Question

If \[A\] varies directly as \[B\] and inversely as \[C\]. \[A\] is 12 when \[B\] is 6 and \[C\] is 2 . What is the value of \[A\] when \[B\] is 12 and \[C\] is 3?
A.4
B.8
C.12
D.16

Answer 1

Hint: We will formulate an equation for \[A\] keeping in mind the conditions given in the question. The equation will contain one unknown variable. We will find the value of this variable and use it to find the value of \[A\] when \[B\] is 12 and \[C\] is 3.

Complete step-by-step answer:
If two variables are directly proportional, we can express them using the symbol of proportion in the following manner:
\[A\]∝\[B\]
When a variable (say \[A\] ) is directly proportional to another variable (say \[B\] ), it can be written as:
\[ \Rightarrow {\rm{A}} = t{\rm{B}}\]
Where \[t\] is any constant.
If two variables are inversely proportional, we can express them using the symbol of proportion in the following manner:
\[A\]∝\[\dfrac{1}{C}\]
When a variable (say \[A\] ) is inversely proportional to another variable (say \[C\] ), it can be written as:
\[ \Rightarrow {\rm{A}} = \dfrac{s}{C}\]
Where \[s\] is any constant.
We will combine the 2 properties that we have mentioned above and express A as:
\[A = k\dfrac{B}{C}\] ……………….\[\left( 1 \right)\]
Where \[k\] is any constant.
Now, we know that \[A\] is 12 when \[B\] is 6 and \[C\] is 2. We will substitute these values in equation (1):
\[\begin{array}{l} \Rightarrow 12 = k\dfrac{6}{2}\\ \Rightarrow 12 = k \cdot 3\end{array}\]
Dividing both side by 3, we get
\[\begin{array}{l} \Rightarrow \dfrac{{12}}{3} = \dfrac{{3k}}{3}\\ \Rightarrow {\rm{ }}4 = k\end{array}\]
We have calculated the value of \[k\] to be 4. We will use this to find value of \[A\] by substituting \[B = 12\] and the value of \[C = 3\] in equation (1):
\[\begin{array}{l} \Rightarrow A = 4\dfrac{{12}}{3}\\ \Rightarrow A = 4 \times 4\end{array}\]
Multiplying the terms, we get
\[ \Rightarrow A = 16\]
$\therefore $ Option D is the correct option

Note: If two things are directly proportional, it means that they behave in a similar manner. That is if one increases the other also increases and vice versa. If two things are inversely proportional, it means that they behave in an opposite manner; if one increases the other decreases and vice versa.
We might make a mistake by reading the given statement wrong and thinking B varies inversely as C, which is wrong. Here variation of A is given with respect to B and C. So A varies directly with B and With C, A varies inversely.