Compound Interest/Future Value

[Admin]

#15266

Compound Interest/Future Value

FV =PV(1 + r)ⁿ

FV = Future Value
PV = Present Value

A = P(1 + r)ⁿ

A = total of principal and interest
P = principal, original amount borrowed or invested
r = interest rate as a decimal ie. divide the percent by a hundred
n = length of the loan or investment

[Admin]

#18592

How long will it take an investment of $10 000 to double at an interest rate of 7.5% p.a. compounded monthly?

20 000 = 10 000(1 + 0.00625)ⁿ
2 = (1 + 0.00625)ⁿ

then start ‘guessing and checking’ values for n to make it work

n = 100: (1 + 0.00625)¹⁰⁰ = 1.86 too low
n = 150: (1 + 0.00625)¹⁵⁰ = 2.54 too high
n = 125: (1 + 0.00625)¹²⁵ = 2.17 too high
n = 115: (1 + 0.00625)¹¹⁵ = 2.04 too high
n = 110: (1 + 0.00625)¹¹⁰ = 1.98 too low
n = 111: (1 + 0.00625)¹¹¹ = 1.997 too low
n = 112: (1 + 0.00625)¹¹² = 2.00 too low

n = 112 months

$$\frac{112}{12}$$
= 9.3 years

[Admin]

#18593

Find the amount $1000 amounts to if it is invested at 10% pa compounded annually for 5 years.
A = 1000(1 + 0.1)5	10% = 10 ÷ 100 = 0.1 as a decimal, r = 0.1, n = 5, P = 1000
A = $1610.51	calculate

[Admin]

#18594

Find the amount $1000 amounts to if it is invested at 12% pa compounded monthly for 5 years.
A = 1000(1.01)60 A = $1816.70	Be careful of the time periods here. We need to change 12% pa into per month so we divide by 12 to get 1% per month. And we need to change 5 years into months by multiplying by 12 to get 60 months. r = 0.01, n = 60, P = 1000

[Admin]

#18595

Find the amount of interest on $6000 if it is invested at 8% pa, for ten years, compounded quarterly.
A = 6000(1.02)40 A = $13248.24	8% pa is 8 ÷ 4 = 2% per quarter and ten years is 10 × 4 = 40 quarters P = 6000, r = 0.02, n = 40
∴ the interest will be $13248.24 – $6000 = $7248.24	A is the amount of the investment = principle + interest, so to find the interest, take the original investment away from the total amount

[Admin]

#18596

What amount of principal must be invested at 5.8% pa for it to grow to $15 000 in 8 years if: a. Simple interest applies? answer to the nearest cent b. Compound interest applies? answer to the nearest cent
a. P + (P × 5.8% × 8) = 15 000 P + (P × 0.058 × 8) = 15 000 P + 0.464P = 15 000 1.464P = 15 000 $$\frac{\cancel{1.464}P}{\cancel{1.464}}=\frac{15,000}{1.46}$$ P = $10 245.90164 ∴ $10,245.90	let the principle be P interest = PRN Interest + Principle = $15 000 PRN + P = 15 000 form an equation and use it to solve for P
b. 15000 = P(1+ 0.058)8 15000 = P(1.058)8 $$\frac{{15,000}}{{{{(1.058)}^8}}} = \frac{{P{{\cancel{(1.058)}^8}}}}{{{{\cancel{(1.058)}^8}}}}$$ P = $9554.4595959 ∴ $9554.46	A = P(1 + r)n where P is unknown, r = 0.058, n = 8 solve the equation for P

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