Calculating Cost for Different Lengths

Oct 13, 2025 by windailys.com 32 views

Introduction

In the world of mathematics, proportions play a vital role in understanding various concepts and relationships between different quantities. Two fundamental types of proportions are direct and inverse proportion. In this article, we will delve into the explanation of these concepts with the help of examples, focusing on the discussion of a maths teacher who uses a real-life scenario to illustrate the concept of direct and inverse proportion.

Direct Proportion

Direct proportion, also known as direct variation, is a relationship between two quantities where one quantity increases or decreases in direct proportion to the other quantity. In other words, as one quantity increases, the other quantity also increases at a constant rate. This relationship can be represented by the equation:

y = kx

where y is the dependent variable, x is the independent variable, and k is the constant of proportionality.

Example: Cost of Cloth

Let's consider an example given by the maths teacher. The cost of 5 metres of a particular quality of cloth is Rs 210. The teacher tabulates the cost of 2, 4, 10, and 13 metres of cloth as follows:

Length of Cloth (m)	Cost (Rs)
2	84
4	168
10	420
13	546

In this example, the cost of the cloth is directly proportional to the length of the cloth. As the length of the cloth increases, the cost also increases at a constant rate. We can represent this relationship using the equation:

Cost = k × Length

To find the constant of proportionality (k), we can use any two values from the table. Let's use the values for 5 metres and Rs 210.

210 = k × 5

k = 210 ÷ 5 k = 42

Now that we have the value of k, we can use it to find the cost of cloth for any given length.

Calculating Cost for Different Lengths

Using the value of k, we can calculate the cost of cloth for different lengths as follows:

For 2 metres: Cost = 42 × 2 = Rs 84
For 4 metres: Cost = 42 × 4 = Rs 168
For 10 metres: Cost = 42 × 10 = Rs 420
For 13 metres: Cost = 42 × 13 = Rs 546

As we can see, the cost of cloth increases at a constant rate as the length of the cloth increases, illustrating the concept of direct proportion.

Inverse Proportion

Inverse proportion, also known as inverse variation, is a relationship between two quantities where one quantity increases or decreases in inverse proportion to the other quantity. In other words, as one quantity increases, the other quantity decreases at a constant rate. This relationship can be represented by the equation:

y = k/x

where y is the dependent variable, x is the independent variable, and k is the constant of proportionality.

Example: Time and Speed

Let's consider another example given by the maths teacher. Suppose a car travels from one city to another at a constant speed. The time taken to travel from one city to another is inversely proportional to the speed of the car. If the car travels at a speed of 60 km/h, it takes 2 hours to cover a distance of 120 km. If the car travels at a speed of 80 km/h, it takes 1.5 hours to cover the same distance.

Speed (km/h)	Time (hours)
60	2
80	1.5

In this example, the time taken to travel from one city to another is inversely proportional to the speed of the car. As the speed of the car increases, the time taken to travel decreases at a constant rate. We can represent this relationship using the equation:

Time = k/Speed

To find the constant of proportionality (k), we can use any two values from the table. Let's use the values for 60 km/h and 2 hours.

2 = k/60

k = 2 × 60 k = 120

Now that we have the value of k, we can use it to find the time taken to travel for any given speed.

Calculating Time for Different Speeds

Using the value of k, we can calculate the time taken to travel for different speeds as follows:

For 60 km/h: Time = 120/60 = 2 hours
For 80 km/h: Time = 120/80 = 1.5 hours
For 100 km/h: Time = 120/100 = 1.2 hours

As we can see, the time taken to travel decreases at a constant rate as the speed of the car increases, illustrating the concept of inverse proportion.

Conclusion

Q: What is direct proportion?

A: Direct proportion, also known as direct variation, is a relationship between two quantities where one quantity increases or decreases in direct proportion to the other quantity. In other words, as one quantity increases, the other quantity also increases at a constant rate.

Q: What is inverse proportion?

A: Inverse proportion, also known as inverse variation, is a relationship between two quantities where one quantity increases or decreases in inverse proportion to the other quantity. In other words, as one quantity increases, the other quantity decreases at a constant rate.

Q: How do I identify direct and inverse proportion in a problem?

A: To identify direct and inverse proportion in a problem, look for the following characteristics:

Direct proportion: If one quantity increases, the other quantity also increases at a constant rate.
Inverse proportion: If one quantity increases, the other quantity decreases at a constant rate.

Q: What is the formula for direct proportion?

A: The formula for direct proportion is:

y = kx

where y is the dependent variable, x is the independent variable, and k is the constant of proportionality.

Q: What is the formula for inverse proportion?

A: The formula for inverse proportion is:

y = k/x

where y is the dependent variable, x is the independent variable, and k is the constant of proportionality.

Q: How do I find the constant of proportionality (k)?

A: To find the constant of proportionality (k), use the following steps:

Identify the two quantities that are in proportion.
Choose two values from the table or data set.
Use the formula for direct or inverse proportion to set up an equation.
Solve for k.

Q: What are some real-life examples of direct and inverse proportion?

A: Some real-life examples of direct and inverse proportion include:

Direct proportion:

The cost of a product increases as the quantity purchased increases.
The time it takes to complete a task decreases as the number of workers increases.

Inverse proportion:

The time it takes to travel from one city to another decreases as the speed of the vehicle increases.
The cost of a product decreases as the quantity purchased increases.

Q: How do I solve problems involving direct and inverse proportion?

A: To solve problems involving direct and inverse proportion, follow these steps:

Identify the two quantities that are in proportion.
Choose two values from the table or data set.
Use the formula for direct or inverse proportion to set up an equation.
Solve for the unknown quantity.

Q: What are some common mistakes to avoid when working with direct and inverse proportion?

A: Some common mistakes to avoid when working with direct and inverse proportion include:

Confusing direct and inverse proportion.
Failing to identify the constant of proportionality (k).
Not using the correct formula for direct or inverse proportion.
Not checking units and making sure they are consistent.

Q: How do I check my work when solving problems involving direct and inverse proportion?

A: To check your work when solving problems involving direct and inverse proportion, follow these steps:

Review the problem and make sure you understand what is being asked.
Check your work for errors in calculation.
Make sure you have used the correct formula for direct or inverse proportion.
Check your units and make sure they are consistent.

By following these steps and avoiding common mistakes, you can confidently solve problems involving direct and inverse proportion.