Representative fraction (RF) is a mathematical concept used to represent the relationship between a map or diagram and the actual size of the object or area it represents. It is a ratio that compares the distance or size on the map to the corresponding distance or size in the real world.
The concept of representative fraction has been used in cartography for centuries. It was first introduced by Gerardus Mercator, a Flemish cartographer, in the 16th century. Mercator developed a projection system that allowed for accurate representation of the Earth's surface on a flat map. The representative fraction was a key component of this system, as it provided a way to express the scale of the map.
Representative fraction (RF) is typically introduced in middle school or early high school mathematics. It is a fundamental concept in geometry and is often included in the curriculum for grades 6-9.
The knowledge points contained in representative fraction (RF) include:
Ratio: RF is a ratio that compares the distance or size on the map to the corresponding distance or size in the real world.
Scale: RF represents the scale of the map or diagram. It tells us how much the distances or sizes on the map have been reduced or enlarged compared to the real world.
Proportional reasoning: RF requires an understanding of proportional reasoning, as it involves comparing two quantities and determining their relationship.
To calculate the representative fraction (RF), follow these steps:
Determine the distance or size on the map that you want to represent.
Measure the corresponding distance or size in the real world.
Divide the distance or size on the map by the distance or size in the real world.
Simplify the resulting ratio, if necessary.
The simplified ratio is the representative fraction (RF) for the given map or diagram.
There are no specific types of representative fraction (RF). However, the RF can vary depending on the scale of the map or diagram. It can be a fraction, such as 1/10,000, or it can be expressed as a ratio, such as 1:10,000.
The properties of representative fraction (RF) include:
RF is always a ratio or fraction.
RF is a unitless quantity, as it represents a relationship between two quantities.
RF is independent of the units used to measure the distances or sizes on the map and in the real world.
RF can be used to determine the actual distance or size corresponding to a given distance or size on the map.
To find or calculate the representative fraction (RF), follow these steps:
Determine the distance or size on the map that you want to represent.
Measure the corresponding distance or size in the real world.
Divide the distance or size on the map by the distance or size in the real world.
Simplify the resulting ratio, if necessary.
The simplified ratio is the representative fraction (RF) for the given map or diagram.
The formula for calculating the representative fraction (RF) is:
RF = Distance or size on the map / Distance or size in the real world
To apply the representative fraction (RF) formula, substitute the values for the distance or size on the map and the corresponding distance or size in the real world into the formula. Then, simplify the resulting ratio to obtain the RF.
For example, if the distance on the map is 5 centimeters and the corresponding distance in the real world is 100 meters, the RF can be calculated as:
RF = 5 cm / 100 m = 1/2000
The symbol or abbreviation for representative fraction (RF) is RF itself. It is commonly used in cartography and map-making.
The methods for determining the representative fraction (RF) include:
Measuring distances or sizes on the map and in the real world and calculating the ratio.
Using a scale bar provided on the map to determine the RF.
Using a measuring tool, such as a ruler or compass, to measure distances on the map and in the real world and calculating the ratio.
Example 1: A map has a scale of 1:50,000. What is the representative fraction (RF) for this map?
Solution: The RF can be calculated by taking the reciprocal of the second number in the ratio. Therefore, the RF is 1/50,000.
Example 2: On a map, the distance between two cities is 10 centimeters. In reality, the distance between the cities is 200 kilometers. What is the representative fraction (RF) for this map?
Solution: The RF can be calculated as follows: RF = 10 cm / 200 km = 1/20,000
Example 3: A diagram of a building has a scale of 1 inch = 10 feet. What is the representative fraction (RF) for this diagram?
Solution: The RF can be calculated by converting the units to a common unit. Since 1 inch is equal to 1/12 feet, the RF is: RF = 1/12
A map has a scale of 1:25,000. What is the RF for this map?
On a map, the length of a river is 8 inches. In reality, the length of the river is 16 miles. What is the RF for this map?
A diagram of a garden has a scale of 1 centimeter = 2 meters. What is the RF for this diagram?
Question: What is representative fraction (RF)? Answer: Representative fraction (RF) is a ratio that compares the distance or size on a map or diagram to the corresponding distance or size in the real world.
Question: How is the representative fraction (RF) calculated? Answer: The RF is calculated by dividing the distance or size on the map by the distance or size in the real world.
Question: What is the significance of representative fraction (RF) in cartography? Answer: RF is essential in cartography as it allows for accurate representation of the Earth's surface on a flat map. It helps in understanding the scale and proportions of the map or diagram.
Question: Can RF be greater than 1? Answer: Yes, RF can be greater than 1. It indicates that the distances or sizes on the map are larger than in the real world.
Question: Can RF be less than 1? Answer: Yes, RF can be less than 1. It indicates that the distances or sizes on the map are smaller than in the real world.