Solutions — Step Graphs — Grade 11 General Maths

Q1 — Parking fee step graph

Fluency

(a) 45 minutes:
45 min = 0.75 hours. 0 < 0.75 ≤ 1, so this falls in the first step.
Cost = $3

(b) Exactly 2 hours:
t = 2 hours. The boundary at t = 2 has a closed dot on the $6 step (1 < t ≤ 2).
Cost = $6

(c) 2 hours 15 minutes:
t = 2.25 hours. 2 < 2.25 ≤ 3, so falls in the third step.
Cost = $9

(d) 3 hours 58 minutes:
t ≈ 3.97 hours. 3 < 3.97 ≤ 4, so falls in the fourth step.
Cost = $12

Q2 — Postage costs

Fluency

(a) 45 g:
0 ≤ 45 ≤ 50, first tier. Postage = $1.20

(b) 100 g:
The boundary at 100 g belongs to the 51–100 g tier (closed at 100 g).
Postage = $2.40

(c) 145 g:
101 ≤ 145 ≤ 200, third tier. Postage = $3.80

(d) 380 g:
201 ≤ 380 ≤ 500, fourth tier. Postage = $5.50

Q3 — Formal step function notation for postage

Fluency

Let P(w) be the postage cost in dollars for a parcel of weight w grams.

                            P(w) = { $1.20   if 0 ≤ w ≤ 50

                                   { $2.40   if 50 < w ≤ 100

                                   { $3.80   if 100 < w ≤ 200

                                   { $5.50   if 200 < w ≤ 500

Note: The open circle at each lower boundary (e.g. w > 50 at the start of the second tier) means that if a parcel weighs exactly 50 g, it pays $1.20 (not $2.40). The closed circle at each upper boundary (e.g. w ≤ 100 for the second tier) means exactly 100 g costs $2.40.

Q4 — Cinema ticket prices for a family

Understanding

(a) Ticket price per person:
Adult (age 38): $38
Adult (age 42): $38
Senior (age 70): $28
Child (age 9): $15
Youth (age 16): $22

(b) Total cost (no discount):
Total = 38 + 38 + 28 + 15 + 22 = $141

(c) With 10% group discount (5 tickets):
Discount = 10% of $141 = $14.10
Final cost = 141 − 14.10 = $126.90

Q5 — Internet data plan costs

Understanding

(a) Cost each month:
Month 1 — 8 GB: 0 ≤ 8 ≤ 10, first tier: $35
Month 2 — 22 GB: 10 < 22 ≤ 30, second tier: $55
Month 3 — 45 GB: 30 < 45 ≤ 60, third tier: $75
Month 4 — 65 GB: > 60, fourth tier: $95

(b) Average monthly cost:
Total over 4 months = 35 + 55 + 75 + 95 = $260
Average = 260 ÷ 4 = $65/month

(c) Comparison with $70/month flat plan:
4 months at $70/month = $280 total.
4 months on step plan = $260 total.
The step plan is cheaper by $20 over 4 months. The family should stay on the step plan (assuming their usage pattern continues). However, if their usage regularly exceeds 60 GB, the flat plan ($70 vs $95) would be more economical in those months.

Q6 — Bus fare zones

Understanding

(a) Single-trip fares:
(i) 2 km: Zone 1 (0–3 km): $3.20
(ii) 5 km: Zone 2 (3–10 km): $5.50
(iii) 15 km: Zone 3 (10–20 km): $7.80
(iv) 28 km: Zone 4 (>20 km): $10.60

(b) Student’s monthly transport cost:
Journey distance: 4 km each way → falls in Zone 2: $5.50 per trip.
Trips per day: 2 (to school and back).
Trips per week: 2 × 5 = 10 trips.
Trips per 4 weeks: 10 × 4 = 40 trips.
Total monthly cost = 40 × $5.50 = $220

(c) Monthly pass comparison:
Monthly pass: $90. Pay-per-trip: $220.
Yes, the monthly pass saves $130/month. The pass is much more economical for a student commuting daily.

Q7 — Movie ticket step graph

Understanding

(a) Table with interval notation:

Age interval	Category	Price	Endpoints
0 ≤ age < 3	Infant	Free ($0)	Closed at 0, open at 3
3 ≤ age ≤ 12	Child	$14	Closed at 3, closed at 12
13 ≤ age ≤ 17	Teen	$18	Closed at 13, closed at 17
18 ≤ age ≤ 60	Adult	$24	Closed at 18, closed at 60
age ≥ 61	Senior	$20	Closed at 61, open at right

(b) Step graph sketch:
The horizontal axis shows age (0 to ~70+) and the vertical axis shows price ($0 to $25). Five horizontal segments appear at heights $0, $14, $18, $24, $20 (note the step drops from $24 to $20 at age 61). Each segment spans its age interval with appropriate open/closed circles at boundaries.

