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Mathematics

Speed, Maps & Unit Conversion

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Amari Cross & Matthew Williams
||11 min read
MapsMeasurementScaleSpeed

Converting units, distance-speed-time problems, scale drawings, and margin of error.

Speed, scale, and unit conversion questions are about keeping quantities consistent. A correct method can still give a wrong answer if kilometres, metres, hours, minutes, centimetres, or millimetres are mixed without conversion.

For CSEC, expect these ideas inside practical contexts: travel, maps, plans, construction, and measurement accuracy. Before using a formula, convert all quantities to compatible units and write the unit beside every answer. This shows comprehension, which is heavily weighted across the exam.

Measurements must often be converted between different units. Let's master the conversions!

Length Conversion

Length conversions change one-dimensional measurements. Move between units by multiplying or dividing by the conversion factor between them.

UnitConversion
1 km1000 m
1 m100 cm
1 cm10 mm
1 inch2.54 cm
1 foot12 inches ≈ 30.48 cm
1 yard3 feet ≈ 0.914 m
1 mile1.609 km
Example

Convert 5 km to m:

5 km=5×1000=5000 m5 \text{ km} = 5 \times 1000 = 5000 \text{ m}

Example

Convert 250 cm to m:

250 cm=250÷100=2.5 m250 \text{ cm} = 250 \div 100 = 2.5 \text{ m}

Example

Convert 8500 mm to cm:

8500 mm=8500÷10=850 cm8500 \text{ mm} = 8500 \div 10 = 850 \text{ cm}

Area Conversion

Area conversions must square the length conversion because area has two dimensions. This is why 1 m21 \text{ m}^2 is 10,000 cm210,000 \text{ cm}^2, not 100 cm2100 \text{ cm}^2.

UnitConversion
1 km²1,000,000 m²
1 m²10,000 cm²
1 cm²100 mm²

Key Rule: When converting area, square the conversion factor!

  • 1 m = 100 cm, so 1 m² = 100² = 10,000 cm²
  • 1 km = 1000 m, so 1 km² = 1000² = 1,000,000 m²
Example

Convert 3 m² to cm²:

3 m2=3×10,000=30,000 cm23 \text{ m}^2 = 3 \times 10,000 = 30,000 \text{ cm}^2

Example

Convert 50,000 cm² to m²:

50,000 cm2=50,000÷10,000=5 m250,000 \text{ cm}^2 = 50,000 \div 10,000 = 5 \text{ m}^2

Volume Conversion

Volume conversions cube the length conversion because volume has three dimensions. This is a common place for exam mistakes.

UnitConversion
1 m³1,000,000 cm³
1 cm³1000 mm³
1 litre1000 cm³ = 1000 mL
1 m³1000 litres

Key Rule: When converting volume, cube the conversion factor!

  • 1 m = 100 cm, so 1 m³ = 100³ = 1,000,000 cm³
Example

Convert 2 m³ to cm³:

2 m3=2×1,000,000=2,000,000 cm32 \text{ m}^3 = 2 \times 1,000,000 = 2,000,000 \text{ cm}^3

Example

Convert 5 litres to mL:

5 litres=5×1000=5000 mL5 \text{ litres} = 5 \times 1000 = 5000 \text{ mL}

Speed Conversion

Speed combines distance and time, so both units may need conversion. Convert the distance unit and the time unit separately before simplifying.

Speed relates distance to time.

FromToConversion
1 km/hm/s÷ 3.6 (or × 5/18)
1 m/skm/h× 3.6
Example

Convert 72 km/h to m/s:

72 km/h=72×10003600=72×518=20 m/s72 \text{ km/h} = 72 \times \frac{1000}{3600} = 72 \times \frac{5}{18} = 20 \text{ m/s}

Exam Tip

Conversion strategy:

  1. Identify what unit you have and what unit you need
  2. Find the conversion factor
  3. For area: multiply/divide by the SQUARE of the length conversion
  4. For volume: multiply/divide by the CUBE of the length conversion
  5. Double-check: does your answer make sense?

Part 7: Time, Distance, and Speed

The Basic Relationship

Distance, speed, and time form one relationship. If you know any two, you can find the third, but the units must match.

Distance=Speed×Time\text{Distance} = \text{Speed} \times \text{Time}

Or rearranged: Speed=DistanceTimeandTime=DistanceSpeed\text{Speed} = \frac{\text{Distance}}{\text{Time}} \quad \text{and} \quad \text{Time} = \frac{\text{Distance}}{\text{Speed}}

Example

A car travels at 60 km/h for 3 hours. How far does it go?

