Question

Beth saves $2,500 a year from age 25 until age 34 (inclusive) and invests the money in an account earning ten percent annually. Beth stops investing at age 34 but does not withdraw the accumulation until age 65. In contrast, Bill saves $2,500 a year from age 35 until age 65 inclusively and invests in a similar account to Beth, earning ten percent annually. Because Bill saved significantly more than Beth, he will have accumulated significantly more than her at age 65.
True or False?

Answer 1

Answer:

False

Bill will have less than Beth

Explanation:

let us compare the amounts accumulated by the two individuals at age 65 to see which is more.

Money accumulated by Bill at age 65:

Since bill saves $2500 a year for 30 years,  amount saved at the end of 30 years = 2500 X 30 = $75000.

Money accumulated by Beth at age 65:

Beth saves $2,500 a year from age 25 until age 34: Amount saved = 2500 X 9 = $22500

Invests the money in an account earning ten percent annually for 31 years.

Assuming it was at a simple interest rate,

The interest at the end of the 31st year will be:

$Interest = \frac{P\times R\times Time}{100}= \frac{22500\times 10\times 31}{100}= 69,750$

Therefore total amount at age 65 = $22, 500 + 69750 = $92,250.

∴ This shows that Bill will have less money than Beth. Primarily due to the fact that he started investing at a lot later time than Beth.

Answer 2

Answer: TRUE

Explanation:

We need to compare the Future Value of Beth's investment at the age 65 with the Future value of Bill's investment at the age of 65 in order to determine if Bill will have accumulated more money than Beth at the age of 65.

Beth's Investment

Beth started by saving $2500 a year from the age of 25 until the age of 34. This tell us that Beth saved for 9 years (from 25 years to 34 years, start counting from 25 to 34) and Beth's savings at the age of 34 were $22500 (9 years x $2500).

Beth Invested all her savings ($22500) at the age of 34, Interest rate is at 10% compounded Annually. The period of this investment is 30 (from Age 34 to Age 65 start counting from 35 to 65)

Savings amount at the age of 34 = $22500

Period (n)= 30 years

Interest rate (r) = 10%

Future Value = Savings amount (1 + r)^n

Future Value = $22500(1 + 0.10)^30

Future Value = 392611.5105

Bill's investment

Bill started investing at the age of 34. Bill pays $2500 a year for 30 years (from the age of 34 to until the age of 65, start counting from 35 to 65). Bill's investment is an annuity because he makes repetitive yearly payments. The interest rate 10% Compounded annually.

Payments = $2500

Interest rate = 10%

period (n) = 30 years

Future Value of an Annuity = Payments x ((1 + r)^n - 1)/r

Future Value of an Annuity = $2500 x ((1 + 0.10)^30 - 1)/0.10

Future Value of an Annuity = $411235.05673.

True : Bill accumulates $411235.05673 and Beth accumulates $392611.5105, therefore Bill accumulates more in his investment than Beth