Question

5.Kate plans to retire at age 60. At that time, she wants to have enough to invest in a travel account so that she can go on a world trip every four years until she reaches 76, i.e. at ages 64, 68, 72 and 76. Each trip will cost $10138. If the interest rate is 4.9% p.a. compounded quarterly, how much should she deposit in the travel account at age 60.

(Give your answer to the nearest cent, omitting the dollar sign.)

6.An investment of $27001, earning compound interest, grows by $2284 in one year. At this rate of growth, how long will it take for the original investment to double?

(Express your answer in years, correct to one decimal point.)

Answer 1

5.

Each trip will cost $10138.

Interest rate p.a. is 4.9% i.e. quarterly interest rate is 1.225% (4.9/4).

Kate ( at the age of years) needs to get $10138 for every16 quarters (i.e 4 years X 4 quarters). First after 16 quarter , Second at after 32 quarters, third after 48 quarters and fourth after 64 quarters.

So, Aggregate of Present values of all these four Amounts gives us the amount required by Kate at the age of 60 to invest in travel account.

Present Value (PV) Formula:

PV = P/(1+r)^-n or PV (1+r)ⁿ = P

Where, P is the periodic payment i.e. $10138; n= number of periods; r is the rate of interst per period

X₁ (1+0.01225)¹⁶ = 10138; X₁ = 10138/1.2151 =8343.35

X₂ (1+0.01225)³² = 10138; X₂ = 10138/1.4764 =6866.70

X₃ (1+0.01225)⁴⁸ = 10138; X₃ = 10138/1.7940 =5651.06

X₄ (1+0.01225)⁶⁴ = 10138; X₄ = 10138/2.1798 =4650.89

Amount to be invested by Kate:                              $25512.00

6.

Original Investment = $27001

Interest compounded for one year and interest earned = $2284

i.e. 27001(1+r) = 27001+2284

1+r = 29285/27001

1+r = 1.0846

r = 0.0846 i.e. r = 8.46%

At this growth rate, it will double in:

27001(1+r)ⁿ = 27001+27001

(1+r)ⁿ = 2

(1+0.0846)ⁿ = 2

By applying trail and error method,

1.0846*1.0846*1.0846*......so on to get 2

we will get 2 i.e. amount will bw doubled in 9 years