D=S×T=60×3=180 kmD = S \times T = 60 \times 3 = 180 \text{ km}

Example

A runner covers 400 m in 50 seconds. What's the speed?

S=DT=40050=8 m/sS = \frac{D}{T} = \frac{400}{50} = 8 \text{ m/s}

Example

How long does it take to travel 250 km at 50 km/h?

T=DS=25050=5 hoursT = \frac{D}{S} = \frac{250}{50} = 5 \text{ hours}

Distance-Time Graphs

On a distance-time graph, the gradient represents speed. A steeper line means faster movement, and a horizontal line means no movement.

A distance-time graph shows how distance changes over time.

Key features:

  • Gradient (slope) = Speed
  • Steeper line = faster speed
  • Horizontal line = stationary (not moving)
  • Curved line = changing speed
Example

A car travels at constant 60 km/h for 5 hours:

Time (h) | Distance (km) 0 | 0 1 | 60 2 | 120 3 | 180 4 | 240 5 | 300

Gradient = 300 ÷ 5 = 60 km/h (the speed!)

Distance-time: constant speed (straight line through origin)
Example

A car accelerates, then maintains constant speed, then brakes:

Distance-time: motion, rest, faster motion

Speed-Time Graphs

On a speed-time graph, the height shows speed at each moment. Changes in height show acceleration or deceleration.

A speed-time graph shows how speed changes over time.

Key features:

  • Gradient (slope) = Acceleration
  • Horizontal line = constant speed
  • Upward line = speeding up (acceleration)
  • Downward line = slowing down (deceleration)
  • Area under curve = Distance traveled
Example

A car accelerates uniformly from 0 to 30 m/s in 10 seconds:

Acceleration = 30 ÷ 10 = 3 m/s²

Distance = Area under graph = ½ × base × height = ½ × 10 × 30 = 150 m

(Triangle area because speed increases linearly)

Speed-time: constant acceleration 0→30 m/s in 10 s
Finding Distance from a Speed-Time Graph

Area Under Speed-Time Curve = Distance

The area under a speed-time graph represents distance because speed multiplied by time gives distance.

The area between the speed-time curve and the time axis equals the distance traveled.

You can break complex shapes into simpler shapes:

Example

A journey with three stages:

Stage 1 (0-5 s): Accelerate from 0 to 20 m/s

  • Shape: Triangle
  • Area = ½ × 5 × 20 = 50 m

Stage 2 (5-12 s): Constant speed at 20 m/s

  • Shape: Rectangle
  • Area = 7 × 20 = 140 m

Stage 3 (12-16 s): Decelerate from 20 to 0 m/s

  • Shape: Triangle
  • Area = ½ × 4 × 20 = 40 m

Total distance = 50 + 140 + 40 = 230 m

Speed-time: accelerate, cruise, decelerate
Breaking a Speed-Time Curve into Shapes

Using Trapezium Formula for Area

If you have a trapezium shape (constant acceleration throughout), use:

Area=12(a+b)×h\text{Area} = \frac{1}{2}(a + b) \times h

Where aa and bb are the two parallel sides (speeds) and hh is the base (time).

Example

A car accelerates from 10 m/s to 30 m/s over 8 seconds:

Distance=12(10+30)×8=12(40)×8=20×8=160 m\text{Distance} = \frac{1}{2}(10 + 30) \times 8 = \frac{1}{2}(40) \times 8 = 20 \times 8 = 160 \text{ m}

Speed-time trapezium: a=10, b=30, h=8

Average Speed

Average speed uses total distance and total time for the whole journey. It is not usually the average of the separate speeds unless the time intervals are equal.

When speed changes during a journey, use average speed:

Average speed=Total distanceTotal time\text{Average speed} = \frac{\text{Total distance}}{\text{Total time}}

Example

A journey has two parts:

  • First 100 km at 50 km/h
  • Next 100 km at 100 km/h

Time for first part: T=10050=2T = \frac{100}{50} = 2 hours Time for second part: T=100100=1T = \frac{100}{100} = 1 hour

Total distance = 200 km Total time = 3 hours

Average speed=2003≈66.7 km/h\text{Average speed} = \frac{200}{3} \approx 66.7 \text{ km/h}

Note: It's NOT (50 + 100) ÷ 2 = 75 km/h!

