Question

First solve the problem, and then enter your answer on the grid provided on the answer sheet. The instructions for entering your answers follow. The variables \(m\) and \(n\) have a directly proportional relationship given by the equation \(m=k n\), where \(k\) is a constant of proportionality. When \(m=10, n=2\). What will be the value of \(n\) if \(m\) equals \(38 ?\)

Short answer

The value of \(n\) is 7.6 when \(m = 38\).

Step-by-step solution

Identify Given Information

We are given that the relationship between \(m\) and \(n\) is directly proportional, which is represented by the equation \(m = k n\). Additionally, we know that when \(m = 10\), \(n = 2\).

Calculate the Constant of Proportionality \(k\)

Using the information \(m = 10\) and \(n = 2\), substitute these values into the equation \( m = k n \) to find \(k\).\[10 = k \times 2 \] Solve for \(k\): \[ k = \frac{10}{2} = 5 \]

Write the Equation With the Constant

Now that we have \(k = 5\), substitute it back into the original equation to express \(m\) in terms of \(n\): \[ m = 5n \]

Solve for \(n\) When \(m = 38\)

Substitute \(m = 38\) into the equation \(m = 5n\) to find \(n\).\[38 = 5n \] Solve for \(n\): \[ n = \frac{38}{5} = 7.6 \]

Check Your Work

Double-check your calculations to ensure accuracy. Substituting \(n = 7.6\) back into \(m = 5n\) should yield \(m = 38\), confirming that the solution is correct.

Key concepts

Constant of Proportionality

In a directly proportional relationship, the constant of proportionality plays a crucial role. It is the factor by which one variable is multiplied to obtain the other. Think of it as a bridge connecting the two variables in a consistent way. In our exercise, the equation given is \( m = k n \), where \( m \) and \( n \) are directly proportional, and \( k \) is the constant of proportionality.
In simple terms, \( k \) is what you multiply \( n \) with to get \( m \). When you have data points, like \( m = 10 \) and \( n = 2 \), you can determine \( k \) by rearranging the equation to \( k = \frac{m}{n} \).
This gives you \( k = \frac{10}{2} = 5 \). Hence, every unit of \( n \) contributes 5 times to \( m \). This constant remains unchanged across the relationship, allowing you to predict one variable when you know the other.

Linear Equations

The equation \( m = k n \) is an example of a linear equation. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. The key characteristic of a linear equation is its graph: a straight line.
In our case, \( m = 5n \) represents a line with a slope of 5. This implies that as \( n \) changes, \( m \) also changes at a consistent rate, which is dictated by the constant \( 5 \).

• The slope of the line is the constant of proportionality \( k \).
• The y-intercept is zero, meaning the line passes through the origin (0,0).

Understanding this relationship as a linear equation allows you to see how changes in \( n \) affect \( m \) and vice versa. Each increase or decrease in \( n \) by one unit results in a 5 unit change in \( m \).

Variable Manipulation

Variable manipulation is a crucial skill in algebra that allows you to solve for unknowns. It involves techniques such as rearranging equations or substituting known values.
In this exercise, once we know the equation is \( m = 5n \), we substitute \( m = 38 \) to find \( n \).
The equation becomes \( 38 = 5n \). To isolate \( n \), we divide both sides by 5:
\[ n = \frac{38}{5} = 7.6 \]
Here is how variable manipulation helps:

• Allows you to "move" elements of the equation to where they're needed for solving.
• Helps you understand relationships in which one element changes based on another.

By mastering variable manipulation, you gain the power to tackle any algebraic problem by simplifying and reorganizing terms to find solutions effectively.