,000, which means 1\:cm1\:cm on the map represents 100,000\:cm100,000\:cm (or 1 km) in reality. The distance between the villages on the map is 36\:cm36\:cm. To find the actual distance, we can use the following proportion:

\frac{Map}{Actual} = \frac{1}{100,000}\frac{Map}{Actual} = \frac{1}{100,000}

We know the distance on the map is 36\:cm36\:cm, so we can set up the equation:

\frac{36\:cm}{Actual} = \frac{1}{100,000}\frac{36\:cm}{Actual} = \frac{1}{100,000}

Now, we want to find the value of the actual distance. To do so, we can cross-multiply and solve for Actual:

Actual = 36\:cm \times 100,000Actual = 36\:cm \times 100,000

Since 1\:cm1\:cm on the map corresponds to 1\:km1\:km in reality:

Actual = 36\:kmActual = 36\:km

So, the actual distance between the two villages is 36\:km36\:km.

Example 3:

The length of a river is 1500 km. Find the length representing the river on a map of scale 1 : 250,000.

Given the scale 1 : 250,0001 : 250,000, we can set up the proportion as follows:

\frac{Map}{Actual} = \frac{1}{250,000\;cm}\frac{Map}{Actual} = \frac{1}{250,000\;cm}

We know the actual length of the river is 1500 km. So, we can plug in the value for Actual:

\frac{Map}{1500\:km} = \frac{1}{250,000\:cm}\frac{Map}{1500\:km} = \frac{1}{250,000\:cm}

To find the length on the map, we can cross-multiply:

Map = \frac{1500\:km}{250,000} = \frac{15}{2500} \times 100,000 cmMap = \frac{1500\:km}{250,000} = \frac{15}{2500} \times 100,000 cm

Since 1\:km = 100,000\:cm1\:km = 100,000\:cm:

Map = 15 \times 40 cm = 600\:cmMap = 15 \times 40 cm = 600\:cm

To express this result in meters (m), we can divide 600 cm by 100, since 1\:m = 100\:cm1\:m = 100\:cm:

Map \text{ distance } = \frac{600}{100} m = 6\:mMap \text{ distance } = \frac{600}{100} m = 6\:m

Example 4:

The actual distance from Brighton to Croydon is 70 km. This is represented by 2 cm on a map. Find the scale that has been used. Considering also that 1 mile is approximately 1.6 km, find the actual distance from Brighton to Croydon in miles.

To find the scale, we can use the given information:

70\:km = 70 \times 1000 \times 100\:cm = 7,000,000\:cm70\:km = 70 \times 1000 \times 100\:cm = 7,000,000\:cm

Scale, Map : Actual = 2 : 7,000,000\:cm= 2 : 7,000,000\:cm or 1 : 3,500,0001 : 3,500,000

Therefore, the scale used is 1 : 3,500,0001 : 3,500,000.

Now, we want to find the distance in miles:

1.6 km \rightarrow 11.6 km \rightarrow 1 mile

1\:km = (1 \div 1.6)1\:km = (1 \div 1.6) miles

70\:km = (1 \div 1.6) × 7070\:km = (1 \div 1.6) × 70 miles

= 43.75 miles

So, the actual distance from Brighton to Croydon is 43.75 miles.