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The balance of a certain loan increases at a rate that is proportional at any time to the balance at that time.
The loan balance is
$1600 initially, and it is
$1920 after one year ( 365 days).
What is the balance of the loan after 90 days?
Choose 1 answer:
(A)
$1529.66
(B)
$1673.57
(C)
$1678.90

The balance of a certain loan increases at a rate that is proportional at any time to the balance at that time. $\newline$ The loan balance is $\$ 1600$ initially, and it is $\$ 1920$ after one year ( $365$ days). $\newline$ What is the balance of the loan after $90$ days? $\newline$ Choose $1$ answer: $\newline$ (A) $\$ 1529.66$ $\newline$ (B) $\$ 1673.57$ $\newline$ (C) $\$ 1678.90$

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Understanding integers

Full solution

Q. The balance of a certain loan increases at a rate that is proportional at any time to the balance at that time.

\newline

The loan balance is

\$ 1600

initially, and it is

\$ 1920

after one year (

365

days).

\newline

What is the balance of the loan after

90

days?

\newline

Choose

1

answer:

\newline

(A)

\$ 1529.66

\newline

(B)

\$ 1673.57

\newline

(C)

\$ 1678.90

Understand problem type: Understand the problem and determine the type of growth. The problem states that the loan balance increases at a rate proportional to the balance at that time. This indicates that the growth is exponential. We are given the initial balance ( $1600) and the balance after one year ($ $1920$ ). We need to find the balance after $90$ days.
Calculate growth rate: Calculate the growth rate. $\newline$ To find the growth rate, we can use the formula for exponential growth: $A = P \cdot e^{rt}$ , where $A$ is the amount after time $t$ , $P$ is the initial amount, $r$ is the rate of growth, and $e$ is the base of the natural logarithm. We can rearrange this formula to solve for $r$ when $t$ is $1$ year ( $365$ days). $\newline$ $A$ $0$ $\newline$ $A$ $1$ $\newline$ $A$ $2$ $\newline$ Taking the natural logarithm of both sides, we get: $\newline$ $A$ $3$ $\newline$ $A$ $4$
Calculate actual rate: Calculate the actual growth rate. $\newline$ Now we will calculate the value of $r$ using the natural logarithm of $1.2$ . $\newline$ $r = \ln(1.2) / 365$ $\newline$ $r \approx 0.001833 / 365$ $\newline$ $r \approx 0.00502$ (rounded to five decimal places)
Calculate balance after $90$ days: Calculate the balance after $90$ days. $\newline$ Now that we have the growth rate, we can find the balance after $90$ days using the same exponential growth formula. $\newline$ $A = 1600 \times e^{r\times90}$ $\newline$ $A = 1600 \times e^{0.00502\times90}$
Perform calculation: Perform the calculation for the balance after $90$ days. $\newline$ $A = 1600 \cdot e^{0.00502\cdot90}$ $\newline$ $A \approx 1600 \cdot e^{0.4518}$ $\newline$ $A \approx 1600 \cdot 1.571$ $\newline$ $A \approx 2513.6$ $\newline$ This answer does not match any of the options provided, which indicates a math error has occurred.

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