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Part 1
Solve the formula for n.
a equals n minus StartFraction n minus f Over p EndFraction t


Let's solve the given formula for \( n \). The formula is:

\[
a = n - \frac{n - f}{p} \cdot t
\]

### Step 1: Distribute \( t \) on the right side
Distribute the \( t \) across the fraction:

\[
a = n - \frac{(n - f) \cdot t}{p}
\]

### Step 2: Isolate terms involving \( n \)
Move the term involving \( n \) to one side. Start by moving the fraction to the left side:

\[
a + \frac{(n - f) \cdot t}{p} = n
\]

### Step 3: Get rid of the fraction
Multiply both sides by \( p \) to eliminate the denominator:

\[
p \cdot a + (n - f) \cdot t = n \cdot p
\]

### Step 4: Expand and collect like terms
Expand the equation:

\[
p \cdot a + n \cdot t - f \cdot t = n \cdot p
\]

Now, bring all terms involving \( n \) to one side:

\[
n \cdot t - n \cdot p = -p \cdot a - f \cdot t
\]

### Step 5: Factor out \( n \)
Factor \( n \) out of the left side:

\[
n \cdot (t - p) = -p \cdot a - f \cdot t
\]

### Step 6: Solve for \( n \)
Finally, solve for \( n \) by dividing both sides by \( (t - p) \):

\[
n = \frac{-p \cdot a - f \cdot t}{t - p}
\]

So, the solution for \( n \) is:

\[
n = \frac{-p \cdot a - f \cdot t}{t - p}
\]

Would you like any further details or have any questions? Here are some related questions:

1. How do you handle special cases in this equation, such as when \( t = p \)?
2. What happens to the formula when \( f = 0 \)?
3. How can you simplify the expression when \( a = 0 \)?
4. How do you interpret the result in a practical context?
5. How would this formula change if we wanted to solve for \( f \) instead of \( n \)?

**Tip:** Always check the domain of the variables after solving an equation to ensure the solution is valid.

Learn how to solve the equation for n in the formula a = n - (n - f) / p * t with detailed step-by-step solutions. This problem involves algebraic manipulation of fractional equations and is suitable for advanced high school students.

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