9.3 Speed, Distance & Map Scale
Key Takeaways
- Speed = distance ÷ time; cover the unknown in the triangle to get distance = speed × time or time = distance ÷ speed
- Average speed uses total distance ÷ total time — never just average the two speeds
- Convert km/h to m/s by dividing by 3.6: 72 km/h = 20 m/s; multiply m/s by 3.6 to go back
- A map scale of 1:100 000 means 1 cm on the map represents 100 000 cm = 1 km in real life
- Scale works both directions: multiply map distance by the scale for real distance, divide real distance by the scale for map distance
Rates are where measurement meets multiplication and division, and ICAS Papers D to J use them to build genuine multi-step problems. The good news: two tools — the speed triangle and the idea of scale — cover almost everything asked.
The Speed Triangle
Speed tells you how much distance is covered per unit of time. The relationship between the three quantities is:
Speed = Distance ÷ Time
Picture a triangle with D on top, S and T on the bottom. Cover the quantity you want:
| You want | Cover it | Formula |
|---|---|---|
| Speed | S | D ÷ T |
| Distance | D | S × T |
| Time | T | D ÷ S |
Worked example: A car travels 150 km in 2 hours. What is its speed? Speed = 150 ÷ 2 = 75 km/h.
Worked example: A cyclist rides at 18 km/h for 2.5 hours. How far does she go? Distance = 18 × 2.5 = 45 km. (18 × 2 = 36, plus half of 18 = 9.)
Worked example: How long does it take to walk 12 km at 4 km/h? Time = 12 ÷ 4 = 3 hours.
Average Speed — the Trap
Average speed is total distance ÷ total time, never the average of the speeds.
Worked example: A family drives 60 km at 60 km/h, then 60 km at 30 km/h. What is the average speed for the whole trip? The slow leg takes 60 ÷ 30 = 2 h and the fast leg 1 h, so total = 120 km in 3 h, giving 120 ÷ 3 = 40 km/h — not 45 km/h, because more time is spent at the slower speed.
Speed Units and Converting Them
The two common units are kilometres per hour (km/h) for vehicles and metres per second (m/s) for science and sport. Because 1 km = 1000 m and 1 h = 3600 s, the conversion factor is 3600 ÷ 1000 = 3.6:
- km/h → m/s: divide by 3.6. Example: 72 km/h ÷ 3.6 = 20 m/s.
- m/s → km/h: multiply by 3.6. Example: 15 m/s × 3.6 = 54 km/h.
Also watch for mixed time units: 90 km in 1 h 30 min means 90 ÷ 1.5 = 60 km/h. Convert minutes to a decimal or fraction of an hour first.
Travel Graphs (Basics)
A distance–time graph has time across the bottom and distance up the side. The steepness (gradient) is the speed: a steeper line means faster travel, and a flat horizontal section means stopped. Papers E–H may show a two-stage journey and ask when the traveller was resting or which leg was faster — just compare the slopes.
Map Scale
A scale of 1:100 000 means every 1 unit on the map stands for 100 000 of the same unit in real life. The handiest version: 1 cm on the map = 100 000 cm = 1000 m = 1 km on the ground.
| Map scale | 1 cm represents | Good for |
|---|---|---|
| 1:1000 | 10 m | a house floor plan |
| 1:25 000 | 250 m | bushwalking maps |
| 1:100 000 | 1 km | regional road maps |
Worked example (real distance): Two towns are 7.5 cm apart on a 1:100 000 map. Real distance = 7.5 × 100 000 = 750 000 cm = 7500 m = 7.5 km.
Worked example (map distance): A school oval is 200 m long. How long is it on a 1:2000 plan? Convert 200 m to 20 000 cm, then 20 000 ÷ 2000 = 10 cm.
Notice the direction logic is identical to unit conversion: going from the small model to big reality you multiply; shrinking reality onto the page you divide.
Multi-Step ICAS Style
On a 1:50 000 map, a trail measures 12 cm. Sam hikes it at an average speed of 3 km/h. How long does the hike take?
- Real distance: 12 × 50 000 = 600 000 cm = 6 km
- Time = distance ÷ speed = 6 ÷ 3 = 2 hours
Break every combined problem into these small conversions, and never mix units inside one calculation.
Checking Answers with Estimation
No calculator means estimation is your proofreader. Before committing to an answer:
- Speed sanity check: cars on a highway do 80–110 km/h, a cyclist 15–25 km/h, a walker 4–6 km/h. An answer of 350 km/h for a school bus must be wrong.
- Unit check: if time came out in hours but the options are in minutes, multiply by 60 before choosing. 0.75 h = 45 min, and ICAS will include 75 as a decoy.
- Scale check: on a 1:100 000 map, a 5 cm gap is a few kilometres — if your answer is 500 m or 500 km, the place-value shift went wrong. Count the zeros: cm → m removes two, m → km removes three more.
Scale Drawings and Plans
Scale drawings use the identical ratio idea for objects rather than maps. An architect's plan at 1:50 means a 4 m wall (400 cm) is drawn 400 ÷ 50 = 8 cm long, while a window drawn 3 cm wide is really 3 × 50 = 150 cm = 1.5 m. If a question gives a drawing length and a real length and asks for the scale, divide real by drawing with matching units: real 3 m (300 cm) drawn as 6 cm gives 300 ÷ 6 = 50, so the scale is 1:50.
One More Travel Graph Example
A distance–time graph shows Mia cycling 10 km in the first 30 minutes, a flat section for 15 minutes, then 5 km home in the final 15 minutes. Her fastest leg is the last one: 5 km in 0.25 h is 5 ÷ 0.25 = 20 km/h, compared with 10 ÷ 0.5 = 20 km/h for the first leg — actually equal speeds, but the steeper-looking second leg fools students who measure with their eyes instead of reading the axes. The flat section (stopped, 15 minutes) is the resting period. Always read actual coordinates off the axes rather than trusting appearances, and convert graph times (30 min = 0.5 h) before applying the triangle.
A runner covers 400 metres in 50 seconds. What is her speed in metres per second?
On a map with scale 1:250 000, two cities measure 8 cm apart. What is the real distance between them?