How much will each withdrawal be in the following case:Manu's family has decided that in three months, they will start depositing $ 160 every three months for three years. The account will earn...

How much will each withdrawal be in the following case:

Manu's family has decided that in three months, they will start depositing $ 160 every three months for three years. The account will earn interest 5.7% per annum, compounded quarterly. Then, three months after the last deposit, they plan to withdraw money every three months for equal 24 payments for music lessons

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justaguide | College Teacher | (Level 2) Distinguished Educator

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Manu's family plans to deposit $160 every three months for three years starting after 3 months. The account in which the deposit is made gives a return of 5.7% per annum compounded quarterly.

The total amount in the account after 3 years can be determined using the formula for compound interest with the interest rate for each term equal to `(5.7%)/4 = 0.01425` . There are 11 deposits in all after 3 months each.

After 3 years the amount in the account is equal to:

`160*(1.01425^10 + 1.01425^9 +...1) `

= `160*1*(1.01425^11 - 1)/(1.01425 - 1)`

= `160*(1.01425^11 - 1)/0.01425`

`~~ 1890.92`

After 3 months, this increases to $1917.86.

Now, the funds are withdrawn in 24 equal payments each after 3 months. Let the payment be denoted by P. As the account continues to earn 5.7% p.a compounded quarterly, the withdrawals have to be discounted at the same rate. This gives:

`P + P/1.01425 + ... + P/1.01425^23 = 1917.86`

=> `P*(1 - 1/1.01425^24)/(1 - 1/1.01425) = 1917.86`

=> `P ~~ 93.58`

Each of the 24 withdrawals is equal to $93.58

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