A rate has two units, one on top and one underneath, and both may need converting. Convert the top and the bottom separately, then combine.
What is the small gap?
A student knows “×1000 for km to m” and stops there, leaving the hours untouched. The result is a number in metres per hour, not metres per second, and the question wanted the latter.
How do you convert step by step?
- Write the rate as a fraction with units, such as 54 km ÷ 1 h.
- Convert the top unit: km to m.
- Convert the bottom unit: h to s.
- Divide and write the new units.
A scaffolded example
An original problem: a school bus travels at 54 km/h. Give the speed in m/s.
Top: 54 km = 54 × 1000 = 54 000 m. Bottom: 1 hour = 60 × 60 = 3600 s.
Speed = 54 000 m ÷ 3600 s = 15 m/s.
Shortcut check: 54 ÷ 3.6 = 15. Size check: 15 m/s is about 15 m per second, which is a sensible bus speed.
What does the common mistake look like?
A student multiplies by 1000 for the top but multiplies by 3600 for the bottom as well: 54 × 1000 × 3600 = 194 400 000. The bottom unit sits under the line, so converting hours into seconds means dividing by 3600, not multiplying.
Seconds are smaller than hours, so an hour holds 3600 of them, and the rate per second must be smaller than the rate per hour in the same length unit.
Self-check
A runner’s speed is 6 m/s. Give it in km/h. Then a liquid has density 0.8 g/cm³, so give it in kg/m³.
Answer
Speed: per hour the runner covers 6 × 3600 = 21 600 m, which is 21.6 km. So 21.6 km/h (or 6 × 3.6).
Density: 0.8 g = 0.0008 kg, and 1 cm³ = 0.000001 m³. So 0.0008 ÷ 0.000001 = 800 kg/m³. Check: water is 1 g/cm³ = 1000 kg/m³, and 0.8 is less, so 800 fits.
Where does this show up in SPM?
In SPM Mathematics, it appears in speed, distance and time questions. In Physics, formulas like v = u + at only work if every quantity is in the same unit system, and density and pressure mix cm³ and m³.
For the same idea with area and volume, read converting square and cubic units. Continue to proportion versus a constant difference or return to the ratio, proportion and rates hub.
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