IB Questionbank

User interface language: English | Español

Date	November 2017	Marks available	3	Reference code	17N.2.AHL.TZ0.H_12
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Show that	Question number	H_12	Adapted from	N/A

Question

Phil takes out a bank loan of $150 000 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.

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.

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

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

[3]

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

[3]

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

[3]

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]

d.i.

Hence or otherwise, find the minimum value of $Q$ that would permit David to withdraw annual amounts of $5000 indefinitely. Give your answer to the nearest dollar.

[3]

d.ii.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

$150000 \times {1.035^{20}}$ (M1)(A1)

$= \$ 298468$ A1

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

[3 marks]

attempt to look for a pattern by considering 1 year, 2 years etc (M1)

recognising a geometric series with first term $P$ and common ratio 1.02 (M1)

EITHER

$P + 1.02P + \ldots + {1.02^{19}}P{\text{ }}\left( { = P(1 + 1.02 + \ldots + {{1.02}^{19}})} \right)$ A1

explicitly identify ${u_1} = P,{\text{ }}r = 1.02$ and $n = 20$ (may be seen as ${S_{20}}$ ). A1

THEN

${s_{20}} = \frac{{({{1.02}^{20}} - 1)P}}{{(1.02 - 1)}}$ AG

[3 marks]

$24.297 \ldots P = 298468$ (M1)(A1)

$P = 12284$ A1

Note: Accept answers which round to 12284.

[3 marks]

METHOD 1

$Q({1.028^n}) = 5000(1 + 1.028 + {1.028^2} + {1.028^3} + \ldots + {1.028^{n - 1}})$ M1A1

$Q = \frac{{5000\left( {1 + 1.028 + {{1.028}^2} + {{1.028}^3} + ... + {{1.028}^{n - 1}}} \right)}}{{{{1.028}^n}}}$ A1

$= \frac{{5000}}{{1.028}} + \frac{{5000}}{{{{1.028}^2}}} + \ldots + \frac{{5000}}{{{{1.028}^n}}}$ AG

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

[3 Marks]

d.i.

sum to infinity is $\frac{{\frac{{5000}}{{1.028}}}}{{1 - \frac{1}{{1.028}}}}$ (M1)(A1)

$= 178571.428 \ldots$

so minimum amount is $178572 A1

Note: Accept answers which round to $178571 or $178572.

[3 Marks]

d.ii.

Examiners report

[N/A]

d.i.

[N/A]

d.ii.

Syllabus sections

Topic 1—Number and algebra » SL 1.4—Financial apps – compound interest, annual depreciation

Show 52 related questions

18M.1.SL.TZ2.T_4a:
Complete the second column of the table by giving one example of a number from each set.
22M.2.SL.TZ2.2d.i:
Calculate the value of $Σ_{n = 1}^{10} (625 \times 1 . 064^{(n - 1)})$ .
SPM.2.SL.TZ0.1a.i:
Find the repayment made each quarter.
18M.2.SL.TZ1.T_3b:
Calculate 14 000 USD in INR.
SPM.2.AHL.TZ0.3a.ii:
Find the total amount paid for the car.
22M.2.SL.TZ2.2d.ii:
Describe what the value in part (d)(i) represents in this context.
22M.1.SL.TZ2.13b:
For option 2, find the minimum value of $x$ that Juliana would need to invest each year. Give your answer to the nearest dollar.
22M.1.SL.TZ2.13a:
For option 1, determine the minimum amount Juliana would need to invest. Give your answer to the nearest dollar.
21N.1.SL.TZ0.10b.i:
Find the amount Raul and Rosy will still owe the bank at the end of the first $10$ years.
17N.2.AHL.TZ0.H_12c:
Given that Phil’s aim is to own the house after 20 years, find the value for $P$ to the nearest dollar.
18M.1.SL.TZ2.T_4b:
Josh states: “Every integer is a natural number”.

Write down whether Josh’s statement is correct. Justify your answer.
19N.1.SL.TZ0.T_8a:
Calculate the value of Siân’s investment after four years. Give your answer correct to two decimal places.
SPM.2.SL.TZ0.1a.iii:
Find the interest paid on the loan.
SPM.2.SL.TZ0.1b.i:
Find the amount to be borrowed for this option.
SPM.2.SL.TZ0.1a.ii:
Find the total amount paid for the car.
SPM.2.AHL.TZ0.3a.iii:
Find the interest paid on the loan.
SPM.2.AHL.TZ0.3a.i:
Find the repayment made each quarter.
EXM.1.SL.TZ0.1a:
Calculate the value of her savings after 10 years.
SPM.2.SL.TZ0.1b.ii:
Find the annual interest rate, $r$ .
21M.2.SL.TZ1.3b:
Find the minimum value of $m$ , where $m \in ℕ$ .
18M.2.SL.TZ1.T_3d:
Harinder chose option B. At the end of five years, Harinder converted this investment back to USD. The exchange rate, at that time, was 1 USD = 67.16 INR.

