1. 1
   A cartographer is designing a map with a given scale and needs to convert between map distances and real-world distances.

   (a)

   On a map, a scale of $1:25000$ means a real distance of 1000 metres is represented on the map by what length?

   [1 mark]

   • A$4$ cm
   • B$40$ cm
   • C$0.4$ cm
   • D$25$ cm

   (b)

   If $y$ varies directly as $x$, and $y = 12$ when $x = 4$, what is $y$ when $x = 10$?

   [1 mark]

   • A$3$
   • B$25$
   • C$30$
   • D$40$

   (c)

   A recipe for 4 people needs 300 g of rice. How much rice is needed for 10 people, assuming direct proportion?

   [1 mark]

   • A$120$ g
   • B$600$ g
   • C$700$ g
   • D$750$ g

   Total for question 1: 3 marks

2. 2
   A landscape designer uses a scale drawing of a garden to plan paths and flower beds before construction begins.

   (a)

   On a scale drawing of a garden, 2 cm represents 5 metres in real life. A path on the drawing measures 7.2 cm. Calculate the real length of the path in metres.

   [2 marks]

   (b)

   A rectangular flower bed in the same garden measures 3.2 cm by 2 cm on the scale drawing (scale 2 cm to 5 metres). Calculate the real area of the flower bed in square metres.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A factory manager is planning machine and staffing requirements as production levels vary, using direct and inverse proportion relationships.

   (a)

   In a factory, the number of identical machines $m$ needed varies directly with the number of units produced per day $u$. When $u = 480$, $m = 6$. Find the formula connecting $m$ and $u$, and calculate the number of machines needed to produce 720 units per day.

   [3 marks]

   (b)

   The number of days $d$ it takes a team to complete an order varies inversely with the number of workers $w$ on the team. When $w = 8$, $d = 15$. Find the formula connecting $d$ and $w$, and calculate how many days it would take a team of 12 workers.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   A group of hikers is using a scale map to plan a trail walk, and wants to compare their planned walking time with the amount of food and water needed, which vary in direct proportion to distance and number of hikers.

   (a)

   On a scale map with scale $1:50000$, a hiking trail measures 8.4 cm. Calculate the real length of the trail in kilometres.

   [4 marks]

   (b)

   The hikers estimate they walk at an average speed that lets them cover 4.2 km in 1 hour 10 minutes (part (a)'s trail). Assuming the same average speed, calculate how long it would take to walk a second trail of length 6 km, giving your answer in hours and minutes.

   [4 marks]

   (c)

   The amount of water needed varies directly with both the number of hikers and the distance walked. For 5 hikers walking 4.2 km, 8.4 litres of water are needed in total. Calculate how much water is needed for 8 hikers walking the 6 km trail from part (b).

   [5 marks]

   Total for question 4: 13 marks

End of questions