Question 11
An annuity with an initial zero balance has $500 deposited at the end of every month. The annuity earns 4.8% p.a. interest, compounding monthly. At the end of the fourth month, the balance is closest to
Not quite. One way to get this is $2000+500\times0.004=2002$: the four deposits plus one month’s interest on a single deposit. In fact the first deposit earns three months of interest, the second earns two and the third earns one. In 2023, 10.9% of Queensland students chose (A).
Not quite. One way to get this is $2000\times1.004=2008$, a month’s interest on the whole $2000. The money arrives $500 at a time, though, and the last deposit lands on the final day, so it earns nothing. In 2023, 18.0% of Queensland students chose (B).
That’s it. Step through the hints below to see why. In 2023, 55.6% of Queensland students chose (C).
Not quite. $2014 has $14 of interest, and four months can’t earn that much here. One way to get close to it is a decimal slip in the monthly rate, using 0.0048 instead of $\frac{0.048}{12}=0.004$, which gives $2014.45. In 2023, 14.5% of Queensland students chose (D).
Nothing is in the account yet. The first deposit arrives at the end of month 1.
The first $500 arrives on the last day of the month, so it hasn’t earned any interest yet.
Deposit 1 earns one month of interest, 0.4% of $500. Then deposit 2 arrives with nothing earned.
Every deposit already in the account grows by another 0.4%. Deposit 3 arrives with nothing earned.
Four deposits put in $2000. Interest adds only about $12, because deposit 1 earned for three months, deposit 2 for two, deposit 3 for one and deposit 4 for none. That balance is option (C).
“Initial zero balance” and “deposited at the end of every month” are the cues. Equal deposits building up from nothing is a future value annuity.
$$i=\frac{0.048}{12}=0.004,\quad n=4,\quad d=500$$
Here $i$ is the rate per month, $n$ is the number of deposits and $d$ is each deposit. The 2023 subject report asks teachers to help students find exactly these cues before choosing a formula, and annuities were its first example.
Use the formula book’s future value annuity. Type it in one line.
$$\begin{aligned}A_{FV}&=d\left(\frac{(1+i)^n-1}{i}\right)\\&=500\left(\frac{1.004^4-1}{0.004}\right)\\&=2012.032\ldots\approx2012\end{aligned}$$
The 2023 formula sheet wrote this as $A=M\left(\frac{(1+i)^n-1}{i}\right)$. $M$ and $d$ are the same regular payment.
Check with the annuity recurrence $A_{n+1}=rA_n+d$, starting from $A_0=0$.
$$\begin{aligned}A_1&=1.004\times0+500=500\\A_2&=1.004\times500+500=1002\\A_3&=1.004\times1002+500=1506.008\\A_4&=1.004\times1506.008+500=2012.032\ldots\end{aligned}$$
Check the size. The deposits alone are $2000, so the interest is about $12. That rules out (A) and (B), which add too little, and (D), which adds too much.
$$\begin{aligned}i&=\frac{0.048}{12}=0.004,\quad n=4,\quad d=500\\A_{FV}&=500\left(\frac{1.004^4-1}{0.004}\right)\\&=2012.032\ldots\approx\text{\textdollar}2012\end{aligned}$$
Four deposits of $500 make $2000, and about $12 of interest has been added. The deposits come at the end of each month, so the last one earns nothing.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2023 QCAA General Mathematics external assessment, © State of Queensland (QCAA) 2023, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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