Question 1209489: Musu Thorsen bought shares worth $10,000 in her employer’s profit-sharing scheme. When the share price rose by $10, she kept 1,000 shares and sold the rest for $11,000. How can you describe her share purchases?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! We can break down Musu Thorsen's share purchases and sales to describe the situation systematically.
### Let:
- \( x \) = the original price per share.
- \( n \) = the total number of shares Musu purchased.
### Step 1: Express the total value of her purchase.
The total value of her purchase was $10,000. Thus:
\[
n \cdot x = 10,000
\]
\[
n = \frac{10,000}{x}
\]
### Step 2: Calculate the number of shares sold.
After the share price rose by $10, the new price per share became \( x + 10 \). Musu kept 1,000 shares, so the number of shares she sold was:
\[
n - 1,000 = \frac{10,000}{x} - 1,000
\]
### Step 3: Express the total value of shares sold.
The total value of the shares sold was $11,000. Therefore:
\[
\left( \frac{10,000}{x} - 1,000 \right)(x + 10) = 11,000
\]
### Step 4: Solve for \( x \), the original share price.
Simplify and solve this equation.
The original price per share (\( x \)) has two solutions:
\[
x = -\frac{11}{2} + \frac{\sqrt{521}}{2} \quad \text{and} \quad x = -\frac{\sqrt{521}}{2} - \frac{11}{2}.
\]
Since share prices must be positive, we consider only the first solution:
\[
x = -5.5 + \frac{\sqrt{521}}{2}.
\]
Let me compute the approximate numerical value of \( x \).
The original price per share was approximately **$5.91**.
### Description of her share purchases:
1. **Total number of shares purchased:**
\[
n = \frac{10,000}{x} \approx \frac{10,000}{5.91} \approx 1,692 \, \text{shares}.
\]
2. **Shares sold:**
She kept 1,000 shares, so the number of shares sold was:
\[
n - 1,000 \approx 1,692 - 1,000 = 692 \, \text{shares}.
\]
3. **New price per share after the $10 increase:**
\[
x + 10 \approx 5.91 + 10 = 15.91 \, \text{dollars per share}.
\]
4. **Value of shares sold:**
The 692 shares she sold at $15.91 each yielded approximately $11,000, as stated.
### Summary:
Musu initially bought about **1,692 shares** at **$5.91** per share. She kept **1,000 shares** and sold the remaining **692 shares** after the price rose to **$15.91**, earning $11,000 from the sale.
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