The time required for the given investment of $5000 to grow to $7400 at an interest rate of 7.5% per year, compounded quarterly is 5 years 3 months i.e., t = 5.27 years.
The formula for calculating the compound interest is
[tex]A = P(1 + \frac{r}{n} )^n^t[/tex]
Where A: Future Amount; P: Current investment; r: Interest rate; n: The number of times the amount is compounded; t: The amount of time
It is given that,
Current investment P = $5000
Future amount A = $7400
Interest rate r = 7.5% = 0.075
The amount is compounded quarterly. So, n = 4
On substituting all these in the formula, we get
[tex]A = P(1 + \frac{r}{n} )^n^t[/tex]
⇒ 7400 = 5000(1 + 0.075/4)^(4t)
⇒ 7400/5000 = (1 + 0.01875)^(4t)
⇒ 1.48 = (1.08175)^(4t)
⇒ (1.08175)^(4t) = 1.48
On applying logarithm on both sides, we get
ln(1.08175)^(4t) = ln(1.48)
⇒ 4t ln(1.08175) = ln(1.48)
⇒ 4t = ln(1.48)/ln(1.08175)
⇒ t = ln(1.48)/4ln(1.08175)
∴ t = 5.27 years
Thus, the time required for the given investment is 5.27 years which is 5 years 3 months.
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