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Word Problem Mastery for O Level & IGCSE Maths (Speed, Distance, Time)

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Fundamental Relationships Every Student Must Memorise

  • Core formula

    • Distance = speed × time

    • Speed = distance ÷ time

    • Time = distance ÷ speed

  • Use consistent units

    • Convert hours ↔ minutes

    • Convert minutes ↔ seconds

    • Convert km ↔ m when necessary

  • Recognise type of motion

    • Constant speed

    • Changing speed (piecewise)

    • Combined journeys

    • Return journeys

  • Speed always positive

  • Time always positive

  • Distance always non-negative

Unit Conversion Essentials for Exam Accuracy

  • 1 hour = 60 minutes

  • 1 minute = 60 seconds

  • 1 km = 1000 m

  • Typical traps

    • Using km/h with minutes

    • Using m/s with hours

  • Convert before substitution

  • Non-calculator exam tip

    • Convert to simpler numbers (e.g., 1.5 h → 90 min)

Direct Speed Problems: Fast and Clean Solution Patterns

  • Identify distance and time quickly

  • Apply formula directly

  • Avoid mixing units

  • Use proportion shortcuts

    • Double distance → double time at same speed

    • Halve speed → double time for same distance

  • Recognise “average speed” requires total distance ÷ total time, NOT mean of speeds

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level & IGCSE Mathematics Free Material

Time Calculation Techniques

  • Time = distance ÷ speed

  • If answer in decimal form, convert properly to minutes

    • 1.2 h → 0.2 h = 12 min → total = 1 h 12 min

  • Compare times across scenarios

  • When speeds vary in different sections

    • Compute each section separately

    • Add times

  • Avoid merging speeds without splitting journey

Distance Calculation Techniques

  • Used in map reading, travel, word problems

  • Distance = speed × time

  • Always check direction irrelevance for scalar distance

  • For two-part journey

    • Find distance for each part separately

    • Add total distance

Return Journey Problems

  • Different speed going vs returning

  • Common structure

    • Distance same both ways

    • Speed different → time different

  • Average speed formula

    • total distance ÷ total time

  • Avoid incorrect averaging

    • (speed₁ + speed₂)/2 is WRONG unless times equal

Relative Speed Problems (Two Objects Moving)

Same direction

  • Relative speed = difference of speeds

  • Useful for

    • Overtaking problems

    • Gap closing problems

  • Distance closed = relative speed × time

Opposite direction

  • Relative speed = sum of speeds

  • Time when meet = total distance ÷ relative speed

Key points

  • Convert all to same unit

  • Determine direction first

  • Identify meeting or overtaking target

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level & IGCSE Mathematics Free Material

Upstream–Downstream (Boat and River) Problems

  • Downstream speed = boat speed + current speed

  • Upstream speed = boat speed − current speed

  • If current stronger than boat speed

    • upstream travel impossible

  • Separate distance/time logic for upstream and downstream

  • Typical mistakes

    • Adding instead of subtracting

    • Subtracting even when downstream

    • Using wrong formula for round trip

Speed Conversions Between km/h and m/s

  • Convert km/h → m/s

    • multiply by 1000/3600

    • or multiply by 5/18

  • Convert m/s → km/h

    • multiply by 18/5

  • Used frequently in acceleration questions and rate contexts

Multi-Stage Journey Problems (Common Exam Type)

  • Journey divided into segments

    • Different speeds

    • Different times

    • Different distances

  • Solve each part separately

  • Combine results

    • Total distance = sum of segment distances

    • Total time = sum of segment times

    • Average speed = total distance ÷ total time

  • Avoid shortcut use unless proportional

Timetable and Schedule Interpretation

  • Convert departure, arrival times into durations

  • Use 24-hour clock where needed

  • Handle AM–PM changes

  • Identify wait time vs travel time

  • Subtraction must consider hour rollover

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level & IGCSE Mathematics Free Material

Word Problems Involving Speed Differences

Chasing problems

  • Faster object chases slower object

  • Initial lead distance matters

  • Time to catch = lead ÷ relative speed

Meeting problems

  • Two objects start from ends moving towards each other

  • Meeting time = total distance ÷ (v₁ + v₂)

Returning problems

  • Use two distances equal

  • Compare time taken with speed differences

Graph Interpretation (Distance–Time and Speed–Time)

Distance–Time Graphs

  • Gradient = speed

  • Straight line → constant speed

  • Curved line → changing speed

  • Horizontal line → rest

  • Steeper line → faster speed

Speed–Time Graphs

  • Area under graph = distance

  • Horizontal line → constant speed

  • Sloping line → acceleration or deceleration

  • Combine geometry with area of shapes

    • rectangles

    • trapezia

    • triangles

Common Speed Word Problem Patterns

  • “A train leaves earlier than another…”

  • “A runner overtakes another…”

  • “A car travels part of the distance at one speed…”

  • “Two vehicles start from opposite ends…”

