Question

If you open a savings account that earns 6.5% simple interest per year, what is the minimum num- ber of years you must wait to triple your balance? Suppose you open another account that earns 6% interest compounded yearly. How many years will it take now to triple your balance?

Answer 1

• 31 years
• 19 years

Step-by-step explanation:

To triple your balance( principal ) using simple interest rate

A = 3 * P = 3P (equation 1)

A = new amount

p = principal

T = x

R = interest rate = 6.5%

to calculate the interest rate =
`(PTR)/(100)` ( equation 2 )

note A = principal + interest rate

A = p +
`(PTR)/(100)` therefore A = p ( 1 +
`(TR)/(100)` ) ( equation 3 )

from (equation 1) A = 3P substitute this into (equation 3)

equation 3 becomes: 3P = P ( 1 +
`(TR)/(100)` ) divide both sides of the equation by P

equation becomes: 3 = ( 1 +
`(T6.5)/(100)` ) therefore 3 = 1 + 0.065T

hence T =
`(3-1)/(0.065)` =
`(2)/(0.065)` = 30.76 ≈ 31 years

To triple your balance using the compound interest

R = 6%

A = 3P

using the compound interest formula

A = P ( 1 +
`(R)/(100)`)^t

3P = P ( 1 +
`(6)/(100)` )^t

3 = ( 1.06 )^t

t =
`(log 3)/(log 1.06)` = 18.85 years ≈ 19 years

Answer 2

6.5% at simple interest requires 30.77 years

6.0% at compounding of 6.5% requires 18.85 years

Step-by-step explanation:

We want a PV of $1 to befome $3 in the future

`1 (1 + r * n) = 3\\(1 + 0.065 * n) = 3\\n = (3 - 1) / 0.065`

n = 30,769

Compounding interest:

`(1+r)^n = 3\\log _(1+r)3 = n\\n = (log 3)/(log (1+r)) \\n = (log 3)/(log (1+0.06))`

n = 18.85417668

n = 18.85