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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 any time, which suggests exponential growth. We can use the formula for exponential growth, which is $A = P \cdot e^{rt}$ , where $A$ is the amount after time $t$ , $P$ is the initial principal balance, $r$ is the rate of growth, and $e$ is the base of the natural logarithm.
Calculate growth rate: Calculate the growth rate based on the information given. $\newline$ We know the initial balance $P$ is $\$1600$ and the balance after one year $t = 1$ is $\$1920$ . We can use these values to find the growth rate $r$ . $\newline$ $\$1920 = \$1600 \times e^{r \times 1}$ $\newline$ To solve for $r$ , we divide both sides by $\$1600$ : $\newline$ $e^r = \frac{\$1920}{\$1600}$ $\newline$ $e^r = 1.2$ $\newline$ Now, we take the natural logarithm of both sides to solve for $r$ : $\newline$ $\$1600$ $1$ $\newline$ $\$1600$ $2$
Calculate natural logarithm: Calculate the natural logarithm of $1.2$ to find the growth rate. $\newline$ $r = \ln(1.2)$ $\newline$ Using a calculator, we find: $\newline$ $r \approx 0.18232$
Use growth rate for balance: Use the growth rate to find the balance after $90$ days. We need to convert $90$ days into years because the growth rate is per year. There are $365$ days in a year, so: $t = \frac{90 \text{ days}}{365 \text{ days/year}} \approx 0.24658 \text{ years}$ Now we can use the exponential growth formula with $P = \$1600$ , $r \approx 0.18232$ , and $t \approx 0.24658$ to find the balance after $90$ days ( $A$ ). $A = \$1600 \times e^{(0.18232 \times 0.24658)}$
Calculate balance after $90$ days: Calculate the balance after $90$ days using the values found. $\newline$ $A \approx \$(1600) \cdot e^{(0.18232 \cdot 0.24658)}$ $\newline$ Using a calculator, we find: $\newline$ $A \approx \$(1600) \cdot e^{(0.04496)}$ $\newline$ $A \approx \$(1600) \cdot 1.04602$ $\newline$ $A \approx \$1673.63$

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