The school canteen records the lunch choices of Year 10 and Year 11 students on one Tuesday, split by jacket potato or panini, in the table below. The total...
Question 1 Report
The school canteen records the lunch choices of Year 10 and Year 11 students on one Tuesday, split by jacket potato or panini, in the table below. The total number of students recorded that day was 96.
| Jacket potato | Panini | |
|---|---|---|
| Year 10 | 2x | 20 |
| Year 11 | 16 | 3x |
- Find the value of \(x\). (2)
- Find the probability that a student chosen at random from those recorded is a Year 11 student who chose a panini. (2)
Answer Details
Every entry in the table must add up to the given total, which turns the table into a linear equation in \(x\); once \(x\) is known, any single cell divided by the grand total gives a probability.
(a) Adding all four entries in the table and setting the total equal to 96 gives \( 2x+20+16+3x=96 \) [1 mark]. Collecting like terms gives \( 5x+36=96 \), so subtracting 36 from both sides gives \( 5x=60 \), and dividing by 5 gives \( x=12 \) [1 mark].
(b) The Year 11 panini entry is \( 3x = 3(12) = 36 \) students. The probability that a randomly selected student is a Year 11 panini student is this count out of the total 96 recorded: \( \dfrac{36}{96} \) [1 mark], which simplifies (dividing top and bottom by 12) to \( \dfrac{3}{8} \) [1 mark].
Substituting \( x=12 \) back into every cell (\( 2x=24 \), \( 3x=36 \)) and checking that \( 24+20+16+36=96 \) confirms the value of \( x \) before it is used to answer part (b).
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