To determine how long it will take an investment of R1,300 to grow to R2,756 with an annual simple interest rate of 8%, we can use the simple interest formula:
[tex]\[ I = P + P \cdot r \cdot t \][/tex]
Where:
- [tex]\( I \)[/tex] is the future value of the investment,
- [tex]\( P \)[/tex] is the principal amount,
- [tex]\( r \)[/tex] is the annual interest rate (expressed as a decimal),
- [tex]\( t \)[/tex] is the time in years.
We need to isolate [tex]\( t \)[/tex] (the time in years). First, let's rewrite the equation:
[tex]\[ I = P(1 + r \cdot t) \][/tex]
Rearranging to solve for [tex]\( t \)[/tex]:
[tex]\[ \frac{I}{P} = 1 + r \cdot t \][/tex]
[tex]\[ \frac{I}{P} - 1 = r \cdot t \][/tex]
[tex]\[ t = \frac{\frac{I}{P} - 1}{r} \][/tex]
Plug in the given values:
- [tex]\( P = 1,300 \)[/tex]
- [tex]\( I = 2,756 \)[/tex]
- [tex]\( r = 0.08 \)[/tex] (8% expressed as a decimal)
Let's substitute these values into the formula:
[tex]\[ t = \frac{\frac{2,756}{1,300} - 1}{0.08} \][/tex]
First, calculate [tex]\( \frac{2,756}{1,300} \)[/tex]:
[tex]\[ \frac{2,756}{1,300} = 2.12 \][/tex]
Now, compute [tex]\( 2.12 - 1 \)[/tex]:
[tex]\[ 2.12 - 1 = 1.12 \][/tex]
Then, divide by the interest rate [tex]\( r \)[/tex]:
[tex]\[ t = \frac{1.12}{0.08} \][/tex]
[tex]\[ t = 14 \][/tex]
Therefore, it will take 14 years for an investment of R1,300 at an 8% annual simple interest rate to grow to R2,756.
