On 1 January 2005, Rebecca invested $30000{\text{ AUD}}$ in a Supersaver account at a nominal annual rate of $2.5\%$ compounded annually. Calculate the amount in the Supersaver account after two years.

On 1 January 2005, Rebecca invested $30000{\text{ AUD}}$ in a Supersaver account at a nominal annual rate of $2.5\%$ compounded annually.

Find the number of complete years since 1 January 2005 it would take for the amount in Rebecca’s account to exceed the amount in Daniel’s account.

On 1 January 2007, Daniel reinvested $80\%$ of the money from the Regular Saver account in an Extra Saver account at a nominal annual rate of $3\%$ compounded quarterly.

(i)     Calculate the amount of money reinvested by Daniel on the 1 January 2007.

(ii)    Find the number of complete years it will take for the amount in Daniel’s Extra Saver account to exceed $30000{\text{ AUD}}$.

${\text{Amount}} = 30000{\left( {1 + \frac{{2.5}}{{100}}} \right)^2}$     (M1)(A1)

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

$31518.75{\text{ AUD}}$     (A1)(G2)

OR

${\text{I}} = 30000{\left( {1 + \frac{{2.5}}{{100}}} \right)^2} - 30000$     (M1)(A1)

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

$31518.75{\text{ AUD}}$     (A1)(G2)

[3 marks]

${\text{Rebecca's amount}} = 30000{\left( {1 + \frac{{2.5}}{{100}}} \right)^n}$
${\text{Daniel's amount}} = 30000 + \frac{{30000 \times 2.75 \times n}}{{100}}$     (M1)(A1)(ft)

Note: Award (M1) for substitution in the correct formula for the two amounts, (A1) for correct substitution. Follow through from their expressions used in part (a) and/or part (b).

OR

2 lists of values seen (at least 2 terms per list)     (M1)

lists of values including at least the terms with $n = 8$ and $n = 9$     (A1)(ft)

For $n = 8$     ${\text{CI}} = 36552.09$     ${\text{SI}} = 36600$

For $n = 9$     ${\text{CI}} = 37465.89$     ${\text{SI}} = 37425$

Note: Follow through from their expressions used in part (a) or/and (b).

OR

Sketch showing 2 graphs, one exponential and the other straight line     (M1)

point of intersection identified     (M1)

Note: Follow through from their expressions used in part (a) or/and (b).

$n = 9$     (A1)(ft)(G2)

Note: Answer $8.57$ without working is awarded (G1).

Note: Accept comparison of interests instead of the total amounts in the two accounts.

[3 marks]

(i)     $0.80 \times 31650 = 25320$     (M1)(A1)(G2)

Note: Award (M1) for correct use of percentages.

(ii)    $25320{\left( {1 + \frac{3}{{4 \times 100}}} \right)^{4n}} > 30000$     (M1)(M1)(ft)

Notes: Award (M1) for correct left-hand side of the inequality, (M1) for comparison to $30000$. Accept equation. Follow through from their answer to part (d) (i).

OR

List of values from their $25320{\left( {1 + \frac{3}{{4 \times 100}}} \right)^{4n}}$ seen (at least 2 terms)     (M1)

Their correct values for $n = 5$ ($29401.18$) and $n = 6$ ($30293$) seen     (A1)(ft)

Note: Follow through from their answer to (d) (i).

OR

Sketch showing 2 graphs – an exponential and a horizontal line     (M1)

Point of intersection identified or vertical line drawn     (M1)

Note: Follow through from their answer to (d) (i).

$n = 6$     (A1)(ft)(G2)

Note: Award (G1) for answer $5.67$ with no working.

[5 marks]

Part b) was well done.

Parts c) and d) were not answered well. Marks were gained by candidates who showed detailed working. Many candidates had difficulty working with the compound interest formula where the interest was compounded quarterly. Correct final answers in parts c) and d) were rare.

Parts c) and d) were not answered well. Marks were gained by candidates who showed detailed working. Many candidates had difficulty working with the compound interest formula where the interest was compounded quarterly. Correct final answers in parts c) and d) were rare.