\begin{tabular}{|c|c|c|c|c|} \hline$P$ & 6 & 8 & 9 & 12 \ \hline$Q$ & 24 & 18 & 16 & 12 \ \hline \end{tabular} Example 7 fy is inversely proportional to $x+2$ and $y=48$ when $x=10$, find $x$ when $y=30$ Solution $y x \frac{1}{x+2}$ or $y=\frac{k}{x+2}$, where $k$ is the constant of proportionality
Answer (2)
Calculate the product of P and Q for each pair of values.
Observe that the product P ⋅ Q is constant and equal to 144.
Conclude that P and Q are inversely proportional.
Express the relationship as Q = P 144 .
The relationship between P and Q is: Q = P 144
Explanation
Understanding the Problem We are given a table of values for P and Q and asked to determine the relationship between them. We are also given an example of an inverse proportion.
Checking for Inverse Proportionality To determine the relationship between P and Q , we can check if their product is constant. If P ⋅ Q = k for some constant k , then P and Q are inversely proportional.
Calculating the Product Let's calculate the product P ⋅ Q for each pair of values in the table:
For P = 6 and Q = 24 , P ⋅ Q = 6 ⋅ 24 = 144 .
For P = 8 and Q = 18 , P ⋅ Q = 8 ⋅ 18 = 144 .
For P = 9 and Q = 16 , P ⋅ Q = 9 ⋅ 16 = 144 .
For P = 12 and Q = 12 , P ⋅ Q = 12 ⋅ 12 = 144 .
Determining the Relationship Since the product P ⋅ Q is constant and equal to 144 for all pairs of values in the table, P and Q are inversely proportional. The constant of proportionality is k = 144 . Therefore, the relationship between P and Q can be expressed as Q = P 144 .
Examples
Understanding inverse proportionality is useful in many real-world scenarios. For example, the time it takes to complete a journey is inversely proportional to the speed at which you travel. If you double your speed, you halve the time it takes to reach your destination, assuming the distance remains constant. Similarly, in economics, the price of a commodity is often inversely proportional to its supply. If the supply increases, the price tends to decrease, and vice versa. These relationships help us make informed decisions in everyday life.
The values of P and Q are inversely proportional since their product is constant at 144 for all pairs. This relationship can be expressed as Q = P 144 . Therefore, if one increases, the other decreases while maintaining the same product.
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