Calculate how much more money, in USD, Harinder earned by choosing option B instead of option A.
17N.2.AHL.TZ0.H_12d.ii:
Hence or otherwise, find the minimum value of $Q$ that would permit David to withdraw annual amounts of $5000 indefinitely. Give your answer to the nearest dollar.
20N.1.SL.TZ0.T_8a:
Calculate the value of Imon’s investment after $5$ years.
SPM.2.AHL.TZ0.3b.i:
Find the amount to be borrowed for this option.
SPM.2.AHL.TZ0.3b.ii:
Find the annual interest rate, $r$ .
19N.1.SL.TZ0.T_8b:
After the four-year period, Siân withdraws $40\,000$ AUD from her savings account and uses this money to buy a car. It is known that the car will depreciate at a rate of $18$ % per year.

The value of the car will be $2500$ AUD after $t$ years.

Find the value of $t$ .
EXN.1.SL.TZ0.9b:
Find an approximation for the real interest rate for the money invested in the account.
SPM.2.SL.TZ0.1c:
State which option Bryan should choose. Justify your answer.
SPM.2.AHL.TZ0.3c:
State which option Bryan should choose. Justify your answer.
17N.2.AHL.TZ0.H_12a:
Find the amount Phil would owe the bank after 20 years. Give your answer to the nearest dollar.
21M.2.SL.TZ1.3d.ii:
the annual interest rate of the loan.
18M.2.SL.TZ1.T_3c:
Calculate the amount of this investment, in INR, in this account after five years.
21M.1.SL.TZ2.10b:
Tommaso wants to invest his money in an account such that his investment will increase to $1.5$ times the initial amount in $5$ years. Assume the account pays a nominal annual interest of $r %$ compounded quarterly.

Determine the value of $r$ .
SPM.2.AHL.TZ0.3d:
Bryan chooses option B. The car dealership invests the money Bryan pays as soon as they receive it.

If they invest it in an account paying 0.4 % interest per month and inflation is 0.1 % per month, calculate the real amount of money the car dealership has received by the end of the 6 year period.
SPM.2.SL.TZ0.1d:
Bryan’s car depreciates at an annual rate of 25 % per year.

Find the value of Bryan’s car six years after it is purchased.
21M.2.SL.TZ1.3c:
Write down the amount of the loan.
21M.2.SL.TZ1.3e:
Find the amount of Daisy’s final payment.
21M.2.SL.TZ1.3d.i:
the amount of interest paid by Daisy.
21M.2.SL.TZ1.3f:
Find how much money Daisy saved by making one final payment after $5$ years.
21M.2.SL.TZ1.3a:
Calculate the value of Daisy’s investment after $2$ years.
21M.1.SL.TZ2.10a:
Calculate the amount Pietro will have in his account after $5$ years. Give your answer correct to $2$ decimal places.
21M.1.AHL.TZ2.5a:
Estimate the value of Roger’s laptop after $5$ years.
21M.1.AHL.TZ2.5c:
Comment on the validity of your answer to part (b).
18M.2.SL.TZ1.T_3a:
Calculate the value of r.
17N.2.AHL.TZ0.H_12d.i:
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

$\frac{{5000}}{{1.028}} + \frac{{5000}}{{{{1.028}^2}}} + \ldots + \frac{{5000}}{{{{1.028}^n}}}$ .
EXM.1.SL.TZ0.1b:
The rate of inflation during this 10 year period is 1.5% per year.

Calculate the real value of her savings after 10 years.
EXN.1.SL.TZ0.9a:
Find the value of her investment after a period of $5$ years.
EXN.1.SL.TZ0.9c:
Hence find the real value of Sophia’s investment at the end of $5$ years.
20N.1.SL.TZ0.T_8b:
At the end of the $5$ years, Imon withdrew $x SGD$ from the fixed deposit account and reinvested this into a super-savings account with a nominal annual interest rate of $5.7 %$ , compounded half-yearly.

The value of the super-savings account increased to $20 000 SGD$ after $18$ months.

Find the value of $x$ .
21M.1.AHL.TZ2.5b:
Find the value of $k$ .
21N.1.SL.TZ0.10a:
Find the amount they will pay the bank each month.
21N.1.SL.TZ0.10b.ii:
Using your answers to parts (a) and (b)(i), calculate how much interest they will have paid in total during the first $10$ years.

Hide related questions

Topic 1—Number and algebra

Question

Markscheme

Examiners report

Syllabus sections

View options