Question

A bank offers all savings accounts 5% interest compounded annually. If one account has a principal of $100 and another

has a principal of $1,000, which account will double first? Explain how you arrived at your answer. Answer in complete
sentences

Answer 1

From the calculation for both the accounts, it is clear that both the account double in the same time period of 14 years 26 days .

Explanation:

Given as :

The principal for the first account = p = $100

The rate of interest = r = 5% compounded annually

The account gets double , so, Amount = A = $200

Let the time after which account gets double = t years

So, From Compound Interest method

Amount = Principal ×
`(1+(\trxtrm rate)/(100))^(\textrm time)`

As amount is double its principal

So, A = 2 × $100 = $200

Or, A = p ×
`(1+(\trxtrm r)/(100))^(\textrm t)`

Or, $200 = $100 ×
`(1+(\trxtrm 5)/(100))^(\textrm t)`

Or,
`(200)/(100)` =
`(1.05)^(\textrm t)`

Or, 2 =
`(1.05)^(\textrm t)`

Taking Log both side

`Log_(10)2` =
`Log_(10)(1.05)^(t)`

Or, 0.3010 = t
`Log_(10)1.05`

Or, 0.3010 = t × 0.0211

∴ t =
`(0.3010)/(0.0211)`

I.e t = 14.26

So, The time period to get account double is 14 years 26 days

Again

Amount = Principal ×
`(1+(\trxtrm rate)/(100))^(\textrm time)`

Or, A = p ×
`(1+(\trxtrm r)/(100))^(\textrm t)`

As amount is double its principal

So, A = 2 × $1000 = $2000

Or, $2000 = $1000 ×
`(1+(\trxtrm 5)/(100))^(\textrm t)`

Or,
`(2000)/(1000)` =
`(1.05)^(\textrm t)`

Or, 2 =
`(1.05)^(\textrm t)`

Taking Log both side

`Log_(10)2` =
`Log_(10)(1.05)^(t)`

Or, 0.3010 = t
`Log_(10)1.05`

Or, 0.3010 = t × 0.0211

∴ t =
`(0.3010)/(0.0211)`

I.e t = 14.26

So, The time period to get account double is 14 years 26 days

Hence From the calculation for both the accounts, it is clear that both the account double in the same time period of 14 years 26 days . Answer