If Clara chooses option 1,

(i) write down the values of the second and third installments;

(ii) calculate the value of the final installment;

(iii) show that the total amount that Clara would pay for the land is \(\$ 88000\).

If Clara chooses option 2,

(i) find the value of the second installment;

(ii) show that the value of the fifth installment is \(\$ 2721\).

The price of the land is \(\$ 80000\). In option 1 her total repayments are \(\$ 88000\) over the 20 months. Find the annual rate of simple interest which gives this total.

Clara knows that the total amount she would pay for the land is not the same for both options. She wants to spend the least amount of money. Find how much she will save by choosing the cheaper option.

(i) Second installment \( = \$ 2700\)     (A1)

Third installment \( = \$ 2900\)     (A1)

(ii) Final installment \( = 2500 + 200 \times 19\)     (M1)(A1)

Note: (M1) for substituting in correct formula or listing, (A1) for correct substitutions.

\( = \$ 6300\)     (A1)(G2)

(iii) Total amount \( = \frac{{20}}{2}(2500 + 6300)\)

OR

\( \frac{{20}}{2}(5000 + 19 \times 200)\)     (M1)(A1)

Note: (M1) for substituting in correct formula or listing, (A1) for correct substitution.

\( = \$ 88000\)     (AG)

Note: Final line must be seen or previous (A1) mark is lost.

[7 marks]

(i) Second installment \(2000 \times 1.08 = \$ 2160\)     (M1)(A1)(G2)

Note: (M1) for multiplying by \(1.08\) or equivalent, (A1) for correct answer.

(ii) Fifth installment \( = 2000 \times {1.08^4} = 2720.98 = \$ 2721\)     (M1)(A1)(AG)

Notes: (M1) for correct formula used with numbers from the problem. (A1) for correct substitution. The \(2720.9 \ldots \) must be seen for the (A1) mark to be awarded. Accept list of 5 correct values. If values are rounded prematurely award (M1)(A0)(AG).

[4 marks]

Interest is \( = \$ 8000\)     (A1)

\(80000 \times \frac{r}{{100}} \times \frac{{20}}{{12}} = 8000\)     (M1)(A1)

Note: (M1) for attempting to substitute in simple interest formula, (A1) for correct substitution.

Simple Interest Rate \( = 6\% \)     (A1)(G3)

Note: Award (G3) for answer of \(6\% \) with no working present if interest is also seen award (A1) for interest and (G2) for correct answer.

[4 marks]

Financial accuracy penalty (FP) is applicable where indicated in the left hand column.

(FP)     Total amount for option 2 \( = 2000\frac{{(1 - {{1.08}^{20}})}}{{(1 - 1.08)}}\)     (M1)(A1)

Note: (M1) for substituting in correct formula, (A1) for correct substitution.

\( = \$ 91523.93\) (\( = \$ 91524\))     (A1)
\(91523.93 - 88000 = \$ 3523.93 = \$ 3524\) to the nearest dollar     (A1)(ft)(G3)

Note: Award (G3) for an answer of \(\$ 3524\) with no working. The difference follows through from the sum, if reasonable. Award a maximum of (M1)(A0)(A0)(A1)(ft) if candidate has treated option 2 as an arithmetic sequence and has followed through into their common difference. Award a maximum of (M1)(A1)(A0)(ft)(A0) if candidate has consistently used \(0.08\) in (b) and (d).

[4 marks]

This question was answered correctly by many. Candidates were able to restart if they failed to complete a particular part. Many candidates wasted much time because their understanding was limited to a recursive method and hence wrote out all the terms rather than using the formula for the nth term or sum. A surprising number of students were not able to use the simple interest formula for a period which was not a whole number of years. Also hardly anyone knew to calculate interest first before substituting into the formula. Many students who attempted part (d) lost a point due to FP. A number of students rounded their answers prematurely to the nearest dollar.

This question was answered correctly by many. Candidates were able to restart if they failed to complete a particular part. Many candidates wasted much time because their understanding was limited to a recursive method and hence wrote out all the terms rather than using the formula for the nth term or sum. A surprising number of students were not able to use the simple interest formula for a period which was not a whole number of years. Also hardly anyone knew to calculate interest first before substituting into the formula. Many students who attempted part (d) lost a point due to FP. A number of students rounded their answers prematurely to the nearest dollar.

This question was answered correctly by many. Candidates were able to restart if they failed to complete a particular part. Many candidates wasted much time because their understanding was limited to a recursive method and hence wrote out all the terms rather than using the formula for the nth term or sum. A surprising number of students were not able to use the simple interest formula for a period which was not a whole number of years. Also hardly anyone knew to calculate interest first before substituting into the formula. Many students who attempted part (d) lost a point due to FP. A number of students rounded their answers prematurely to the nearest dollar.

This question was answered correctly by many. Candidates were able to restart if they failed to complete a particular part. Many candidates wasted much time because their understanding was limited to a recursive method and hence wrote out all the terms rather than using the formula for the nth term or sum. A surprising number of students were not able to use the simple interest formula for a period which was not a whole number of years. Also hardly anyone knew to calculate interest first before substituting into the formula. Many students who attempted part (d) lost a point due to FP. A number of students rounded their answers prematurely to the nearest dollar.