  • “A cyclist slows down after halfway…”

Solution strategy

  • Draw quick diagram

  • Label distances and speeds

  • Identify whether difference or sum of speeds needed

  • Split journey into parts

  • Use relative speed when objects interact

Non-Calculator Shortcuts

  • Divide by 4 or 8 using halving repeatedly

  • Multiply by 1.5 using ×3 ÷2

  • Convert decimals into easier fractions

  • Convert hours to minutes before dividing

Two-Vehicle Problems With Different Start Times

  • Identify time gap

  • Distance gap = earlier speed × time gap

  • Use relative speed to determine catch-up time

  • Total time = catch-up time + initial offset depending on required answer

Real-World Interpretations

  • Train schedules

  • Bus journeys

  • Race timing

  • Motion comparisons

  • Journey planning

  • Airplane travel with wind speed

  • River current scenarios

  • Delivery route problems

Exam-Type Traps to Avoid

  • Forgetting unit conversion

  • Using “average of speeds” instead of total distance ÷ total time

  • Mixing minutes and seconds

  • Assuming equal times when not stated

  • Ignoring rest time

  • Using upstream formula incorrectly

  • Forgetting gap distance in overtaking problems

  • Mixing relative speed signs

  • Reading graph incorrectly

  • Assuming distance from gradient without unit consideration

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level & IGCSE Mathematics Free Material

Distance–Time vs Speed–Time Logic Clarification

Distance–Time

  • Gradient shows speed

  • Curved line means variable speed

  • Vertical line impossible

Speed–Time

  • Area shows distance

  • Negative speed does not occur

  • Horizontal zone shows constant speed

Ratio Use in Speed Problems

  • If speed ratios known

    • time inversely proportional

  • Example

    • Speed ratio 2 : 3

    • Time ratio 3 : 2

  • Useful for solving travel comparison questions

  • Also used for fuel consumption patterns

Average Speed Deep Dive

  • Average speed = total distance ÷ total time

  • NOT (speed₁ + speed₂)/2

  • NOT average of individual speeds unless distances equal

  • Use piecewise summation for multi-segment journeys

Journey Reversals and Opposite Directions

  • Cars starting from same point in opposite directions

  • Use sum of speeds

  • Distance between grows = relative speed × time

  • Opposite direction movement always increases separation

Wind and Current Adjusted Speed

  • Plane with wind

    • Ground speed = aircraft speed ± wind

  • Boat with current

    • Effective speed = boat speed ± current

  • Speed sign depends on direction match

  • Calculate each leg separately

Tabular Organisation for Word Problems

  • Organise S, D, T for each segment

  • Avoid confusion when multiple speeds involved

  • Table automatically forces consistency

  • Prevents missing one part of journey

Using Graphs to Solve Journey Word Problems

  • Estimate values from grid

  • Interpret slopes and curvature

  • Determine rest periods

  • Compare speeds visually

  • Read distances using area under speed–time graph

  • Identify acceleration and deceleration zones

Piecewise Journey With Unknown Variables

  • Represent distances symbolically

  • Solve system of equations

  • Use S = D/T arrangement

  • Compare times for two travellers

  • Identify variable using relative speed logic

Check Techniques Before Final Answer

  • Substitute values back into original situation

  • Verify unit correctness

  • Check if time realistic

  • Check if speed reasonable

  • Ensure direction logic consistent

  • Ensure no negative values produced

  • Recalculate each part independently

A Star Habits for Word Problem Mastery

  • Draw diagrams always

  • Convert units before substituting

  • Split complex journeys into segments

  • Use relative speed formula instantly

  • Always compute average speed correctly

  • Use table representation for clarity

  • Check realism of answer

  • Master conversion between km/h and m/s

  • Combine algebra with motion logic

  • Interpret motion graphs confidently

  • Recognise misdirection patterns in wording

  • Predict solution shape before calculating

Hunain Zia

Hunain Zia has previously achieved considerable success in the high-school educational stream, gaining a massive 154 Total A Grades and 7 Major Distinctions, a process in which he broke/ set 11 different world records, including the most A grades ever achieved, most subjects ever appeared in, most distinctions, gaining distinctions across two separate boards (Pearson Edexcel and CAIE) within the same year/ ever, and even 19 hours of constant examination. All records stand intact today. Hunain pursued his Honors Accounting Degree from the prestigious Bentley University, and then completed an LLB (in Honors) from University of London. Currently, he manages multiple social and commercial projects, has founded digital streams and works tirelessly in the education sector.

Educate A Change focuses on truly bringing change in the educational sector worldwide. All our courses are designed to ensure success, and provide complete preparation without requiring any other external material. Therefore, it takes away the expense of academies, books, past paper compilations and other miscellaneous items. Instead, by just the courses we offer, the student can prepare fully for the best grades in their CAIEs, Edexcel and other high-school exams. The earnings from our course-sales are invested in social projects and growing the free educational activities of Educate A Change.