Question

A loan of $36,000 is made at 6.75% interest, compounded annually. After how many years will the amount due reach $57,000 or more?

Answer 1

After 7.04 years the amount will reach $57,000 or more

Explanation:

The rule of the compound interest is
`A=P(1+(r)/(n))^(nt)` , where

• A is the new value

• P is the initial value

• r is the rate in decimal

• n is the period of the time

• t is the time

∵ A loan of $36,000 is made at 6.75% interest, compounded annually

∴ P = 36,000

∴ r = 6.75% = 6.75 ÷ 100 = 0.0675

∴ n = 1 ⇒ compounded annually

∵ The amount after t years will reach $57,000 or more

∴ A = 57,000

→ To find t substitute these values in the rule above

∵ 57,000 = 36,000
`(1+(0.0675)/(1))^((1)(t))`

∴ 57,000 = 36,000
`(1.0675)^(t)`

→ Divide both sides by 36,000

∵
`(19)/(12)` =
`(1.0675)^(t)`

→ Insert ㏒ in both sides

∴ ㏒(
`(19)/(12)` ) = ㏒
`(1.0675)^(t)`

→ Remember ㏒
`a^(n)` = n ㏒(
`a`)

∵ ㏒(
`(19)/(12)` ) = t ㏒(1.0675)

→ Divide both sides by ㏒(1.0675)

∴ 7.035151337 = t

∴ t ≅ 7.04

∴ After 7.04 years the amount will reach $57,000 or more