INTERPRETING TABLES AND GRAPHS

QUESTION 1 [3 marks] — Criterion A Medium

The travel graph shows a car's journey.

a. Find the distance travelled and the time taken.

[1]

Show Solution

Distance $=80$ km, time $=2$ hours.

b. Calculate the car's average speed for the journey.

[2]

Show Solution

$$\text{speed} = \frac{80}{2} = 40 \text{ km/h}$$

QUESTION 2 [5 marks] — Criterion A Medium

The travel graph shows a cyclist's journey, including a rest stop.

a. Find the distance travelled in the first hour, and the cyclist's speed during that hour.

[2]

Show Solution

Distance $=40$ km in 1 hour, so speed $=40$ km/h.

b. Describe what happens between 1 hour and 1.5 hours, using the shape of the graph.

[1]

Show Solution

The graph is flat (horizontal) here — the cyclist has stopped, remaining at 40 km for 30 minutes.

c. Find the cyclist's speed during the final stage (1.5 h to 2.5 h), and compare it to the speed in the first stage.

[2]

Show Solution

Distance covered: $90-40=50$ km in $2.5-1.5=1$ hour, so speed $=50$ km/h — faster than the first stage's 40 km/h.

QUESTION 3 [7 marks] — Criterion B Medium

Investigate what different sections of a travel (distance-time) graph tell us about a journey, using the full 5-stage journey shown.

a. Identify which stage(s) of the journey represent the traveller being STOPPED (not moving), and justify your answer using the graph's shape.

[2]

Show Solution

Stage 2 (1h to 1.5h) and Stage 4 (2.5h to 3h) are flat/horizontal sections — since distance is not changing over time in these intervals, the traveller must be stationary.

b. Calculate the speed during Stage 5 (3h to 4.5h), and explain what the DOWNWARD direction of this section of the graph means about the direction of travel.

[3]

Show Solution

Distance covered: $90-0=90$ km over $4.5-3=1.5$ hours, so speed $=90\div1.5=60$ km/h. The downward slope means distance from the starting point is DECREASING — the traveller is returning back toward the start.

c. Compare the speeds of all 3 moving stages (Stage 1: 40 km/h, Stage 3: 50 km/h, Stage 5: 60 km/h) and state which was fastest, connecting this to the steepness of each section on the graph.

[2]

Show Solution

Stage 5 (60 km/h) was the fastest, followed by Stage 3 (50 km/h), then Stage 1 (40 km/h). This matches the graph — Stage 5 has the steepest line, confirming that steeper sections of a travel graph always represent faster speeds.

QUESTION 4 [4 marks] — Criterion C Medium

A classmate looks at the downward-sloping Stage 5 of the journey graph above and says: 'the car is going backwards in time, which doesn't make sense.'

a. Explain the classmate's misunderstanding — what does a downward slope on a DISTANCE-time graph actually represent?

[2]

Show Solution

Time always moves forward (left to right) on this type of graph — a downward slope does NOT mean 'going backwards in time'. It means the DISTANCE from the starting point is decreasing as time moves forward, i.e. the traveller is heading back toward where they started.

b. Explain why a travel graph could never have a section that goes 'backwards' along the time axis (right to left), regardless of what the traveller is doing.

[2]

Show Solution

Time is the independent variable on the horizontal axis, and it only ever increases — a traveller can change their DISTANCE (moving closer to or further from the start), but they can never make time itself run backwards, so the graph must always move rightward as time progresses, even while distance goes up or down.

QUESTION 5 [6 marks] — Criterion D Medium

A delivery driver's morning route is shown in the travel graph.

a. Find the driver's average speed for the ENTIRE journey (from start to the final point shown), including the rest stop time.

[3]

Show Solution

Total distance $=140$ km. Total time $=3.5$ hours. Average speed $=140\div3.5=40$ km/h.

b. The driver's manager wants to know the speed while ACTUALLY DRIVING (excluding the stopped time). Find this, and explain why it's different from your answer in (a).

[3]

Show Solution

Driving time only $=1.5+1.5=3$ hours (excluding the 0.5 hour stop). Speed while driving $=140\div3\approx46.7$ km/h. This is higher than the overall average (40 km/h) because the overall average also 'spreads' the stopped time across the whole journey, effectively counting it as if the driver was moving very slowly during that period too.

QUESTION 6 [6 marks] — Criterion A Medium

The travel graph shows a delivery truck's journey, including a rest stop.

a. Find the truck's speed during the FIRST stage (0 to 1.5 hours).

[2]

Show Solution

$$60\div1.5=40\text{ km/h}$$

b. Find the truck's speed during the THIRD stage (2 to 3.5 hours).

