11.1-Introduction to Direct and Inverse Proportions

Direct and inverse proportions are mathematical relationships between two variables where a change in one variable affects the other in a specific way. These concepts are fundamental in understanding how quantities relate to each other in various real-life situations, such as speed, time, distance, and other physical phenomena.

1. Direct Proportion

Two quantities are said to be in direct proportion if an increase in one quantity leads to an increase in the other, or a decrease in one leads to a decrease in the other. In other words, the ratio between the two quantities remains constant.

Mathematically, we say that quantities $x$ and $y$ are directly proportional if:

Formula: $x \propto y$ or $x = ky$, where $k$ is a constant.

For example, if the number of workers increases, the amount of work done increases proportionally, provided the amount of work per worker remains the same. This is a direct proportion.

Example: If the time taken to travel a fixed distance is directly proportional to the speed of the vehicle, then:

If $x$ is the time taken and $y$ is the speed, then: $x \propto \frac{1}{y}$, where $k$ is the constant of proportionality.

2. Inverse Proportion

Two quantities are said to be in inverse proportion if an increase in one quantity leads to a decrease in the other, or vice versa, such that the product of the two quantities remains constant.

Mathematically, we say that quantities $x$ and $y$ are inversely proportional if:

Formula: $x \propto \frac{1}{y}$ or $xy = k$, where $k$ is a constant.

For example, if the number of workers increases, the time taken to complete the same amount of work decreases, provided the amount of work per worker remains constant. This is an inverse proportion.

Example: If the time taken to travel a fixed distance is inversely proportional to the speed, then:

If $x$ is the time and $y$ is the speed, then: $xy = k$, where $k$ is the constant of proportionality.

3. Solving Problems Involving Proportions

In both direct and inverse proportions, we can solve problems by first identifying the type of relationship (direct or inverse) and then using the corresponding formula to find the unknown quantity. The constant of proportionality, $k$, can be determined from known values in the problem.

For direct proportion problems, we use the formula $x = ky$. For inverse proportion problems, we use $xy = k$ to find the missing quantity. It is important to carefully read the problem and identify the variables that change together.