2024 Paper 2, Q1
Each of the 60 performers in a music and dance concert is either a Year 11 or Year 12 student and either a musician or a dancer.
There are four more Year 11 students than Year 12 students. One quarter of the Year 11 students are dancers and half of the Year 12 students are dancers.
| Year 11 | Year 12 | Total | |
|---|---|---|---|
| Musician | |||
| Dancer | |||
| Total | \(60\) |
Complete the two-way frequency table to calculate the percentage of students who are musicians.
Fill the table one fact at a time
Only the grand total is given. Each sentence in the question fills one or two cells, and every cell you fill makes the next one possible.
The order matters
Year 11 and Year 12 add to 60 and differ by 4. Halve the 56 that is left after taking off the extra 4, so Year 12 is 28 and Year 11 is 32.
"One quarter of the Year 11 students" means a quarter of 32. It is easy to take a quarter of 60 by mistake.
Every student is either a musician or a dancer, so musicians are the column total minus the dancers.
Complete the table yourself
Type a number into any cell. A cell gets a blue border when it is right. If you get stuck, ask for the next cell and the explanation appears below.
| Year 11 | Year 12 | Total | |
|---|---|---|---|
| Musician | |||
| Dancer | |||
| Total | 60 |
Year 11 has four more students than Year 12. How many are in Year 12?
Then Year 11 would be 30 and the total 56. Check by adding the two back up.
Yes. \(28+32=60\) and \(32-28=4\). Both conditions hold.
That is half of 60, which would make the two years equal. Year 12 has to be smaller.
QCAA marking guide · 5 marks
\[\text{Year 11}=32,\qquad\text{Year 12}=28\]
Marker: correctly calculates the frequencies for total Year 11 students and total Year 12 students.
\[\tfrac14\times32=8,\qquad\tfrac12\times28=14\]
Marker: calculates the frequencies for dancers in Year 11 and dancers in Year 12.
\[32-8=24,\qquad 28-14=14\]
Marker: calculates the frequencies for musicians in Year 11 and in Year 12. Follow-through marks are allowed from here on.
\[\text{musicians}=24+14=38,\qquad\text{dancers}=8+14=22\]
Marker: calculates the frequencies for total musicians and total dancers.
\[\frac{38}{60}\times100\%=63.3\%\]
63.3% of the students are musicians.
Marker: calculates the percentage of students who are musicians.
Checking the table
\(24+8=32\), \(14+14=28\), \(38+22=60\). If any total fails, one of the cells is wrong.
The question asks for musicians as a percentage of all students, so divide by 60. Dividing 24 by 38, or by 32, answers a different question.
Completing a two-way table is familiar. Here almost nothing is given directly. The column totals come from a sentence that needs a small piece of algebra, and the fractions apply to different totals in different columns. I want you to write a short reason next to each number as you fill the table, so you can see which fact each one came from.
Question wording and marking-guide steps are from the 2024 QCAA General Mathematics external assessment, © State of Queensland (QCAA) 2024, licensed under CC BY 4.0, and have been adapted. Tangent Tuition is not affiliated with the QCAA.
Spot a mistake on this page? Report it