IB Maths AA HL 1.4 Financial applications

[Maximum number: 12]

Phil takes out a bank loan of $\$ 150000$ to buy a house, at an annual interest rate of 3.5 %. The interest is calculated at the end of each year and added to the amount outstanding.

(a)

Find the amount Phil would owe the bank after 20 years. Give your answer to the nearest dollar.

To pay off the loan, Phil makes annual deposits of $\$ P$ at the end of every year in a savings account, paying an annual interest rate of 2 %. He makes his first deposit at the end of the first year after taking out the loan.

[ 3 ]

$150000 \times 1.035^{20}$
= $298468

Note: Only accept answers to the nearest dollar. Accept $\$ 298469$ .

(b)

Show that the total value of Phil's savings after 20 years is $\frac{\left(1.02^{20}-1\right) P}{(1.02-1)}$ .

[ 3 ]

attempt to look for a pattern by considering 1 year, 2 years etc
recognising a geometric series with first term P and common ratio 1.02

EITHER

P+1.02 P+\ldots+1.02^{19} P\left(=P\left(1+1.02+\ldots+1.02^{19}\right)\right)

OR
explicitly identify $u_{1}=P, r=1.02$ and n=20 (may be seen as $S_{20}$ ).

THEN
$S_{20}=\frac{\left(1.02^{20}-1\right) P}{(1.02-1)}$

(c)

Given that Phil's aim is to own the house after 20 years, find the value for P to the nearest dollar.

David visits a different bank and makes a single deposit of $\$ Q$ , the annual interest rate being 2.8 \%.

[ 3 ]

$\quad 24.297 \ldots P=298468$
P=12284

Note: Accept answers which round to 12284.

(d)

David wishes to withdraw $\$ 5000$ at the end of each year for a period of n years. Show that an expression for the minimum value of Q is

\frac{5000}{1.028}+\frac{5000}{1.028^{2}}+\ldots+\frac{5000}{1.028^{n}}

[ 3 ]

METHOD 1

METHOD 2
the initial value of the first withdrawal is $\frac{5000}{1.028}$
A1
the initial value of the second withdrawal is $\frac{5000}{1.028^{2}}$
R1
the investment required for these two withdrawals is $\frac{5000}{1.028}+\frac{5000}{1.028^{2}}$
R1
$Q=\frac{5000}{1.028}+\frac{5000}{1.028^{2}}+\ldots+\frac{5000}{1.028^{n}}$
AG

SL 1.4—Financial applications question 1

Question (a)

Question (b)

Question (c)

Question (d)