Mathematics, three units · Grade 11
When do you use a distance-speed-time table and when the part done in one hour?
For motion problems build a table of distance, speed and time with one row per traveller, and use distance equals speed times time. Work-rate problems have no distance, so switch to the fraction of the job done in one hour: someone who finishes in 6 hours does one sixth an hour, and those fractions add.
Learning objectives
- Fill a distance, speed and time table before writing any equation
- Set up a pair of equations in two unknowns from a buying or selling situation
- Solve a work rate problem by working with the part completed in one hour
- Say in words what each solution means back in the situation described
The table comes before the equation
For every motion problem, draw a three-column table — distance, speed, time — with one row per moving body or per leg of the journey. Fill in what is known and mark the unknown as x.
The only relation you need is distance equals speed times time, and therefore time equals distance over speed. Every third cell in a row follows from the other two.
The equation comes from the sentence that links the rows: the same distance, or an hour's difference in the times, or the two distances adding to the gap between the towns. The table does not replace that sentence — it only guarantees that every number in it is right.
The three standing traps
Units. If the speed is in kilometres per hour and the time is given in minutes, convert before, not after. 45 minutes is 0.75 of an hour, not 0.45.
Current and wind. A boat going downstream moves at its own speed plus the current, and upstream at its speed minus the current. The same holds for a plane with a tailwind and a headwind.
Stops. If the car took a half-hour break, the travelling time is not the journey time. Separate the two into distinct rows, or the speed you compute will come out lower than the real one.
Work rate and buying
Work problems have no distance, so the table does not help. Ask instead what fraction of the job is done in one hour. Someone who finishes alone in 6 hours does one sixth of it per hour.
When two people work together the hourly fractions add: one sixth plus one eighth is seven twenty-fourths per hour. The joint time is then the reciprocal: twenty-four sevenths of an hour.
Buying and selling problems usually have two unknowns — a quantity and a price — and two sentences: one about the quantity and one about the total. Two sentences give two equations, and the system is solved by substitution.
Worked examples
A car travelled 180 km. Had it gone 15 km/h faster it would have arrived an hour earlier. What was its speed?
- Let x be the original speed, so the time is 180 over x
- At the faster speed the time is 180 over (x plus 15)
- The linking sentence: the second time is one hour less
- 180 over x minus 180 over (x plus 15) equals 1
Answer: 45 km/h. The negative root is rejected because a speed cannot be negative
One worker finishes a job in 6 hours and another in 3. How long together?
- The first does one sixth of the job per hour
- The second does one third per hour
- Together: one sixth plus one third, which is one half per hour
- If half is done in an hour, the whole takes twice as long
Answer: Two hours
A boat covers 24 km downstream in 2 hours and the same distance upstream in 3. Find the speed of the current.
- Downstream: 24 over 2, which is 12 km/h
- Upstream: 24 over 3, which is 8 km/h
- Downstream is boat plus current, upstream is boat minus current
- The difference between 12 and 8 is twice the current
Answer: The current is 2 km/h and the boat's own speed is 10 km/h
Common mistakes
- Converting minutes to hours by moving a decimal point
- 45 minutes is 0.75 of an hour, not 0.45. The conversion is a division by 60, and an error here carries a wrong number through every remaining line.
- Adding times instead of hourly fractions
- In a work problem nothing adds except the fraction completed per unit of time. Adding the hours — six plus three — produces a number with no meaning at all.
- Treating journey time as travelling time
- A break is not travel. If there was a stop, separate the two in the table; otherwise the speed that comes out will be lower than the true one.
What to remember
- Table first, equation second.
- Distance equals speed times time, always.
- For work, switch to the part done in one hour.
- Current and wind add downstream and subtract upstream.
More in Mathematics, three units
- How do you turn a word problem into an equation without going wrong at the start?
- How do you find the equation of the line through two given points?
- How do you sketch a parabola without computing a whole table of values?
- Why does a 10 percent rise followed by a 10 percent fall not return the original price?
- Two classes share an average — what else do you need to compare them?
- How do you find an average from a frequency table rather than from a list?