9 A loan grows to $3740 after 1 year and $4114 after 2 years with compound interest which is computed annually. Find (a) the interest rate per annum, (b) the original loan. 340^(@)

Question

9 A loan grows to  $3740 after 1 year and  $4114 after 2 years with compound interest which is computed annually. Find (a) the interest rate per annum, (b) the original loan.  340^(@)

Bytelearn · Accepted Answer

Establish Relationship Using Compound Interest Formula: First, let's establish the relationship between the amounts after each year and the interest rate. We can use the compound interest formula, which is A = P(1 + r)^n, where A is the amount of money accumulated after n years, including interest, P is the principal amount (the initial amount of money), r is the annual interest rate (decimal), and n is the number of years the money is invested or borrowed for.Set Up Equations for Each Year: We know the loan amount after 1 year (A1) is $3740, and after 2 years (A2) is $4114. We can set up two equations using the compound interest formula for each year:  For the end of the first year: A1 = P(1 + r)^1  For the end of the second year: A2 = P(1 + r)^2Express P in Terms of A1 and r: We can use the first equation to express P in terms of A1 and r: P = A1 / (1 + r)Substitute P into Second Equation: Now we can substitute this expression for P into the second equation to find r: A2 = (A1 / (1 + r)) * (1 + r)^2Simplify Equation and Solve for r: Simplify the equation by canceling out (1 + r) in the numerator and denominator: A2 = A1 * (1 + r)Calculate r Value: Now we can solve for r by dividing both sides by A1 and then subtracting 1: (1 + r) = A2 / A1 r = (A2 / A1) - 1Express r as Percentage: Plug in the values for A1 and A2 to calculate r: r = (4114 / 3740) - 1 r ≈ 1.10026737967914 - 1 r ≈ 0.10026737967914Find Original Loan Amount: To express r as a percentage, we multiply by 100: r = 0.10026737967914 * 100 r ≈ 10.03%Calculate P Using Values for A1 and r: Now that we have the annual interest rate, we can find the original loan amount (P) using the first year's equation: P = A1 / (1 + r)Substitute the values for A1 and r (as a decimal) to calculate P: P = 3740 / (1 + 0.10026737967914) P ≈ 3740 / 1.10026737967914 P ≈ 3399.97

A loan grows to $\$ 3740$ after $1$ year and $\$ 4114$ after $2$ years with compound interest which is computed annually. Find $\newline$ (a) the interest rate per annum, $\newline$ (b) the original loan.

Full solution

More problems from Compound interest

Related topics

A loan grows to $3740 \$ 3740 $3740 after 111 year and $4114 \$ 4114 $4114 after 222 years with compound interest which is computed annually. Find\newline(a) the interest rate per annum,\newline(b) the original loan.

A loan grows to $\$ 3740$ after $1$ year and $\$ 4114$ after $2$ years with compound interest which is computed annually. Find $\newline$ (a) the interest rate per annum, $\newline$ (b) the original loan.