IB Questionbank

Updates to Questionbank

User interface language: English | Español

Date	May 2021	Marks available	2	Reference code	21M.2.SL.TZ1.7
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 1
Command term	Determine	Question number	7	Adapted from	N/A

Question

Two friends Amelia and Bill, each set themselves a target of saving $$ 20 000$ . They each have $$ 9000$ to invest.

Amelia invests her $$ 9000$ in an account that offers an interest rate of $7 %$ per annum compounded annually.

A third friend Chris also wants to reach the $$ 20 000$ target. He puts his money in a safe where he does not earn any interest. His system is to add more money to this safe each year. Each year he will add half the amount added in the previous year.

Find the value of Amelia’s investment after $5$ years to the nearest hundred dollars.

[3]

a.i.

Determine the number of years required for Amelia’s investment to reach the target.

[2]

a.ii.

Bill invests his $$ 9000$ in an account that offers an interest rate of $r %$ per annum compounded monthly, where $r$ is set to two decimal places.

Find the minimum value of $r$ needed for Bill to reach the target after $10$ years.

[3]

b.

Show that Chris will never reach the target if his initial deposit is $$ 9000$ .

[5]

c.i.

Find the amount Chris needs to deposit initially in order to reach the target after $5$ years. Give your answer to the nearest dollar.

[3]

c.ii.

Markscheme

EITHER

$9000 \times {(1 + \frac{7}{100})}^{5}$ (A1)

$12622.965 \dots$ (A1)

OR

$n = 5$
$I % = 7$
$PV = \mp 9000$
$P/Y = 1$
$C/Y = 1$ (A1)
$\pm 12622.965 \dots$ (A1)

THEN

$($) 12600$ A1

[3 marks]

a.i.

EITHER

$9000 {(1 + \frac{7}{100})}^{x} = 20000$ (A1)

OR

$I % = 7$
$PV = \mp 9000$
$FV = \pm 20000$
$P/Y = 1$
$C/Y = 1$ (A1)

THEN

$= 12$ (years) A1

[2 marks]

a.ii.

METHOD 1

attempt to substitute into compound interest formula (condone absence of compounding periods) (M1)

$9000 {(1 + \frac{r}{100 \times 12})}^{12 \times 10} = 20000$

$8.01170 \dots$ (A1)

$r = 8.02 (%)$ A1

METHOD 2

$n = 10$
$PV = \pm 9000$
$FV = \mp 20000$
$P/Y = 1$
$C/Y = 12$
$r = 8.01170 \dots$ (M1)(A1)

Note: Award M1 for an attempt to use a financial app in their technology, award A1 for $(r =) 8.01170 \dots$

$r = 8.02 (%)$ A1

[3 marks]

b.

recognising geometric series (seen anywhere) (M1)

$r = \frac{4500}{9000} (= \frac{1}{2})$ (A1)

EITHER

considering $S_{\infty}$ (M1)

$\frac{9000}{1 - 0.5} (= 18000)$ A1

correct reasoning that $18000 < 20000$ R1

Note: Accept $S_{\infty} < 20000$ only if $S_{\infty}$ has been calculated.

OR

considering $S_{n}$ for a large value of $n, n \geq 80$ (M1)

Note: Award M1 only if the candidate gives a valid reason for choosing a value of $n$ , where $50 \leq n < 80$ .

correct value of $S_{n}$ for their $n$ A1

valid reason why Chris will not reach the target, which involves their choice of $n$ , their value of $S_{n}$ and Chris’ age OR using two large values of $n$ to recognize asymptotic behaviour of $S_{n}$ as $n \to \infty$ . R1

Note: Do not award the R mark without the preceding A mark.

THEN

Therefore, Chris will never reach the target. AG

[5 marks]

c.i.

recognising geometric sum M1

$\frac{u_{1} (1 - 0 . 5^{5})}{0.5} = 20000$ (A1)

$10322.58 \dots$

$($) 10323$ A1

[3 marks]

c.ii.

Examiners report

[N/A]

a.i.

[N/A]

a.ii.

[N/A]

b.

