Question

at time 0, cheryl deposits x into a bank account that credits interest at an annual effective rate of . at time 3, gomer deposits 1,000 into a different bank account that credits simple interest at an annual rate of y%. at time 5, the annual forces of interest on the two accounts are equal, and gomer's account has accumulated to z.

Answer 1

To solve this problem, we can set up equations that represent the growth of Cheryl's account and Gomer's account over time. We then find the value of x in terms of y using the equation (1 + r)^5 * x = 1000 + 30y.

Step-by-step explanation:

To solve this problem, we need to set up equations that represent the growth of Cheryl's account and Gomer's account over time. Let x represent the initial deposit in Cheryl's account.

After 5 years, Cheryl's account will have grown to (1 + r)^5 * x, where r is the annual effective interest rate.

Gomer's account, on the other hand, grows at a simple interest rate of y%. After 3 years, Gomer's account will have accumulated to z = 1000 + (1000 * y/100 * 3) = 1000 + 30y.

Since the annual forces of interest on both accounts are equal at time 5, we can set up the equation (1 + r)^5 * x = 1000 + 30y and solve for x in terms of y.

By rearranging the equation, we have x = (1000 + 30y) / (1 + r)^5. This equation relates the initial deposit x to the simple interest rate y for Gomer's account and the annual effective interest rate r for Cheryl's account.

Answer 2

The question requires understanding of simple and compound interest calculations. The simple interest of a $5,000 loan at 6% for three years is $900, and the required interest rate to earn $500 from $10,000 over five years is 1%. To have $10,000 in ten years at 10% interest compounded annually, one must deposit approximately $3855.43.

Step-by-step explanation:

The student is asking about simple and compound interest calculations, which is a typical high school-level mathematics question. The annual effective rate is used for the compound interest, while the annual rate denoted by y% is used for simple interest. To find the total amount of interest from a $5,000 loan after three years with a simple interest rate of 6%, you would use the formula:

I = P × r × t

Where I is the interest, P is the principal amount ($5,000), r is the annual interest rate (6% or 0.06), and t is the time in years (3).

Interest = $5,000 × 0.06 × 3 = $900

To determine the interest rate charged if you receive $500 in simple interest on a loan that you made for $10,000 for five years, rearrange the formula to solve for r:

r = I / (P × t)

r = $500 / ($10,000 × 5)

Interest rate = 0.01 or 1%

For the scenario of depositing an amount to get $10,000 after ten years at 10% interest compounded annually, you would use the compound interest formula:

P = A / (1 + r)^n

Where P is the principal, A is the amount of money accumulated after n years, including interest (here, $10,000), r is the annual interest rate (10% or 0.1), and n is the number of years (10).

Principal = $10,000 / (1 + 0.1)^10

Principal = $10,000 / (1.1)^10

Principal = $10,000 / 2.5937424601

Principal ≈ $3855.43