key term - Directly Proportional

Definition

A direct proportion is a relationship between two variables where one variable increases or decreases at a constant rate as the other variable changes. When two quantities are directly proportional, an increase in one quantity results in a proportional increase in the other quantity, and a decrease in one quantity results in a proportional decrease in the other quantity.

5 Must Know Facts For Your Next Test

1. In a direct proportion, the ratio between the two variables remains constant as the variables change.
2. The equation for a direct proportion is $y = kx$, where $k$ is the constant of proportionality.
3. Direct proportions are often used to model real-world situations, such as the relationship between the cost and quantity of an item, or the relationship between the distance traveled and the time taken.
4. Solving proportions involves finding the missing value in a proportion by using the property that the cross-products of a proportion are equal.
5. Direct proportions are an important concept in solving applications and word problems involving ratios, rates, and scaling.

Review Questions

• Explain how the constant of proportionality relates to a direct proportion.
  • The constant of proportionality, $k$, is the value that represents the rate at which one variable changes in relation to the other variable in a direct proportion. It is the value that makes the equation $y = kx$ true, where $x$ and $y$ are the two variables in the direct proportion. The constant of proportionality is the same for any pair of corresponding values of $x$ and $y$ in the direct proportion, and it can be found by dividing any value of $y$ by the corresponding value of $x$.
• Describe how to solve a proportion problem involving direct proportions.
  • To solve a proportion problem involving direct proportions, you can use the property that the cross-products of a proportion are equal. This means that if $a/b = c/d$, then $ad = bc$. In the context of a direct proportion, you can set up a proportion with the known values and then solve for the missing value. For example, if you know that 3 apples cost $\$1.50$, and you want to find the cost of 5 apples, you can set up the proportion $3/1.50 = 5/x$ and solve for $x$ to find the cost of 5 apples.
• Analyze how direct proportions can be used to model real-world situations and solve related problems.
  • Direct proportions are widely used to model various real-world situations, such as the relationship between the cost and quantity of an item, the relationship between the distance traveled and the time taken, or the relationship between the volume of a container and the amount of liquid it can hold. By recognizing the direct proportion in these situations, you can set up and solve proportions to find missing values, make predictions, or scale measurements. For example, if you know that 10 gallons of paint can cover 200 square feet, you can use a direct proportion to find how much paint is needed to cover a 400 square foot area. This allows you to apply the concept of direct proportions to solve practical problems encountered in everyday life and various fields of study.

Related terms

Proportion:

A proportion is a statement of equality between two ratios, indicating that the relationship between the quantities in one ratio is the same as the relationship between the quantities in the other ratio.

Inverse Proportion: An inverse proportion is a relationship between two variables where one variable increases as the other variable decreases, and vice versa, at a constant rate.

Constant of Proportionality: The constant of proportionality is the value that represents the rate at which one variable changes in relation to the other variable in a direct proportion.