Understanding Proportionality Relationships

This question asks us to determine the relationship between different variables based on given proportionalities. We are given the following relationships:

• A is directly proportional to B
• B is inversely proportional to C
• C is directly proportional to D

Let's translate these proportional relationships into mathematical equations using constants.

Direct Proportionality: If X is directly proportional to Y (X ∝ Y), it means X = kY for some constant k.

Inverse Proportionality: If X is inversely proportional to Y (X ∝ \(\frac{1}{Y}\)), it means X = \(\frac{k}{Y}\) for some constant k.

Translating Given Proportionalities to Equations

Based on the definitions:

• A ∝ B \(\implies\) A = k\(_1\)B (Equation 1, where k\(_1\) is a constant)
• B ∝ \(\frac{1}{C}\) \(\implies\) B = \(\frac{k_2}{C}\) (Equation 2, where k\(_2\) is a constant)
• C ∝ D \(\implies\) C = k\(_3\)D (Equation 3, where k\(_3\) is a constant)

Finding the Relationship Between A and D

We want to find how A relates to D. We can substitute the expressions for variables from one equation into another.

1. From Equation 3, we know C = k\(_3\)D.
2. Substitute this expression for C into Equation 2: \[B = \frac{k_2}{C} = \frac{k_2}{k_3D}\] \[B = \left(\frac{k_2}{k_3}\right) \times \frac{1}{D}\] Let k\(_4\) = \(\frac{k_2}{k_3}\). Since k\(_2\) and k\(_3\) are constants, k\(_4\) is also a constant. \[B = k_4 \times \frac{1}{D}\] This shows B is inversely proportional to D (B ∝ \(\frac{1}{D}\)).
3. Now, substitute this expression for B into Equation 1: \[A = k_1B = k_1 \times \left(k_4 \times \frac{1}{D}\right)\] \[A = (k_1k_4) \times \frac{1}{D}\] Let k\(_5\) = k\(_1\)k\(_4\). Since k\(_1\) and k\(_4\) are constants, k\(_5\) is also a constant. \[A = k_5 \times \frac{1}{D}\] This equation shows that A is inversely proportional to D.

So, we conclude that A is inversely proportional to D.

Finding the Relationship Between A and C

We want to find how A relates to C. We already have relationships involving A, B, and C:

• A ∝ B (Equation 1)
• B ∝ \(\frac{1}{C}\) (Equation 2)

Since A is proportional to B, and B is proportional to the inverse of C, we can chain these relationships:

A ∝ B and B ∝ \(\frac{1}{C}\)

This directly implies that A must be proportional to \(\frac{1}{C}\).

A ∝ \(\frac{1}{C}\)

So, we conclude that A is inversely proportional to C.

Evaluating the Given Statements

Let's look at the statements provided in the question based on our findings:

A) A is inversely proportional to D

Our derivation showed A ∝ \(\frac{1}{D}\). This statement is Correct.

B) A is directly proportional to D

Our derivation showed A is inversely proportional to D. This statement is Incorrect.

C) A is directly proportional to C

Our derivation showed A is inversely proportional to C. This statement is Incorrect.

D) A is inversely proportional to C

Our derivation showed A ∝ \(\frac{1}{C}\). This statement is Correct.

Matching Statements to Options

The correct statements are A and D. We need to choose the option that says "A and D". Looking at the options provided:

Option 1 correctly identifies statements A and D as the true ones.

Revision Table: Proportionality Summary

Additional Information: Chain Rule of Proportionality

When you have a chain of proportionalities, you can link them together. For example, if \(X \propto Y\) and \(Y \propto Z\), you can conclude that \(X \propto Z\). However, you must be careful with inverse relationships.

• If \(X \propto Y\) and \(Y \propto \frac{1}{Z}\), then \(X \propto \frac{1}{Z}\). (Direct then Inverse = Inverse)
• If \(X \propto \frac{1}{Y}\) and \(Y \propto Z\), then \(X \propto \frac{1}{Z}\). (Inverse then Direct = Inverse)
• If \(X \propto \frac{1}{Y}\) and \(Y \propto \frac{1}{Z}\), then \(X \propto Z\). (Inverse then Inverse = Direct)

In our problem:

• A ∝ B (Direct)
• B ∝ \(\frac{1}{C}\) (Inverse)
• C ∝ D (Direct)

From A ∝ B and B ∝ \(\frac{1}{C}\), we get A ∝ \(\frac{1}{C}\) (Direct followed by Inverse gives Inverse).

From B ∝ \(\frac{1}{C}\) and C ∝ D, we can find the relationship between B and D:

B ∝ \(\frac{1}{C}\) and C ∝ D implies B ∝ \(\frac{1}{D}\) (Inverse followed by Direct gives Inverse).

Finally, from A ∝ B and B ∝ \(\frac{1}{D}\), we get A ∝ \(\frac{1}{D}\) (Direct followed by Inverse gives Inverse).

These chain rule observations confirm our step-by-step substitution results: A is inversely proportional to C, and A is inversely proportional to D.