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?

Answer 1

A coin collector has ​$2.00 in dimes and nickels. She has two more dimes than nickels. How many nickels does she​ have?

Answer 2

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%

Answer 3

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**.