Question
An artist invests in a mutual fund account paying 6% and $5000 more than four times as much in a mutual fund account paying 4%. Her total annual interest income from the investments is $1410. How much does she invest at each rate?
Answers
GPT-4o mini
Let \( x \) be the amount the artist invests at 6%, and let \( y \) be the amount she invests at 4%. From the problem, we know:
1. \( y = 4x + 5000 \) (the amount invested at 4% is $5000 more than four times the amount invested at 6%)
2. The total annual interest from both investments is $1410, represented by the equation:
\[
0.06x + 0.04y = 1410
\]
Now, we can substitute \( y \) from the first equation into the second equation.
Substituting \( y \):
\[
0.06x + 0.04(4x + 5000) = 1410
\]
Expanding the equation:
\[
0.06x + 0.16x + 200 = 1410
\]
Combine like terms:
\[
0.22x + 200 = 1410
\]
Subtracting 200 from both sides gives:
\[
0.22x = 1210
\]
Now, divide both sides by 0.22:
\[
x = \frac{1210}{0.22} \approx 5500
\]
Now we can find \( y \):
\[
y = 4(5500) + 5000 = 22000 + 5000 = 27000
\]
Therefore, the artist invests:
- At 6%: **$5500**
- At 4%: **$27000**
To check the total interest:
- Interest from 6% investment: \( 0.06 \times 5500 = 330 \)
- Interest from 4% investment: \( 0.04 \times 27000 = 1080 \)
- Total interest: \( 330 + 1080 = 1410 \)
Thus, the amounts are confirmed to be correct.
So, the artist invests:
- **$5500** at 6%
- **$27000** at 4%
1. \( y = 4x + 5000 \) (the amount invested at 4% is $5000 more than four times the amount invested at 6%)
2. The total annual interest from both investments is $1410, represented by the equation:
\[
0.06x + 0.04y = 1410
\]
Now, we can substitute \( y \) from the first equation into the second equation.
Substituting \( y \):
\[
0.06x + 0.04(4x + 5000) = 1410
\]
Expanding the equation:
\[
0.06x + 0.16x + 200 = 1410
\]
Combine like terms:
\[
0.22x + 200 = 1410
\]
Subtracting 200 from both sides gives:
\[
0.22x = 1210
\]
Now, divide both sides by 0.22:
\[
x = \frac{1210}{0.22} \approx 5500
\]
Now we can find \( y \):
\[
y = 4(5500) + 5000 = 22000 + 5000 = 27000
\]
Therefore, the artist invests:
- At 6%: **$5500**
- At 4%: **$27000**
To check the total interest:
- Interest from 6% investment: \( 0.06 \times 5500 = 330 \)
- Interest from 4% investment: \( 0.04 \times 27000 = 1080 \)
- Total interest: \( 330 + 1080 = 1410 \)
Thus, the amounts are confirmed to be correct.
So, the artist invests:
- **$5500** at 6%
- **$27000** at 4%
A coin collector has $2.00 in dimes and nickels. She has two more dimes than nickels. How many nickels does she have?
GPT-4o mini
Let \( n \) be the number of nickels the coin collector has. Since she has two more dimes than nickels, the number of dimes can be expressed as \( n + 2 \).
The total value of the nickels and dimes is $2.00, which is equal to 200 cents. The value of the nickels is \( 5n \) cents (since each nickel is worth 5 cents), and the value of the dimes is \( 10(n + 2) \) cents (since each dime is worth 10 cents).
We can set up the following equation based on the total value:
\[
5n + 10(n + 2) = 200
\]
Now, let's solve the equation step by step:
1. Distribute \( 10 \) in the equation:
\[
5n + 10n + 20 = 200
\]
2. Combine like terms:
\[
15n + 20 = 200
\]
3. Subtract \( 20 \) from both sides:
\[
15n = 180
\]
4. Divide by \( 15 \):
\[
n = 12
\]
Thus, the coin collector has **12 nickels**.
To double-check the calculation:
- Number of dimes: \( n + 2 = 12 + 2 = 14 \)
- Value of nickels: \( 12 \times 5 = 60 \) cents
- Value of dimes: \( 14 \times 10 = 140 \) cents
- Total value: \( 60 + 140 = 200 \) cents or $2.00
This confirms that the solution is correct. The collector has **12 nickels**.
The total value of the nickels and dimes is $2.00, which is equal to 200 cents. The value of the nickels is \( 5n \) cents (since each nickel is worth 5 cents), and the value of the dimes is \( 10(n + 2) \) cents (since each dime is worth 10 cents).
We can set up the following equation based on the total value:
\[
5n + 10(n + 2) = 200
\]
Now, let's solve the equation step by step:
1. Distribute \( 10 \) in the equation:
\[
5n + 10n + 20 = 200
\]
2. Combine like terms:
\[
15n + 20 = 200
\]
3. Subtract \( 20 \) from both sides:
\[
15n = 180
\]
4. Divide by \( 15 \):
\[
n = 12
\]
Thus, the coin collector has **12 nickels**.
To double-check the calculation:
- Number of dimes: \( n + 2 = 12 + 2 = 14 \)
- Value of nickels: \( 12 \times 5 = 60 \) cents
- Value of dimes: \( 14 \times 10 = 140 \) cents
- Total value: \( 60 + 140 = 200 \) cents or $2.00
This confirms that the solution is correct. The collector has **12 nickels**.
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