Rate and Speed

Not in PSLE 2026: Speed and Rate topics have been removed from the PSLE Mathematics syllabus under the 2021 MOE curriculum. Students sitting PSLE from 2026 onwards do not need to study this topic — it has moved to Secondary 1. This page is kept for reference only.

Key Concepts

Rate

• A rate compares a quantity to a unit of something else (often time).

• It tells us how much of something happens per unit.

  $$\text{Rate} = \frac{\text{Total quantity}}{\text{Time}}$$

• Examples of rates:

  • $15/hour (money earned per hour)
  • 3 litres/minute (water flowing per minute)
  • 60 words/minute (typing speed)

• Speed is a specific type of rate — it is distance per unit time.

  • All speeds are rates, but not all rates are speeds.

Worked Example: Rate

A tap fills a tank at 4 litres per minute. How long does it take to fill a 60-litre tank?

Answer: 15 minutes

• Speed tells us how fast something is moving.

• It is found using the formula:

  $$\text{Speed} = \frac{\text{Distance}}{\text{Time}}$$

• This means:

  • if something travels a longer distance in the same time, it has a greater speed
  • if something takes more time to travel the same distance, it has a lower speed

The 3 related quantities

• Distance

  • the length of the path travelled by an object
  • common units:
    • metre (m)
    • kilometre (km)

• Time

  • how long the object takes to travel
  • common units:
    • second (s)
    • minute (min)
    • hour (h)

• Speed

  • the distance travelled in a unit of time
  • common units:
    • metres per second (m/s)
    • kilometres per hour (km/h)

Formula triangle idea

You can remember the relationship between speed, distance and time like this:

• Speed = Distance ÷ Time
• Distance = Speed × Time
• Time = Distance ÷ Speed

Average speed

• Sometimes an object does not move at the same speed throughout the whole journey.
• It may:
  • speed up
  • slow down
  • stop for a while
• In such cases, we use average speed.

• Important:
  • use the total distance
  • use the total time
  • include any stopping time if it is part of the journey time given

Converting units of speed

To calculate speed correctly, the units of distance and time must match the unit of speed required.

Common conversions

• 1 km = 1000 m
• 1 h = 60 min
• 1 min = 60 s
• 1 h = 3600 s

Converting speed units

• To convert km/h to m/s:
  • multiply by 1000
  • divide by 3600
  • so overall, divide by 3.6

• To convert m/s to km/h:
  • multiply by 3600
  • divide by 1000
  • so overall, multiply by 3.6

Distance-time graphs

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

• The horizontal axis (x-axis) shows time
• The vertical axis (y-axis) shows distance

What the graph tells us

• A straight sloping line means the object is moving at constant speed
• A steeper line means greater speed
• A less steep line means lower speed
• A horizontal line means the object is not moving because distance does not change with time

Comparing speeds on a graph

• The gradient or steepness of the line shows speed
• Greater gradient = greater speed

Solving speed word problems

When solving word problems:

1. Identify the distance, time, and speed
2. Decide which formula to use
3. Convert units first if needed
4. Substitute the values carefully
5. Write the answer with the correct unit

Real-life examples of speed

• walking speed may be measured in m/s
• car speed is usually measured in km/h
• athletes in races may have speeds compared using the same formula

Important Definitions

• Speed: the distance travelled per unit time.
• Distance: the length of the path travelled by an object.
• Time: the duration taken for an event or journey.
• Average speed: the total distance travelled divided by the total time taken.
• Constant speed: a speed that does not change over time.
• Unit: a standard way of measuring a quantity, such as metre, second, or kilometre per hour.
• Distance-time graph: a graph that shows how distance changes with time.
• Gradient / Steepness: how steep a line is on a graph; on a distance-time graph, it shows speed.
• Stationary: not moving.
• Metres per second (m/s): the speed of an object travelling a certain number of metres each second.
• Kilometres per hour (km/h): the speed of an object travelling a certain number of kilometres each hour.

Worked Examples

Example 1: Finding speed

A boy cycles 600 m in 3 min. Find his speed in m/min.

Step 1: Write the formula

Step 2: Substitute the values

Step 3: Calculate

Step 4: Write the unit

Speed = 200 m/min

Example 2: Finding average speed

A van travels 40 km in the first hour and 20 km in the second hour. Find its average speed for the whole journey.

Step 1: Find total distance

Step 2: Find total time

Step 3: Use the average speed formula

Step 4: Calculate

Answer: 30 km/h

Example 3: Converting units and finding speed

A runner covers 1800 m in 6 min. Find his speed in m/s.

Step 1: Check the units

Distance is in metres, but time is in minutes.
Since the answer must be in m/s, convert minutes to seconds.

Step 2: Use the formula

Step 3: Calculate

Step 4: Write the unit

Answer: 5 m/s

Common Mistakes to Avoid

• Using the wrong formula
  • for example, multiplying distance and time instead of dividing
• Forgetting to convert units before calculating
  • e.g. using km with minutes and giving answer in km/h
• Leaving out the unit in the final answer
• Mixing up average speed with one part of the journey
  • average speed must use total distance and total time
• Forgetting to include stopping time when finding average speed
• Reading a distance-time graph wrongly
  • a horizontal line means stationary, not moving at constant speed
• Thinking that a line higher on the graph always means faster
  • it is the steepness, not the height, that shows speed
• Converting speed units incorrectly
  • remember:
    • km/h to m/s → divide by 3.6
    • m/s to km/h → multiply by 3.6
• Copying numbers wrongly from the question
• Giving answers in the wrong form or wrong unit asked in the question

Exam Tips

• Always underline or note the 3 important quantities:
  • distance
  • time
  • speed
• Check what the question is asking for before calculating.
• Write the formula first:
  • Speed = Distance ÷ Time
  • Distance = Speed × Time
  • Time = Distance ÷ Speed
• If units are different, convert them before substituting into the formula.
• For average speed questions, use the phrase:
  • total distance travelled
  • total time taken
• For distance-time graph questions, describe the line clearly:
  • The object moved at constant speed
  • The object was stationary
  • The object moved faster because the line is steeper
• When comparing two objects on a graph:
  • mention steeper line = greater speed
• Always include units such as:
  • m/s
  • km/h
  • m/min
• Recheck whether your answer is reasonable:
  • a person usually does not walk at 100 km/h
  • a car usually does not move at 2 m/h
• If the answer looks strange, check your unit conversion again.

Quick Summary

• Speed tells how fast an object moves.
• Formula: Speed = Distance ÷ Time
• Rearranged formulas:
  • Distance = Speed × Time
  • Time = Distance ÷ Speed
• Common distance units: m, km
• Common time units: s, min, h
• Common speed units: m/s, km/h
• Average speed = total distance ÷ total time
• Include stopping time when finding average speed if it is part of the whole journey.
• Convert units before calculating when needed.
• 1 km = 1000 m
• 1 h = 60 min = 3600 s
• On a distance-time graph:
  • sloping straight line = constant speed
  • steeper line = faster
  • horizontal line = stationary
• In graphs, speed depends on the steepness of the line, not how high the line is.
• Always give the final answer with the correct unit.