(c) Group total (ages 2, 7, 14, 25, 42, 65):
Age 2: under 3 → $0
Age 7: 3–12 → $14
Age 14: 13–17 → $18
Age 25: 18–60 → $24
Age 42: 18–60 → $24
Age 65: 61+ → $20
Total = 0 + 14 + 18 + 24 + 24 + 20 = $100

(d) With 15% group discount (6+ people):
Discount = 15% of $100 = $15
Final price = 100 − 15 = $85

Q8 — Worker daily earnings

Understanding

(a) 6-hour weekday shift:
All 6 hours at standard rate $28/hr (no overtime since ≤ 8 hrs).
Earnings = 6 × $28 = $168

(b) 10-hour weekday shift:
First 8 hours at $28/hr: 8 × 28 = $224
Hours 9 and 10 (2 overtime hours) at $42/hr: 2 × 42 = $84
Total earnings = 224 + 84 = $308

(c) 8-hour Saturday shift:
All hours on a weekend are paid at the weekend rate: $56/hr.
Earnings = 8 × $56 = $448

(d) 5-hour shift starting 9 pm (weekday):
9 pm to 10 pm: 1 hour at standard rate $28/hr = $28
(Note: 2 hours were expected at standard time, but the shift is already on a weekday. Re-reading: the worker starts at 9 pm, works until 2 am. From 9 pm to 10 pm = 1 hour at $28/hr. From 10 pm to 2 am = 4 hours at night shift rate $35/hr.)
Correction: question states 2 hours normal + 3 hours night shift within the 5-hour shift.
Hours 1–2 (9 pm to 11 pm): 2 × $28 = $56
Actually from 9 pm to 10 pm = 1 hr standard; 10 pm to 2 am = 4 hrs night. But question specifies 2 hrs standard + 3 hrs night. Use as given.
2 hours at $28/hr = $56
3 hours at $35/hr = $105
Total = 56 + 105 = $161

Q9 — Income tax calculation

Problem Solving

(a) Income $15 000:
$15 000 ≤ $18 200 → first bracket (nil tax).
Tax = $0

(b) Income $32 000:
$18 201 ≤ $32 000 ≤ $45 000 → second bracket.
Tax = 19c × ($32 000 − $18 200) = 0.19 × $13 800 = $2 622

(c) Income $85 000:
$45 001 ≤ $85 000 ≤ $120 000 → third bracket.
Tax = $5 092 + 32.5c × ($85 000 − $45 000)
= $5 092 + 0.325 × $40 000
= $5 092 + $13 000
= $18 092

(d) Income $140 000:
$120 001 ≤ $140 000 ≤ $180 000 → fourth bracket.
Tax = $29 467 + 37c × ($140 000 − $120 000)
= $29 467 + 0.37 × $20 000
= $29 467 + $7 400
= $36 867

Q10 — Shipping charges

Problem Solving

Key rule: weight is rounded up to the nearest 500 g.

(a) 800 g:
800 g rounded up to nearest 500 g = 1 000 g = 1.0 kg.
Cost from table at 1.0 kg: $12.00

(b) 2 000 g = exactly 2.0 kg:
Exactly on the boundary at 2.0 kg. The 2.0 kg step applies directly.
Cost = $18.00

(c) 4.3 kg:
4.3 kg rounded up to nearest 500 g = 4.5 kg.
4.5 kg is above 3.0 kg. Extra above 3.0 kg = 4.5 − 3.0 = 1.5 kg = three 500 g increments.
Cost = $21.00 + 3 × $3.00 = $21.00 + $9.00 = $30.00

(d) Maximum weight for $30:
At cost = $30: using the formula for ≥ 3 kg: $21 + $3 × n = $30, where n = number of extra 500 g increments.
3n = 9, n = 3.
Maximum weight = 3.0 kg + 3 × 0.5 kg = 3.0 + 1.5 = 4.5 kg.
Check: actual parcel weight must round up to 4.5 kg, so maximum actual weight is anything up to and including 4.5 kg (since it rounds up to 4.5 kg exactly).
Maximum weight: 4.5 kg (or any parcel up to 4.5 kg)
(A parcel of 4.6 kg would round up to 5.0 kg, costing $21 + 4 × $3 = $33 > $30.)