To solve for \( c \) in the equation \( \frac{k(m+c)}{m} = \frac{4}{x} \), follow these steps:

### Step 1: Eliminate the denominators
Multiply both sides of the equation by \( m \) to get rid of the denominator on the left-hand side:

\[
k(m+c) = \frac{4m}{x}
\]

### Step 2: Expand the left-hand side
Distribute \( k \) on the left-hand side:

\[
km + kc = \frac{4m}{x}
\]

### Step 3: Isolate the term with \( c \)
Subtract \( km \) from both sides to isolate the \( kc \) term:

\[
kc = \frac{4m}{x} - km
\]

### Step 4: Solve for \( c \)
Now, divide both sides by \( k \):

\[
c = \frac{\frac{4m}{x} - km}{k}
\]

This simplifies to:

\[
c = \frac{4m - kmx}{kx}
\]

So, the final expression for \( c \) is:

\[
c = \frac{m(4 - kx)}{kx}
\]

Would you like further clarification on any of the steps?

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Here are five related questions to expand on this topic:

1. How would the equation change if \( m \) were a constant value?
2. Can you solve this equation for \( m \) instead of \( c \)?
3. How do you handle similar equations if fractions or higher powers are involved?
4. What happens to the solution when \( k = 0 \)?
5. How does this result change if \( k = x \)?

**Tip:** Always check if multiplying through by variables involves any conditions, such as ensuring the variable is not zero to avoid division by zero errors.

Learn how to rearrange and solve for c in the equation k(m+c)/m = 4/x. This detailed guide covers key algebraic techniques, such as distributing, isolating variables, and simplifying equations. Suitable for students in Grades 9-11.

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