Find y for a 20.12% increase in damage per bullet when compared to 100 bullets for y.

We will use the given variables: y= bullets fired x = 145 z = 100 Total damage = [(y-18)*122.5]+2025 Total damage/y = Damage per bullet

To solve for \( y \) given the conditions, let's break down the problem step by step.

First, we'll express the total damage per bullet for both scenarios (100 bullets and \( y \) bullets) and set up an equation to find \( y \).

### Step 1: Calculate the Damage per Bullet for 100 Bullets

Given:
- \( y = 100 \)
- \( \text{Total damage} = [(100 - 18) * 122.5] + 2025 \)

Calculate the total damage for 100 bullets:

\[
\text{Total damage}_{100} = (82 \times 122.5) + 2025
\]

\[
\text{Total damage}_{100} = 10045 + 2025
\]

\[
\text{Total damage}_{100} = 12070
\]

Now, calculate the damage per bullet for 100 bullets:

\[
\text{Damage per bullet}_{100} = \frac{12070}{100} = 120.7
\]

### Step 2: Calculate the Damage per Bullet for \( y \) Bullets

Given that the damage per bullet increases by 20.12% when using \( y \) bullets, we express this increase as:

\[
\text{Damage per bullet}_{y} = 120.7 \times (1 + 0.2012) = 120.7 \times 1.2012 = 145
\]

### Step 3: Set up the Equation for Total Damage for \( y \) Bullets

We need to find \( y \) such that the damage per bullet equals 145:

\[
\text{Total damage}_{y} = [(y - 18) \times 122.5] + 2025
\]

The damage per bullet for \( y \) bullets is:

\[
\frac{\text{Total damage}_{y}}{y} = 145
\]

### Step 4: Solve for \( y \)

Set up the equation:

\[
\frac{[(y - 18) \times 122.5] + 2025}{y} = 145
\]

Multiply both sides by \( y \):

\[
[(y - 18) \times 122.5] + 2025 = 145y
\]

Expand and simplify:

\[
122.5y - 2205 + 2025 = 145y
\]

Combine like terms:

\[
122.5y - 180 = 145y
\]

Isolate \( y \):

\[
-180 = 145y - 122.5y
\]

\[
-180 = 22.5y
\]

Solve for \( y \):

\[
y = \frac{-180}{22.5}
\]

\[
y = -8
\]

Since \( y \) should be a positive number, it indicates an error in our problem setup or calculations. Let's verify the given conditions and equations again. Assuming there might be a logical or typographical error, please check if any conditions have been misinterpreted or if additional context is needed.

Do you want me to review the steps again or provide additional context for clarification? 

Would you like to go through the detailed steps for a better understanding? 

### Relative Questions

1. What could be the potential sources of error in the calculation?
2. How does the increase in damage per bullet affect the total damage calculation?
3. What are the implications if \( y \) turns out to be a negative number?
4. Can we verify the initial conditions and given variables again for accuracy?
5. How do we interpret the 20.12% increase in a real-world scenario?
6. Is there an alternative method to solve for \( y \)?
7. Can we use graphing techniques to visualize the solution?
8. How would changes in other parameters, like the initial damage or bullets fired, affect the outcome?

### Tip

Always double-check the problem statement and conditions to ensure accuracy before solving complex equations.

Learn how to calculate a 20.12% increase in damage per bullet compared to 100 bullets. Solve for y using algebraic equations and percentage increase formulas. Suitable for high school students and above.

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