Question 5
The table lists the number of books sold per month by an online bookstore. If the simple 3-point moving average in October is 54, what is the simple 3-point moving average in May?
| Jan | Feb | Mar | Apr | May | Jun | Jul | Aug | Sep | Oct | Nov | Dec |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 45 | 52 | 68 | 65 | 89 | 65 | 53 | 33 | 40 | 45 | 77 | 92 |
Not quite. One way to get 69 is to average May, June and July, a window that starts at May. Test that window on October first: October, November and December average 71.3, not 54. In 2022, 4.1% of Queensland students chose (A).
That’s it. Step through the hints below to see why. In 2022, 88.7% of Queensland students chose (B).
Not quite. One way to get 74 is to average March, April and May, a window that ends at May. The same kind of window on October (August, September, October) averages 39.3, not the 54 the question gives. In 2022, 5.2% of Queensland students chose (C).
Not quite. 89 is May’s own sales figure, read straight from the table. A 3-point moving average is the mean of three months, so it smooths out May’s spike. In 2022, 1.2% of Queensland students chose (D).
A 3-point window can sit three ways around a month. The question’s October value tells you which one it uses. Choose a month and a window, and check.
Orange: the three months being averaged. Blue outline: the month the average belongs to.
The QCAA gave you October’s average so you can test where the window sits. Try the three ways a 3-month window can include October:
$$\begin{aligned}\text{Sep, Oct, Nov}&:\ \dfrac{40+45+77}{3}=54\\\text{Aug, Sep, Oct}&:\ \dfrac{33+40+45}{3}=39.3\ldots\\\text{Oct, Nov, Dec}&:\ \dfrac{45+77+92}{3}=71.3\ldots\end{aligned}$$
Only the centred window gives 54. So each month’s average uses that month and one month either side.
Do the same for May: April, May and June.
$$\frac{65+89+65}{3}=\frac{219}{3}=73$$
Type it in one line with brackets around the sum, or the calculator only divides the last number by 3.
Each wrong option is a window in the wrong place, or no window at all:
- 69: May, June, July, a window that starts at May.
- 74: March, April, May, a window that ends at May.
- 89: May on its own.
Neither of the shifted windows survives the October test, which is exactly why the question gives you October.
October’s average of 54 uses September, October and November, so the window is centred:
$$\frac{40+45+77}{3}=54\ \checkmark$$
May’s centred window is April, May and June:
$$\frac{65+89+65}{3}=73$$
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2022 QCAA General Mathematics external assessment, © State of Queensland (QCAA) 2022, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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