# How long will it take for 500 to accumulate to 2,000 at 12% compounded semi-annually?

Answers: 1

## Answers

How long will it take for 500 to accumulate to 2,000 at 12% compounded semi-annually?...

Answers: 1

How long will it take for 500 to accumulate to 2,000 at 12% compounded semi-annually?...

It will take roughly 11 years, 10 months, and 24 days to accumulate 2,000 at 12% rate.

Step-by-step explanation:

Compound Interest Formula:

Future Value = Present Value (1 + r/n)^(tn)Where:

Future Value = maturity or accumulated value ⇒ 2,000

Present Value = principal value, the basis of computation ⇒ 500

r = rate of interest ⇒ 12% or 0.12

n = compounding period in a year ⇒ 2 (semi-annual)

t = years ⇒ Unknown

x = t

Find the number of years to accumulate 2,000:

Let x = t (time in years)

2,000 = 500 (1 + 0.12/2)²ˣ2,000 = 500 (1 + 0.06)²ˣ

2,000/500 = (1.06)²ˣ

4 = (1.06)²ˣ

To remove the variable x from the exponent of base 1.06, take the natural logarithm of both sides:

ln(1.06²ˣ) = ln(4)2xln(1.06) = ln(4)

2xln(1.06)/2ln(1.06) = ln(4)/2ln(1.06)

x = ln(2)/ln(1.06)

Use a calculator to find ln(2) and ln(1.06):

ln(2) = 0.693

ln(1.06) = 0.0583

x ≈ 0.693/0.0583

x ≈ 11.89 or 11.9 years or 11 ⁹/₁₀ yearsConvert 11 ⁹/₁₀ to years,months,days:

11 years + (9/10)(12 months)

11 years, 10 4/5 months

11 years, 10 months and (4/5)(30 days)

11 years, 10 months and 24 days.

x ≈ 11 years, 10 months, 24 days.