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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.

The variable c is directly proportional to the variable b. Write an equation to represent this relationship. State what happens to: a) c if b is doubled, b) b if c is tripled, c) c if b is multiplied by 2/3, d) c if 4 is added to b.

Learn how to solve a problem involving direct proportionality between variables c and b. This detailed solution explores the effects of doubling, tripling, and other changes on the relationship between these variables. Suitable for students in Grades 8-10.

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