Given: { R $}$ Is Inversely Proportional To { S $}$. If { R = 16 $}$ When { S = 3 $}$, Then Write The Formula For The Relation Between { R $}$ And { S $} . A . \[ .A. \[ . A . \[ R = \frac{s}{48}
Introduction
In mathematics, proportionality is a fundamental concept that describes the relationship between two or more variables. In this article, we will explore the concept of inverse proportionality, where one variable is inversely proportional to another. We will use a real-world example to illustrate the concept and derive the formula for the relation between the two variables.
What is Inverse Proportionality?
Inverse proportionality is a relationship between two variables where one variable is inversely proportional to the other. This means that as one variable increases, the other variable decreases, and vice versa. In mathematical terms, if two variables x and y are inversely proportional, then their product remains constant.
Mathematical Representation
The mathematical representation of inverse proportionality is given by the equation:
y = k/x
where k is a constant of proportionality.
Given Problem
We are given that r is inversely proportional to s. This means that the product of r and s remains constant. Mathematically, we can represent this as:
r × s = k
where k is a constant of proportionality.
Using the Given Values
We are given that r = 16 when s = 3. We can use these values to find the constant of proportionality, k.
r × s = k 16 × 3 = k 48 = k
Deriving the Formula
Now that we have found the constant of proportionality, k, we can derive the formula for the relation between r and s.
r × s = k r × s = 48
To isolate r, we can divide both sides of the equation by s:
r = 48/s
Simplifying the Formula
We can simplify the formula by rewriting it as:
r = s/48
Conclusion
In this article, we have explored the concept of inverse proportionality and derived the formula for the relation between two variables. We have used a real-world example to illustrate the concept and have shown that the product of the two variables remains constant. We have also simplified the formula to make it easier to use.
Example Problems
- If x is inversely proportional to y, and x = 12 when y = 4, find the formula for the relation between x and y.
- If a is inversely proportional to b, and a = 20 when b = 5, find the formula for the relation between a and b.
Answer Key
- a = 48/b
- a = b/4
Tips and Tricks
- When dealing with inverse proportionality, always remember that the product of the two variables remains constant.
- Use the given values to find the constant of proportionality, k.
- Derive the formula by isolating one variable and then simplifying the equation.
Further Reading
- Inverse proportionality is a fundamental concept in mathematics and is used in many real-world applications, such as physics, engineering, and economics.
- To learn more about inverse proportionality, we recommend checking out the following resources:
- Khan Academy: Inverse Proportionality
- Mathway: Inverse Proportionality
- Wolfram Alpha: Inverse Proportionality
Inverse Proportionality Q&A =============================
Frequently Asked Questions
Q: What is inverse proportionality?
A: Inverse proportionality is a relationship between two variables where one variable is inversely proportional to the other. This means that as one variable increases, the other variable decreases, and vice versa.
Q: How is inverse proportionality represented mathematically?
A: The mathematical representation of inverse proportionality is given by the equation:
y = k/x
where k is a constant of proportionality.
Q: What is the constant of proportionality?
A: The constant of proportionality, k, is a value that remains constant in an inverse proportionality relationship. It is a measure of the strength of the relationship between the two variables.
Q: How do I find the constant of proportionality?
A: To find the constant of proportionality, you can use the given values of the two variables. For example, if r = 16 when s = 3, you can find k by multiplying r and s:
k = r × s k = 16 × 3 k = 48
Q: How do I derive the formula for the relation between two variables?
A: To derive the formula, you can start with the equation:
r × s = k
Then, isolate one variable by dividing both sides of the equation by the other variable:
r = k/s
Q: Can I simplify the formula?
A: Yes, you can simplify the formula by rewriting it in a more convenient form. For example, if the formula is:
r = k/s
You can rewrite it as:
r = s/k
Q: What are some real-world applications of inverse proportionality?
A: Inverse proportionality has many real-world applications, including:
- Physics: The force of gravity between two objects is inversely proportional to the square of the distance between them.
- Engineering: The resistance of a wire is inversely proportional to its cross-sectional area.
- Economics: The price of a commodity is inversely proportional to its supply.
Q: How do I determine if two variables are inversely proportional?
A: To determine if two variables are inversely proportional, you can plot a graph of the two variables and check if the points on the graph lie on a straight line that passes through the origin. You can also use the following criteria:
- The product of the two variables remains constant.
- As one variable increases, the other variable decreases.
- The relationship is not linear.
Q: What are some common mistakes to avoid when working with inverse proportionality?
A: Some common mistakes to avoid when working with inverse proportionality include:
- Assuming that the relationship is linear when it is actually inverse.
- Failing to check if the product of the two variables remains constant.
- Not isolating one variable to derive the formula.
Q: How do I check if my answer is correct?
A: To check if your answer is correct, you can use the following methods:
- Plug in the values of the two variables into the formula and check if the product of the two variables remains constant.
- Graph the two variables and check if the points on the graph lie on a straight line that passes through the origin.
- Use a calculator or computer program to check if the formula is correct.