How Long Will It Be?

Question

If Marsha has $100 to invest at 10% per annum compounded monthly, how long will it be before she has $175? If the compounding is continuously, quarterly and daily, how long will it be?

Answer 1

monthly,
175 = 100(1 + 0.10/12)^(12t)
1.75 = (1.0083333)^(12t)
ln 1.75 = 12t ln 1.0083333
t = (ln 1.75)/(12 ln 1.0083333)
t = 5.62 yr

quarterly,
1.75 = (1.025)^(4t)
t = (ln 1.75)/(4 ln 1.025)
t = 5.67 yr

daily,
1.75 = (1.000273973)^(365t)
t = (ln 1.75)/(365 ln 1.000273973)
t = 5.597 yr

continuously,
1.75 = e^(0.1t)
ln 1.75 = 0.1t
t = (ln 1.75)/0.1 = 5.596 yr.

Answer 2

Answer is 68 months or 5 yrs, 8 months for monthly compounding.

Answer is 23 quarters or 5 yrs, 9 months for quarterly compounding.

Answer 3

175 = 100(1+.1/12)^(12t), compounded monthly
Solve for t,
t = ln1.75/[12*)ln(1+.1/12)] = 5.62 years

Since the compound is calculated only at the end of each month, 5.62 years = 67.44 months. Therefore, 68 months are needed.

Could you finish the remaining part in a similar way?

Answer 4

5.87 years @10% gives $174.97
If compounded quarterly amount would be $178.57
If compounded daily amount would be $179.84
If compounded continuously amount would be $179.86

As you can see, compounding is n't a big deal.