Question

If \(y\) is inversely proportional to \(x,\) how does \(y\) change when \(x\) is doubled?

Short answer

When \(x\) is doubled, \(y\) becomes half of its original value.

Step-by-step solution

Understanding Inverse Proportionality

Recognize that when two quantities are inversely proportional, it means that when one quantity increases, the other decreases in such a way that the product of the two quantities remains constant. This relationship can be expressed as: if \(y\) is inversely proportional to \(x\), there exists a constant \(k\) such that \(y = \frac{k}{x}\).

Setting Up the Initial Relationship

Let's write the initial relationship with a constant of proportionality \(k\): \(y_1 = \frac{k}{x_1}\).

Doubling the Value of \(x\)

Now double the value of \(x\) to get \(x_2 = 2x_1\), and express the new value of \(y\), called \(y_2\), in terms of \(k\) and \(x_2\): \(y_2 = \frac{k}{x_2} = \frac{k}{2x_1}\).

Comparing the New Value of \(y\) with the Initial Value

Comparing the new value of \(y\) to the original, we have \(y_2 = \frac{k}{2x_1} = \frac{1}{2} \cdot \frac{k}{x_1} = \frac{1}{2}y_1\). Thus, when \(x\) is doubled, \(y\) is reduced to half of its original value.

Key concepts

Proportional Relationships

When we talk about proportional relationships in mathematics, we're referring to a connection between two variables where their ratios remain consistent. Specifically, in an inverse proportional relationship, as one variable increases, the other variable decreases at a rate that maintains a constant product between them.

Think of it as a seesaw balance – when one side goes up, the other must go down to keep it balanced, except in this case, the 'balance' is a specific numerical value, the constant of proportionality. For example, in the relationship between time and work done, if two workers take 4 hours to complete a task, a single worker would take 8 hours for the same task, assuming they work at the same rate. The product of 'number of workers' and 'time taken' remains constant, highlighting an inverse proportional relationship.

Constant of Proportionality

The constant of proportionality is the fixed number that relates two variables in a proportional relationship. In the context of inverse proportionality, this constant is the product of the two variables involved. So, for two values, x and y, if they are inversely proportional, their constant of proportionality, denoted as k, would be found by multiplying them together (i.e., k = x * y).

This constant is literally 'constant', meaning that no matter how x and y change, as long as they are inversely proportional, the product will always equal k. In our previous seesaw example, if one worker finishes the task in 8 hours, the constant of proportionality is 8 (time) times 1 (worker), which equals 8. If two workers tackle the job, halving the time taken to 4 hours, the constant remains the same, at 4 (time) times 2 (workers), again equaling 8.

Variables in Proportionality

In the equations representing proportional relationships, the two key players are the variables – typically x and y. When the variables are in inverse proportion, as we increase one, we must decrease the other to keep the constant of proportionality unchanged. These variables are often represented in equations like y = k/x, where y and x are inversely proportional.

If we consider an example where the variable x is the distance you travel and y represents the amount of fuel left in your vehicle, an increase in x (more distance covered) would naturally lead to a decrease in y (less fuel). However, if we were to calculate the constant of proportionality, which could represent the fuel efficiency of the vehicle, it would remain unchanged regardless of the variations in distance and fuel level.