[2]

Show Solution

$$(150-60)\div1.5=60\text{ km/h}$$

c. Find the truck's OVERALL average speed for the entire journey (including the rest stop time).

[2]

Show Solution

$$150\div3.5\approx42.9\text{ km/h}$$

QUESTION 7 [2 marks] — Criterion A Easy

The travel graph shows a train's journey between two cities, with a station stop.

a. Find the total journey time and total distance.

[2]

Show Solution

Total time: 4 hours. Total distance: 220 km.

QUESTION 8 [6 marks] — Criterion B Medium

Investigate which stage of the journey shown represents the FASTEST travel, and connect this to the visual STEEPNESS of the graph.

a. Calculate the speed during Stage 1 (0-1h) and Stage 3 (1.5-2.5h).

[3]

Show Solution

Stage 1: $50\div1=50$km/h. Stage 3: $(130-50)\div1=80$km/h.

b. Which stage is faster, and how does this show up VISUALLY in the graph (referring to the steepness/gradient of each line segment)?

[3]

Show Solution

Stage 3 (80km/h) is faster than Stage 1 (50km/h). This is shown visually by Stage 3's line segment being STEEPER (rising more sharply per unit of time) than Stage 1's segment — a general rule for travel graphs is that greater steepness always corresponds to greater speed.

QUESTION 9 [5 marks] — Criterion B Medium

Investigate what a LONG horizontal (flat) section of a travel graph — much longer than the moving sections — might suggest about a journey, using the graph shown.

a. Compare the DURATION of the moving stage (0-1h) to the duration of the stopped stage (1h to 3h). Which is longer?

[2]

Show Solution

Moving stage: 1 hour. Stopped stage: 2 hours — the stop is TWICE as long as the travel itself.

b. Suggest a plausible real-world scenario that would explain a stop lasting LONGER than the actual travel time, and discuss whether this journey pattern (short travel, long stop) seems unusual or could be entirely normal depending on context.

[3]

Show Solution

A plausible scenario: a delivery driver travels a short distance to a client site (1 hour), then spends 2 hours actually completing a job/installation/meeting before returning. This pattern isn't unusual at all in many real contexts (e.g. service calls, business meetings, medical appointments) — a long 'stop' relative to short travel time often reflects the actual PURPOSE of the trip being time-consuming, with the travel itself being just a small part of the overall outing.

QUESTION 10 [5 marks] — Criterion B Medium

A classmate looks at the graph shown (distance rising then falling back to 0) and says: 'the traveller went 80km then came back, so total distance travelled is 0km since they ended where they started.'

a. Explain the classmate's error, distinguishing between DISPLACEMENT (net change in position, which could be 0) and TOTAL DISTANCE TRAVELLED (which accounts for the whole journey, including the return trip).

[3]

Show Solution

The classmate is confusing 'ending back at the start' (which does mean the NET displacement is 0km) with 'total distance travelled' (which must count BOTH the outward AND return legs of the journey). Since the traveller went 80km out AND 80km back, the total distance travelled is $80+80=160$km, even though their final position is the same as where they started.

b. Calculate the correct total distance travelled for this journey.

[2]

Show Solution

$$80+80=160\text{ km}$$

QUESTION 11 [3 marks] — Criterion A Medium

A cyclist's journey is shown, with speeds of 60km/h in Stage 1 and Stage 3 (equal speeds, different stages).

a. Verify that the speeds in Stage 1 (0-0.5h) and Stage 3 (1.5-2.5h) are indeed both 60km/h, showing your calculation for each.

[3]

Show Solution

Stage 1: $30\div0.5=60$km/h. Stage 3: $(90-30)\div1=60$km/h — confirmed both are 60km/h.

QUESTION 12 [5 marks] — Criterion B Hard

Investigate whether TWO different stages of a travel graph can have the SAME speed but look VISUALLY DIFFERENT on the graph, using the cyclist graph above (Stage 1: 0.5h duration, Stage 3: 1h duration, both at 60km/h).

a. Even though both stages have the SAME speed (60km/h), explain why their line segments on the graph do NOT look identical — what differs about them?

[2]

Show Solution

While both segments have the SAME STEEPNESS (gradient, since steepness represents speed), they differ in LENGTH along the graph — Stage 1 is a shorter segment (only 0.5 hours, covering 30km) while Stage 3 is a longer segment (1 hour, covering 60km), even though both segments rise at the exact same rate.

b. Explain the general principle: what property of a travel-graph line segment represents SPEED, and what property represents DURATION/DISTANCE covered — are these the same thing or different?

[3]

Show Solution

The STEEPNESS (gradient/slope) of a segment represents SPEED — two segments with the same steepness always represent the same speed, regardless of their length. The LENGTH of the segment (how far it extends along the graph) represents the DURATION and total DISTANCE covered during that stage — these are different properties: two segments can have identical steepness (same speed) while having very different lengths (different durations/distances), exactly as shown here.

