Solved: Two friends Olivia and Jack, each set themselves a target of saving \$20,000.... [IB Mathematics Analysis and Approaches (AA)]

Question

SLPaper 2

Two friends Olivia and Jack, each set themselves a target of saving $20,000. They each have $9,000 to invest.

Olivia invests her $9,000 in an account that offers an interest rate of 7% per annum compounded annually.

A third friend Mia also wants to reach the $20,000 target. She puts her money in a safe where she does not earn any interest. Her system is to add more money to this safe each year. Each year she will add half the amount added in the previous year.

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

[3]

Verified

Solution

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

$12622.965\ldots$ (A1)

$n=5$

$I\%=7$

$PV=\pm9000$

$P/Y=1$

$C/Y=1$ (A1)

$\pm12622.965\ldots$ (A1)

[3 marks]

Determine the number of years required for Olivia's investment to reach the target.

[2]

Verified

Solution

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

${I\%=7}$

${PV=\pm9000}$

${FV=\pm20000}$

${P/Y=1}$

${C/Y=1}$ (A1)

THEN ${=12}$ (years) A1

[2 marks]

Jack 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 Jack to reach the target after ${10}$ years.

[3]

Verified

Solution

METHOD 1

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

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

$8.01170...$ (A1)

$r = 8.02\%$ A1

METHOD 2

$n = 10$

$PV = \pm 9000$

$FV = \mp 20000$

$P/Y = 1$

$C/Y = 12$

$r = 8.01170...$ (M1)(A1)

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

$r = 8.02\%$ A1

[3 marks]

Show that Mia will never reach the target if her initial deposit is $9000.

[5]

Verified

Solution

recognising geometric series (seen anywhere) (M1)

$r=\frac{4500}{9000}$ (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.

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 Mia will not reach the target, which involves their choice of $n$ , their value of $S_n$ and Mia’s 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, Mia will never reach the target. (AG)

[5 marks]

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

[3]

Verified

Solution

recognising geometric sum M1

${\frac{{u_1(1-0.5^5)}}{{0.5}}}=20000$ (A1)

${10322.58...}$

$\$10323$ A1

[3 marks]

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