[N/A]

c.i.

[N/A]

c.ii.

Syllabus sections

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

Show 31 related questions

22M.2.SL.TZ1.1c:
Find the interest David will earn over the $10$ years.
18M.1.SL.TZ2.T_5a.ii:
Write down the amount of interest he earned after 5 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.
17M.1.SL.TZ2.T_14a:
Write down the amount of money Jashanti saves per month.
17N.1.SL.TZ0.T_9a:
Calculate the original price of the bicycle.
SPM.2.SL.TZ0.3a:
Find the value of r, giving your answer to four significant figures.
17N.2.AHL.TZ0.H_12b:
Show that the total value of Phil’s savings after 20 years is $\frac{{({{1.02}^{20}} - 1)P}}{{(1.02 - 1)}}$ .
17N.1.SL.TZ0.T_9b:
Calculate the difference between the original price of the bicycle and the total amount Juan will pay.
22M.2.SL.TZ1.1b:
Find the value of $r$ required so that the amount in David’s account after $10$ years will be equal to the amount in Sam’s account.
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.
EXN.2.SL.TZ0.7c.i:
Find the value of $N$ .
EXN.2.SL.TZ0.7b:
Given that $J_{n}$ follows a geometric sequence, state the value of the common ratio, $r$ .
EXN.2.SL.TZ0.7d:
Find Jane’s total earnings at the start of her $10$ th year of employment. Give your answer correct to the nearest dollar.
21M.2.SL.TZ1.7c.ii:
Find the amount Chris needs to deposit initially in order to reach the target after $5$ years. Give your answer to the nearest dollar.
21M.2.SL.TZ1.7c.i:
Show that Chris will never reach the target if his initial deposit is $$ 9000$ .
18M.1.SL.TZ2.T_5b:
Karl decides to donate this interest to a charity in France. The charity receives 170 euros (EUR). The exchange rate is 1 USD = t EUR.

Calculate the value of t.
SPM.2.SL.TZ0.3b:
Laurie makes no further deposits to or withdrawals from the account.

Find the year in which the amount of money in Laurie’s account will become double the amount she invested.
EXN.2.SL.TZ0.7a:
Show that $H_{n} = 2400 n + 67 600$ .
17N.2.AHL.TZ0.H_12a:
Find the amount Phil would owe the bank after 20 years. Give your answer to the nearest dollar.
EXN.2.SL.TZ0.7c.ii:
For the value of $N$ found in part (c) (i), state Helen’s annual salary and Jane’s annual salary, correct to the nearest dollar.
22M.2.SL.TZ1.1a:
Find the amount that Sam will have in his account after $10$ years.
22M.2.SL.TZ2.3a:
Find Gemma’s annual salary for the year 2021, to the nearest dollar.
19M.2.SL.TZ2.T_6a:
Determine the amount that he will have in his account after 3 years. Give your answer correct to two decimal places.
19M.2.SL.TZ2.T_6c:
After $m$ complete months Tommaso will, for the first time, have enough money in his account to buy the bicycle.

Find the value of $m$ .
20N.1.SL.TZ0.T_8a:
Calculate the value of Imon’s investment after $5$ years.
21M.2.SL.TZ1.7a.i:
Find the value of Amelia’s investment after $5$ years to the nearest hundred dollars.
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 $x$ .
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

$\frac{{5000}}{{1.028}} + \frac{{5000}}{{{{1.028}^2}}} + \ldots + \frac{{5000}}{{{{1.028}^n}}}$ .
21M.2.SL.TZ1.7b:
Bill invests his $$ 9000$ in an account that offers an interest rate of $r %$ per annum compounded monthly, where $r$ is set to two decimal places.

Find the minimum value of $r$ needed for Bill to reach the target after $10$ years.
19M.2.SL.TZ2.T_6b:
Find the difference between the cost of the bicycle and the amount of money in Tommaso’s account after 3 years. Give your answer correct to two decimal places.
18M.1.SL.TZ2.T_5a.i:
Calculate the amount of money he has in the account after 5 years.

Hide related questions

Topic 1—Number and algebra