To solve the equation \( A = b\left(\frac{1}{c} - \frac{1}{d}\right) \) for \( c \), follow these steps:

### Step 1: Expand the expression
First, expand the right-hand side of the equation:
\[
A = b\left(\frac{1}{c} - \frac{1}{d}\right)
\]
\[
A = b\left(\frac{d - c}{cd}\right)
\]
### Step 2: Isolate the fraction involving \( c \)
Divide both sides by \( b \):
\[
\frac{A}{b} = \frac{d - c}{cd}
\]

### Step 3: Multiply both sides by \( cd \) to eliminate the denominator:
\[
Acd = b(d - c)
\]

### Step 4: Distribute \( b \) on the right-hand side:
\[
Acd = bd - bc
\]

### Step 5: Move terms involving \( c \) to one side
Add \( bc \) to both sides:
\[
Acd + bc = bd
\]

### Step 6: Factor out \( c \)
\[
c(Ad + b) = bd
\]

### Step 7: Solve for \( c \)
Divide both sides by \( (Ad + b) \):
\[
c = \frac{bd}{Ad + b}
\]

Thus, the solution for \( c \) is:
\[
c = \frac{bd}{Ad + b}
\]

Would you like any more details or clarifications?

Here are 5 related questions to explore:
1. How do you solve for \( d \) in the same equation?
2. What happens if \( A = 0 \)? How does that affect \( c \)?
3. Can this formula be applied in real-world problems involving proportions?
4. What is the significance of isolating \( c \) in terms of its role in the equation?
5. How does the equation change if \( b \) is negative?

**Tip:** Always factor out the variable you are solving for to make the isolation step easier.

Explore how to solve the rational equation A = b(1/c - 1/d) for c with a step-by-step breakdown. This detailed solution involves algebraic manipulation and rational expressions, perfect for students in Grades 10-12.

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