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

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 2

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

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.