QUESTION 13 [4 marks] — Criterion C Medium

A student solving a travel graph problem writes only: '80/2=40'.

a. Explain why this single unlabeled line, despite possibly being mathematically correct, is inadequate mathematical communication for a travel graph problem.

[2]

Show Solution

Without any labels or context, a reader cannot tell WHAT the 80 and 2 represent (distance in km? time in hours? something else?), which STAGE of the journey is being analyzed, or what the resulting 40 actually MEANS (a speed? in what units?) — the calculation might be correct, but it communicates nothing meaningful on its own.

b. Rewrite this as a properly labelled solution, inventing a plausible travel graph context (e.g. a car travelling 80km in 2 hours during one stage of a journey).

[2]

Show Solution

Example: 'During Stage 2 of the journey (from $t=1$h to $t=3$h), the car travelled from the 40km mark to the 120km mark — a distance of $120-40=80$km over $3-1=2$ hours. Speed during this stage: $80\div2=40$km/h.'

QUESTION 14 [4 marks] — Criterion C Medium

A student calculates a NEGATIVE speed (e.g. '-60km/h') for a downward-sloping section of a travel graph, and is confused about what this means.

a. Explain what a negative 'speed' actually represents in the context of a downward-sloping travel graph segment (referring to direction of travel, not an impossible physical speed).

[2]

Show Solution

A negative value here doesn't mean an impossible negative SPEED (speed itself — how fast something moves — is always positive or zero) — it reflects that DISTANCE FROM THE STARTING POINT is DECREASING, meaning the traveller is moving in the OPPOSITE direction (returning toward the start). The actual speed (magnitude, ignoring direction) would be the positive value $|{-60}|=60$km/h; the negative sign specifically indicates the direction of travel relative to the starting point, not the speed itself being negative.

b. State how a student should correctly communicate this in a written answer, distinguishing between 'speed' (always positive) and the signed 'rate of change of distance from start' (which can be negative).

[2]

Show Solution

A well-communicated answer should state: 'The rate of change of distance from the starting point is $-60$km/h, meaning the traveller is moving TOWARD the starting point at a speed of 60km/h' — explicitly separating the concept of speed (magnitude, always non-negative) from the signed rate that indicates direction.

QUESTION 15 [8 marks] — Criterion D Hard

A logistics company plans delivery routes. The graph shows a driver's journey to a delivery point (90km away), followed by a REQUIRED 2-hour delivery/unloading stop (company policy), before the driver's shift ends (driver stays at the delivery point).

a. Find the driver's speed during the travel stage, and the total time from departure to the end of the required stop.

[3]

Show Solution

Speed: $90\div1.5=60$km/h. Total time (travel + stop): $1.5+2=3.5$ hours.

b. The company wants to know if a driver could complete TWO such delivery trips (each with full 90km travel + 2hr stop, but with an IMMEDIATE 90km return leg at the same 60km/h speed after the stop) within a standard 8-hour shift. Calculate the total time for ONE complete round trip (there and back, with the stop), and determine how many such ROUND TRIPS fit within 8 hours.

[5]

Show Solution

One round trip: travel there ($1.5$h) + stop ($2$h) + travel back ($90\div60=1.5$h) $=1.5+2+1.5=5$ hours. Within an 8-hour shift: $8\div5=1.6$, so only 1 COMPLETE round trip fits (a second round trip would need 5 more hours, totalling 10 hours, exceeding the 8-hour shift).

QUESTION 16 [2 marks] — Criterion D Easy

An airline's flight path is simplified to a single travel graph segment: a flight covering 2,400km in 3 hours (cruise speed only, ignoring takeoff/landing).

a. Find the average cruise speed in km/h.

[2]

Show Solution

$$2400\div3=800\text{ km/h}$$

QUESTION 17 [5 marks] — Criterion D Hard

A hiker's journey up a mountain trail is tracked, measuring cumulative trail distance covered.

a. Assuming the vertical axis represents cumulative trail distance (km) and NOT elevation, find the hiker's pace (km/h) for the first stage.

[2]

Show Solution

$$160\div2=80\text{ km/h — this is unrealistically fast for hiking, suggesting the units in this simplified model don't represent a literal hiking scenario.}$$

b. Given your answer to (a) seems physically unrealistic for a hiker (80 km/h is faster than a car on many roads), identify what is likely WRONG with the problem's stated units or numbers, and suggest a more realistic total distance for a 2-hour hiking stage.

