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6. The amount of money, [tex]P[/tex], in a bank account earning simple interest can be determined using the formula [tex]P = I(1 + rt)[/tex], where [tex]I[/tex] is the initial amount, [tex]r[/tex] is the rate of interest earned, and [tex]t[/tex] is the time in years. Which of the following formulas shows the value of the rate [tex]r[/tex] in terms of the account balance [tex]P[/tex], initial amount [tex]I[/tex], and time [tex]t[/tex]?

A) [tex]r = \frac{P - I}{It}[/tex]
B) [tex]r = \frac{P + I}{It}[/tex]
C) [tex]r = \frac{PI}{I + t}[/tex]
D) [tex]r = \frac{P - I}{I - t}[/tex]



Answer :

GinnyAnswer
To solve for the rate of interest [tex]\( r \)[/tex] in terms of the account balance [tex]\( P \)[/tex], the initial amount [tex]\( I \)[/tex], and time [tex]\( t \)[/tex] in years, we start from the given formula for simple interest:

[tex]\[ P = I (1 + rt) \][/tex]

We need to isolate [tex]\( r \)[/tex] in this equation. Here are the detailed steps:

1. Start with the given formula:
[tex]\[ P = I (1 + rt) \][/tex]

2. Divide both sides by [tex]\( I \)[/tex] to isolate the term containing [tex]\( r \)[/tex]:
[tex]\[ \frac{P}{I} = 1 + rt \][/tex]

3. Subtract 1 from both sides to further isolate the term containing [tex]\( r \)[/tex]:
[tex]\[ \frac{P}{I} - 1 = rt \][/tex]

4. Recognize that [tex]\((\frac{P}{I} - 1)\)[/tex] is a common denominator and can be simplified:
[tex]\[ \frac{P - I}{I} = rt \][/tex]

5. Divide both sides by [tex]\( t \)[/tex] to solve for [tex]\( r \)[/tex]:
[tex]\[ r = \frac{\frac{P - I}{I}}{t} \][/tex]

6. Simplify the fraction by multiplying the numerator and the denominator by [tex]\( I \)[/tex]:
[tex]\[ r = \frac{P - I}{I t} \][/tex]

Thus, the correct formula to solve for the rate [tex]\( r \)[/tex] in terms of the account balance [tex]\( P \)[/tex], the initial amount [tex]\( I \)[/tex], and time [tex]\( t \)[/tex] is:

[tex]\[ r = \frac{P - I}{I t} \][/tex]

The corresponding choice is:

A) [tex]\( r = \frac{P - I}{I t} \)[/tex]

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