Let's break down this problem step-by-step.

Problem Statement

1. The variable $c$ is directly proportional to the variable $b$. We need to:
   • Write an equation representing this relationship.
   • Determine how $c$ changes in different scenarios:
     • (a) If $b$ is doubled.
     • (b) If $c$ is tripled.
     • (c) If $b$ is multiplied by $\frac{2}{3}$.
     • (d) If 4 is added to $b$.

Solution

1. Direct Proportionality Equation

Since $c$ is directly proportional to $b$, we can express this relationship with the equation: $c = k \cdot b$ where $k$ is the constant of proportionality.

2. Analyzing the Scenarios

Let's apply each scenario to this equation.

(a) If $b$ is doubled: $b' = 2b$ $c' = k \cdot b' = k \cdot (2b) = 2 \cdot (k \cdot b) = 2c$ So, if $b$ is doubled, $c$ will also be doubled.

(b) If $c$ is tripled: $c' = 3c$ If we look at our proportionality equation $c = k \cdot b$: $3c = 3(k \cdot b) = k \cdot (3b)$ So, $b$ must be tripled to make $c$ tripled. Therefore, $b$ should be tripled.

(c) If $b$ is multiplied by $\frac{2}{3}$: $b' = \frac{2}{3}b$ $c' = k \cdot b' = k \cdot \left(\frac{2}{3}b\right) = \frac{2}{3} \cdot (k \cdot b) = \frac{2}{3}c$ So, if $b$ is multiplied by $\frac{2}{3}$, $c$ will also be multiplied by $\frac{2}{3}$.

(d) If 4 is added to $b$: $b' = b + 4$ $c' = k \cdot b' = k \cdot (b + 4) = k \cdot b + k \cdot 4 = c + 4k$ So, if 4 is added to $b$, $c$ will increase by $4k$ (not simply adding 4 to $c$).

Would you like any more details on these solutions or have any other questions?

Additional Questions

1. How do you determine the constant of proportionality $k$ in real-world scenarios?
2. What are the differences between direct and inverse proportionality?
3. How does the value of $k$ affect the relationship between $c$ and $b$?
4. Can direct proportionality involve negative values for $k$?
5. How can we represent inverse proportionality using an equation?

Tip

When dealing with proportional relationships, changes in one variable directly impact the other. It's important to understand how multiplication, division, addition, and subtraction affect these relationships.