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.

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Grade 8

Compound interest

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Q. 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.

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 $A_1$ is $\$3740$ , and after $2$ years $A_2$ is $\$4114$ . We can set up two equations using the compound interest formula for each year: $\newline$ For the end of the first year: $\newline$ $A_1 = P(1 + r)^1$ $\newline$ For the end of the second year: $\newline$ $A_2 = P(1 + r)^2$
Express $P$ in Terms of $A_1$ and $r$ : We can use the first equation to express $P$ in terms of $A_1$ and $r$ : $P = \frac{A_1}{(1 + r)}$
Substitute $P$ into Second Equation: Now we can substitute this expression for $P$ into the second equation to find $r$ : $A_2 = \left(\frac{A_1}{1 + r}\right) \cdot (1 + r)^2$
Simplify Equation and Solve for $r$ : Simplify the equation by canceling out $(1 + r)$ in the numerator and denominator: $\newline$ $A_2 = A_1 \times (1 + r)$
Calculate r Value: Now we can solve for $r$ by dividing both sides by $A_1$ and then subtracting $1$ :
$(1 + r) = \frac{A_2}{A_1}$
$r = \left(\frac{A_2}{A_1}\right) - 1$
Express $r$ as Percentage: Plug in the values for $A_1$ and $A_2$ to calculate $r$ :
$r = \frac{4114}{3740} - 1$
$r \approx 1.10026737967914 - 1$
$r \approx 0.10026737967914$
Find Original Loan Amount: To express $r$ as a percentage, we multiply by $100$ : $r = 0.10026737967914 \times 100$ $r \approx 10.03\%$
Calculate $P$ Using Values for $A_1$ and $r$ : Now that we have the annual interest rate, we can find the original loan amount ( $P$ ) using the first year's equation: $\newline$ $P = \frac{A_1}{(1 + r)}$
Calculate $P$ Using Values for $A_1$ and $r$ : Now that we have the annual interest rate, we can find the original loan amount ( $P$ ) using the first year's equation: $\newline$ $P = \frac{A_1}{(1 + r)}$ Substitute the values for $A_1$ and $r$ (as a decimal) to calculate $P$ : $\newline$ $P = \frac{3740}{(1 + 0.10026737967914)}$ $\newline$ $P \approx \frac{3740}{1.10026737967914}$ $\newline$ $P \approx 3399.97$