Suppose A is directly proportional to B, B is inversely proportional to C and C is inversely proportional to D. Determine whether A and D are directly proportional, inversely proportional, or neither.
C is inversely proportional to D ---> C = p/D
B is inversely proportional to C ---> B = qC
so B = q/(p/D) = (q/p)D
A is directly proportional to B --> A = rB
A = r(q/p) D
looks like A is directly proportional to D
To determine the relationship between A and D, we can analyze the given information about their proportional relationships with B and C.
Given:
1. A is directly proportional to B.
2. B is inversely proportional to C.
3. C is inversely proportional to D.
Let's break down each of these proportional relationships:
1. A is directly proportional to B:
When two quantities are directly proportional, it means that as one quantity increases, the other also increases in the same proportion, and vice versa. Mathematically, it can be expressed as A = k * B, where k is a constant of proportionality.
2. B is inversely proportional to C:
In an inverse proportional relationship, as one quantity increases, the other decreases in the same proportion, and vice versa. Mathematically, it can be expressed as B = k / C, where k is a constant of proportionality.
3. C is inversely proportional to D:
Similar to the second relationship, C and D are also inversely proportional. If C increases, D decreases in the same proportion, and vice versa. Mathematically, it can be expressed as C = k / D, where k is a constant of proportionality.
Now, to determine the relationship between A and D, we can rearrange the equations for B and C in terms of A and D, respectively:
From the first relationship: A = k * B
Rearranging for B: B = A / k
From the third relationship: C = k / D
Rearranging for C: D = k / C
Substituting the expressions for B and C into the equation for A, we get:
A = k * (A / k) * (k / C)
Simplifying, we have:
A = A * (k^2 / (k * C))
Since A is present on both sides of the equation, we can cancel it out:
1 = k^2 / (k * C)
Rearranging this equation, we have:
C = k * k
From this equation, we can see that C is directly proportional to the constant of proportionality squared (k^2). Therefore, A and D are directly proportional to each other.
In conclusion, A and D are directly proportional.