[3]

Show Solution

80km/h is wildly unrealistic for hiking (a fast human hiking pace is typically 3-6 km/h) — this suggests the graph's numbers (160 units in 2 hours) are almost certainly NOT meant to represent kilometres for a hiking context; a more realistic distance for a 2-hour hiking stage might be around 6-10km, meaning the graph's '160' value likely represents a different unit entirely (e.g. metres of elevation gain, not km of trail distance), and the problem as stated contains an inconsistency between its claimed units and realistic hiking speeds.

QUESTION 18 [6 marks] — Criterion B Medium

Investigate whether a travel graph with only TWO moving stages (no stops) can still show a clear change in speed.

a. Find the speed during Stage 1 (0-0.5h) and Stage 2 (0.5-1.5h).

[3]

Show Solution

Stage 1: $40\div0.5=80$km/h. Stage 2: $(100-40)\div1=60$km/h.

b. Even though there's no flat (stopped) section anywhere in this graph, explain how you can still tell the traveller's speed CHANGED partway through the journey, just from the shape of the two connected segments.

[3]

Show Solution

The two segments have DIFFERENT steepness (gradients) — Stage 1 is steeper (80km/h) than Stage 2 (60km/h) — this visible 'kink' or change in slope at the point where the segments meet ($t=0.5$h) directly shows the speed changed at that moment, even without any flat section indicating a stop; a travel graph doesn't need a stop to show varying speed, just a change in the line's steepness.

QUESTION 19 [3 marks] — Criterion C Medium

A student writes a travel graph solution as: 'the graph shows the car went fast then slow', with no supporting calculations at all.

a. Explain what specific pieces of quantitative evidence (numbers) this description is missing, and why 'fast' and 'slow' alone are not acceptable mathematical communication.

[3]

Show Solution

The description gives no actual SPEED VALUES (in km/h or similar units), no indication of WHEN the change from fast to slow occurred, and no distances or times to support the claim — 'fast' and 'slow' are subjective, relative terms with no fixed meaning, whereas proper mathematical communication requires specific, calculated, and clearly labelled quantities that another person could verify.

QUESTION 20 [3 marks] — Criterion C Medium

Two students both correctly calculate a journey's average speed as 55km/h, but Student 1 shows full working with labelled steps, while Student 2 writes only the final answer '55km/h' with a small unlabeled calculation scribbled beside it.

a. Even though both reach the correct answer, explain why Student 1's response would likely be assessed as demonstrating STRONGER mathematical communication, referencing what a reader can verify or understand from each response.

[3]

Show Solution

A reader examining Student 1's labelled, step-by-step work can VERIFY the reasoning is sound, follow exactly how the 55km/h was derived, and check for any errors in the process — with Student 2's response, a reader has no way to confirm whether the correct answer was reached through valid reasoning or through a lucky guess, memorized shortcut, or even a calculator error that happened to cancel out. Communication quality is judged by how well the REASONING is conveyed, not just whether the final number happens to be correct.

QUESTION 21 [9 marks] — Criterion D Hard

A courier company evaluates a driver's route efficiency using the travel graph of a single delivery run.

a. Find the speed during each of the two DRIVING stages (ignoring the brief stop).

[3]

Show Solution

Stage 1 (0-1h): $70\div1=70$km/h. Stage 2 (1.25-2.75h): $(190-70)\div1.5=80$km/h.

b. The company's fuel-efficiency guideline recommends drivers maintain speeds under 75km/h whenever possible, since fuel efficiency drops significantly above this threshold. Identify which stage(s) VIOLATE this guideline, and estimate the ADDITIONAL fuel cost if that stage used 15% more fuel per km than a compliant stage would, given the violating stage covers 120km and fuel costs \$0.12 per km at the efficient rate.

[6]

Show Solution

Stage 2 (80km/h) VIOLATES the 75km/h guideline; Stage 1 (70km/h) complies. Extra fuel cost for Stage 2: efficient cost would be $120\times0.12=\$14.40$; at 15% more, actual cost $\approx14.40\times1.15=\$16.56$. Additional cost due to speeding: $16.56-14.40=\$2.16$ for this one delivery run.

QUESTION 22 [4 marks] — Criterion D Medium

A high-speed rail service covers 320km in 4 hours on a particular route.

a. Find the average speed, and compare it to a typical car journey covering the same distance at 100km/h, finding how much LONGER the car journey would take.

[4]

Show Solution

Train speed: $320\div4=80$km/h. Car time: $320\div100=3.2$hours. Interesting — the TRAIN is actually SLOWER on average in this scenario (80km/h vs the car's 100km/h), meaning the car would take LESS time ($3.2$h vs the train's $4$h), a difference of $4-3.2=0.8$ hours (48 minutes) — the car arrives faster, assuming no traffic delays, contrary to the common assumption that trains are always faster than driving.

INTERPRETING TABLES AND GRAPHS

Travel graphs