Question
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Part 1
Susan Marciano invested part of her $ 12 comma 000 bonus in a fund that paid a 9 % profit and invested the rest in stock that suffered a 4 % loss. Find the amount of each investment if her overall net profit was $ 300.
Question content area bottom
Part 1
The amount invested at 9 % is $
enter your response here.
The amount invested in stock is $
enter your response here
Part 1
Susan Marciano invested part of her $ 12 comma 000 bonus in a fund that paid a 9 % profit and invested the rest in stock that suffered a 4 % loss. Find the amount of each investment if her overall net profit was $ 300.
Question content area bottom
Part 1
The amount invested at 9 % is $
enter your response here.
The amount invested in stock is $
enter your response here
Answers
GPT-4o mini
To find out how much Susan Marciano invested in each option, we can set up the problem using algebra.
Let \( x \) be the amount invested in the fund that paid a 9% profit. Then, the amount invested in stock that suffered a 4% loss would be \( 12000 - x \).
The total profit from the investment in the fund is:
\[
0.09x
\]
The total loss from the investment in stock is:
\[
0.04(12000 - x) = 4800 - 0.04x
\]
The overall profit is the profit from the fund minus the loss from the stock:
\[
0.09x - (4800 - 0.04x) = 300
\]
Now, we can simplify this equation:
\[
0.09x - 4800 + 0.04x = 300
\]
Combining like terms gives:
\[
0.09x + 0.04x - 4800 = 300
\]
\[
0.13x - 4800 = 300
\]
Next, add 4800 to both sides:
\[
0.13x = 5100
\]
Now, divide by 0.13:
\[
x = \frac{5100}{0.13} \approx 39230.77
\]
This value seems incorrect because it exceeds her total investment. Let's quickly verify the calculations to isolate errors.
We need to set up the equation again step by step:
1. Given the total investment of \( \$12,000 \):
2. Profit from the fund: \( 0.09x \)
3. Loss from the stock: \( 0.04(12000 - x) = 4800 - 0.04x \)
4. Total profit from both investments gives us:
\[
0.09x - (4800 - 0.04x) = 300
\]
Simplifying:
\[
0.09x + 0.04x - 4800 = 300
\]
\[
0.13x - 4800 = 300
\]
\[
0.13x = 5100
\]
\[
x = \frac{5100}{0.13} \approx 39230.77 (seems incorrect)
\]
Let's switch gears.
---
1. Instead,
Let \( y \) be the amount invested at a 9% profit and \( z \) be the amount invested at a 4% loss:
\[
y + z = 12000
\]
\[
0.09y - 0.04z = 300
\]
Substituting \( z = 12000 - y \) into the second equation:
\[
0.09y - 0.04(12000 - y) = 300
\]
\[
0.09y - 480 + 0.04y = 300
\]
\[
0.13y - 480 = 300
\]
\[
0.13y = 780
\]
\[
y = \frac{780}{0.13} = 6000
\]
Put back into the investment:
\[
z = 12000 - 6000 = 6000
\]
Thus, the amounts are:
- Amount invested at 9%: **$6000**
- Amount invested in stock: **$6000**
Answers:
- The amount invested at 9% is **$6000**.
- The amount invested in stock is **$6000**.
Let \( x \) be the amount invested in the fund that paid a 9% profit. Then, the amount invested in stock that suffered a 4% loss would be \( 12000 - x \).
The total profit from the investment in the fund is:
\[
0.09x
\]
The total loss from the investment in stock is:
\[
0.04(12000 - x) = 4800 - 0.04x
\]
The overall profit is the profit from the fund minus the loss from the stock:
\[
0.09x - (4800 - 0.04x) = 300
\]
Now, we can simplify this equation:
\[
0.09x - 4800 + 0.04x = 300
\]
Combining like terms gives:
\[
0.09x + 0.04x - 4800 = 300
\]
\[
0.13x - 4800 = 300
\]
Next, add 4800 to both sides:
\[
0.13x = 5100
\]
Now, divide by 0.13:
\[
x = \frac{5100}{0.13} \approx 39230.77
\]
This value seems incorrect because it exceeds her total investment. Let's quickly verify the calculations to isolate errors.
We need to set up the equation again step by step:
1. Given the total investment of \( \$12,000 \):
2. Profit from the fund: \( 0.09x \)
3. Loss from the stock: \( 0.04(12000 - x) = 4800 - 0.04x \)
4. Total profit from both investments gives us:
\[
0.09x - (4800 - 0.04x) = 300
\]
Simplifying:
\[
0.09x + 0.04x - 4800 = 300
\]
\[
0.13x - 4800 = 300
\]
\[
0.13x = 5100
\]
\[
x = \frac{5100}{0.13} \approx 39230.77 (seems incorrect)
\]
Let's switch gears.
---
1. Instead,
Let \( y \) be the amount invested at a 9% profit and \( z \) be the amount invested at a 4% loss:
\[
y + z = 12000
\]
\[
0.09y - 0.04z = 300
\]
Substituting \( z = 12000 - y \) into the second equation:
\[
0.09y - 0.04(12000 - y) = 300
\]
\[
0.09y - 480 + 0.04y = 300
\]
\[
0.13y - 480 = 300
\]
\[
0.13y = 780
\]
\[
y = \frac{780}{0.13} = 6000
\]
Put back into the investment:
\[
z = 12000 - 6000 = 6000
\]
Thus, the amounts are:
- Amount invested at 9%: **$6000**
- Amount invested in stock: **$6000**
Answers:
- The amount invested at 9% is **$6000**.
- The amount invested in stock is **$6000**.