To determine how long it would take for [tex]$75 to double its value at a 5% simple interest rate, we can use the simple interest formula and solve for the time.
Here is a detailed, step-by-step solution:
1. Identify the given values:
- Initial amount (principal), \( P \) = $[/tex]75
- Interest rate, [tex]\( R \)[/tex] = 5% per year = 0.05 (as a decimal)
- Final amount, [tex]\( A \)[/tex] = 2 * [tex]$75 = $[/tex]150 (since we want the amount to double)
2. Write the formula for simple interest:
Simple interest [tex]\( I \)[/tex] can be calculated using:
[tex]\[
I = PRT
\][/tex]
where, [tex]\( I \)[/tex] is the interest earned, [tex]\( P \)[/tex] is the initial amount, [tex]\( R \)[/tex] is the interest rate, and [tex]\( T \)[/tex] is the time in years.
3. Express the final amount in terms of initial amount and interest:
The final amount [tex]\( A \)[/tex] is given by the sum of the initial amount and the interest earned:
[tex]\[
A = P + I
\][/tex]
Substituting the interest formula into the equation, we get:
[tex]\[
A = P + PRT
\][/tex]
4. Rearrange to solve for time [tex]\( T \)[/tex]:
Substitute the given values of [tex]\( A \)[/tex], [tex]\( P \)[/tex], and [tex]\( R \)[/tex]:
[tex]\[
150 = 75 + (75 \times 0.05 \times T)
\][/tex]
Simplify the equation:
[tex]\[
150 = 75 + 3.75T
\][/tex]
Subtract 75 from both sides:
[tex]\[
75 = 3.75T
\][/tex]
Divide both sides by 3.75:
[tex]\[
T = \frac{75}{3.75} = 20
\][/tex]
Therefore, it would take 20 years for $75 to double its value at a 5% simple interest rate.
