How long will it take to have R30 000 for a trip to Europe if I invest R20 000 now at 15% per year, compounded quarterly?

Compound  interest  formula [tex]A=P(1+ \frac{r}{n} )^{nt} [/tex]   Where
A= Future value
P = the Principal (the initial amount of money) 
r = annual interest rate t = time n= number of times compounded in one t






Remark
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r is generally a percentage like 3%, 7% etc and are applied in the formula as 0.03, 0.07..., 

the interest is compounded generally annually (n=1), quarterly (n=4), monthly (n=12), etc...

t is in years,


In our problem:

A= 30 000P =20 000r = 15%=0.15time = t = ?n= 4

applying the formula:

[tex]A=P(1+ \frac{r}{n} )^{nt}\\\\30,000=20,000(1+ \frac{0.15}{4} )^{4t}\\\\ \displaystyle{\frac{30,000}{20,000} =(1.0375)^{4t}[/tex]

[tex] \displaystyle{ 1.5={(1.0375^4)}^t\\\\[/tex]

[tex] 1.5=1.159^t\\\\ \displaystyle { t=log_{1.159}1.5=2.75[/tex]

75% of 12 months is 3/4 of 12 months, which is 9 months

Answer: 2 years, 9 months