Part 8: Margin of Error

All measurements have some error. The actual value could be slightly higher or lower than the measured value.

Relative Accuracy

Relative accuracy describes how much uncertainty a measurement has. A small absolute error may still matter if the measurement itself is small.

The margin of error depends on the precision of the measuring instrument.

PrecisionExampleRange
Nearest millimetre7 mm± 0.5 mm (6.5 to 7.5 mm)
Nearest centimetre12 cm± 0.5 cm (11.5 to 12.5 cm)
Nearest metre25 m± 0.5 m (24.5 to 25.5 m)

General rule: Margin of error = ± (Half the measuring unit)

Example

A length is measured as 8.5 cm to the nearest mm.

Margin of error = ± 0.5 mm = ± 0.05 cm

Actual length is between:

  • Minimum: 8.5 - 0.05 = 8.45 cm
  • Maximum: 8.5 + 0.05 = 8.55 cm

Maximum and Minimum Values

Rounded measurements represent a range of possible actual values. Maximum and minimum values help you find the largest or smallest possible result.

When you calculate with measured values, the answer also has a range.

Example

Rectangle: length 10 cm ± 0.5 cm, width 6 cm ± 0.5 cm

Maximum area: (10.5)×(6.5)=68.25 cm2(10.5) \times (6.5) = 68.25 \text{ cm}^2

Minimum area: (9.5)×(5.5)=52.25 cm2(9.5) \times (5.5) = 52.25 \text{ cm}^2

Calculated area (using measured values): 10×6=60 cm210 \times 6 = 60 \text{ cm}^2

So the area is between 52.25 and 68.25 cm².

Part 9: Maps and Scale Drawings

A scale shows the ratio between distances on a map/drawing and real distances.

Understanding Scale

A scale drawing keeps shape the same while changing size. The scale tells how a length on the drawing corresponds to a real length.

ScaleMap DistanceReal Distance
1:1001 cm100 cm = 1 m
1:10001 cm1000 cm = 10 m
1:500001 cm50 km

Note: Larger second number = smaller map (less detailed)

Example

On a map with scale 1:50000, the distance between two towns is 8 cm.

Real distance = 8 × 50000 = 400,000 cm = 4 km

Example

Two cities are 30 km apart in reality. On a map with scale 1:100000, how far apart are they?

Map distance = 30 km ÷ 100000 = 3,000,000 cm ÷ 100000 = 30 cm

Scale Areas

Area scale factors are squared because area is two-dimensional. If lengths are multiplied by 3, areas are multiplied by 9.

Important: If scale is 1:n for length, then area scale is 1:n².

Example

On a scale 1:100 map, a field measures 4 cm × 3 cm.

Length scale: 1:100 Area scale: 1:10000 (because 100² = 10000)

Map area = 4 × 3 = 12 cm² Real area = 12 × 10000 = 120,000 cm² = 12 m²

Or: Real dimensions are 4 × 100 = 400 cm = 4 m and 3 × 100 = 300 cm = 3 m Real area = 4 × 3 = 12 m² ✓

Map scale bar (1 cm represents 100 m)

Part 10: Problem-Solving with Measurement

Real-world problems combine multiple concepts.

Example

A cylindrical swimming pool has radius 4 m and depth 1.5 m. How many litres of water does it hold?

Volume = πr2h=π(4)2(1.5)=24π≈75.4\pi r^2 h = \pi(4)^2(1.5) = 24\pi \approx 75.4 m³

Convert to litres: 75.4 m³ × 1000 litres/m³ = 75,400 litres

Example

A car travels at 80 km/h for 2 hours, then at 100 km/h for 1.5 hours.

Distance = (80 × 2) + (100 × 1.5) = 160 + 150 = 310 km Total time = 2 + 1.5 = 3.5 hours Average speed = 310 ÷ 3.5 ≈ 88.6 km/h

Exam Tip

CSEC Measurement exam tips:

  1. Always show units in your answer
  2. Use appropriate precision — if given 2 decimal places, give answer to 2 d.p.
  3. Watch for "perimeter vs. area" — they're different!
  4. For 3D shapes, know the formulas — no cheating!
  5. Check your unit conversions — common mistake area
  6. For scale drawings, remember 1:n means ÷n or ×n
  7. For combined shapes, break them into simple parts