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Date	May 2022	Marks available	3	Reference code	22M.2.SL.TZ1.1
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 1
Command term	Find	Question number	1	Adapted from	N/A

Question

In this question, give all answers correct to two decimal places.

Sam invests $$ 1700$ in a savings account that pays a nominal annual rate of interest of $2.74 %$ , compounded half-yearly. Sam makes no further payments to, or withdrawals from, this account.

David also invests $$ 1700$ in a savings account that pays an annual rate of interest of $r %$ , compounded yearly. David makes no further payments or withdrawals from this account.

Find the amount that Sam will have in his account after $10$ years.

[3]

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.

[2]

Find the interest David will earn over the $10$ years.

[1]

Markscheme

Note: The first time an answer is not given to two decimal places, the final A1 in that part is not awarded.

EITHER

$N = 10$ OR $N = 20$

$I % = 2.74$ $I % = 2.74$

$P V = (\mp) 1700$ $P V = (\mp) 1700$

$P / Y = 1$ $P / Y = 2$

$C / Y = 2$ $C / Y = 2$ (M1)(A1)

Note: Award (M1) for an attempt to use a financial app in their technology with at least two entries seen, and award (A1) for all entries correct. Accept a positive or negative value for $P V$ .

$1700 {(1 + \frac{0.0274}{2})}^{2 \times 10}$ (M1)(A1)

Note: Award (M1) for substitution into compound interest formula.
Award (A1) for correct substitution.

THEN

$$ 2231.71$ A1

[3 marks]

Note: The first time an answer is not given to two decimal places, the final A1 in that part is not awarded.

EITHER

$N = 10$

$P V = \mp 1700$

$F V = \pm 2231.71 \dots$

$P / Y = 1$

$C / Y = 1$ (M1)

Note: Award (M1) for an attempt to use a financial app in their technology with at least two entries seen.

$1700 {(1 + \frac{r}{100})}^{10} = 2231.71 \dots$ (M1)

THEN

$r = 2.75876 \dots$

$r = 2.76$ A1

Note: Ignore omission of opposite signs for $P V$ and $F V$ if $r = 2.76$ is obtained.

[2 marks]

Note: The first time an answer is not given to two decimal places, the final A1 in that part is not awarded.

$$ 531.71$ A1

[1 mark]

Examiners report

This was the only short question which was not common to both HL and SL papers.

Candidates found this question a straightforward start, with the majority being able to attempt all three parts. While most candidates initially gave answers to two decimal places as required, many subsequently rounded their answers to three significant figures. In parts (a) and (b), most used the compound interest formula rather than the finance app on their GDC. Difficulty using the formula arose in determining the value of r (0.0274 was often seen, rather than 2.74) and the value of k. Many candidates struggled to correctly interpret 'half-yearly', and attempted to use either $k = 6$ or $k = \frac{1}{2}$ in the formula.

In part (b), a correct equation was usually seen, but lengthy analytical methods to solving it were favoured, with varying degrees of success, over an approach using the GDC. A common error was an attempt to use logarithms rather than the tenth root. It was not uncommon to see a final answer of either 2.76% or 0.0276, rather than 2.76. Many did not know what was meant by 'interest' in part (c). Those that did often recalculated David's amount using their value of r, rather than subtract $1700 from Sam's amount.

The candidates who attempted to use a finance app on their GDC in this question were generally able to give accurate and concise answers. In the case where they obtained an incorrect answer, most gave sufficient detail of the values they were using in the app, to be awarded the method mark. The most common error seen was the use of an incorrect value for the number of payments per year or the number of compounding periods per year.

[N/A]

Syllabus sections

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

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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.TZ1.7a.ii:
Determine the number of years required for Amelia’s investment to reach the target.
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.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.

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