lesson
A fixed journey: more speed, less time
1 · Learn
Imagine a 12 km journey travelled at a constant positive speed v km/h. Time t in hours is distance divided by speed: t = 12/v. Speed and time are inversely proportional because v × t stays 12. Doubling the speed halves the journey time; it does not subtract a fixed number of hours.
The graph shows speeds from 1 to 12 km/h. At speeds 1, 2, 3, 4, 6 and 12, the times are 12, 6, 4, 3, 2 and 1 hours. These points form a decreasing curve, not a straight line. Between the points the rule still applies because speed can take fractional values.
To estimate the speed for a 5-hour journey, start at t = 5 on the vertical axis, move across to the curve and then down to speed. Read about 2.4 km/h. Checking the equation gives 12/5 = 2.4 exactly for this ideal model. A coarse graph gives only an estimate.
Zero speed is outside this rule's domain: division by zero is undefined, and a stationary traveller does not finish a 12 km journey. A finite positive speed never gives zero time. Do not join the curve to the origin. This simple model leaves out stops and changes of speed.
2 · Worked example
At v = 8 km/h the graph gives about 1.5 hours. The calculation confirms t = 12/8 = 1.5 h. The point (8,1.5) lies below (6,2), consistent with a shorter time at greater speed.
3 · Your turn
Use the supplied curve to estimate the time at 5 km/h, then calculate it. Estimate the speed needed for a 3-hour journey. Why is (0,0) not on this graph?
Check your answer
About 2.4 h; 12/5 = 2.4 h. About 4 km/h; 12/3 = 4 km/h. At zero speed the traveller does not complete the journey, and 12/0 is undefined.
4 · Apply your learning
On squared paper plot a second fixed-distance model t = 24/v using the same positive speeds. Compare times at the same speed and explain why changing the distance changes the curve.
Adult guidance and safety
Check axis quantities, units and scales before reading a curve. Supply squared paper for plotting. Distinguish an estimate from an exact calculation, and a stated mathematical model from measured evidence. Do not infer behaviour outside its domain.
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Check a calculation · Year 5 onward
Use this only when your teacher or adult allows a calculator for the task. Show your thinking first. It does not replace mental or written arithmetic and does not mark your lesson answer.
Powers, roots and trigonometry · Year 7 onward
Use only when the task allows a calculator. Choose one operation at a time and record your working. Trigonometry here always uses degrees (DEG), not radians. Inverse sine, cosine and tangent return an angle, not a reciprocal. Keep exact fractions and π in answers when requested.
Approximate numerical results use up to 12 significant digits. Inverse sine returns −90° to 90°, inverse cosine 0° to 180°, and inverse tangent −90° to 90°; these are principal values, not every possible solution of a trigonometric equation.
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