Unit and temperature conversions

Unit conversion is proportion wearing a different hat. Whenever a measurement of the same physical quantity is re-expressed in different units — kilometres to metres, hours to seconds, kilograms to grams — the new value is always the old value multiplied by one fixed number, the conversion factor $k$:

where: $x$ is the measurement in the old unit, $y$ is the same measurement in the new unit, and $k$ is the constant conversion factor fixed by how the two units relate. Because $k$ never changes, doubling $x$ always doubles $y$ — the defining test of direct proportion.

Building the speed factor from definitions. Speed conversion between km/h and m/s is the workhorse of this topic. Since $1\ \text{km} = 1000\ \text{m}$ and $1\ \text{hour} = 3600\ \text{s}$,

So converting a speed from km/h to m/s means multiplying by $\dfrac{5}{18}$, and converting back from m/s to km/h means multiplying by the reciprocal $\dfrac{18}{5}$:

Other everyday same-quantity conversions — $1\,\text{kg} = 1000\,\text{g}$, $1\,\text{km} = 1000\,\text{m}$, $1\,\text{hour} = 60\,\text{min} = 3600\,\text{s}$ — are all the same story: a single multiplying factor, no addition, no subtraction.

Temperature conversion is the non-example that sharpens the definition. Celsius and Fahrenheit are related by

where: $C$ is the temperature in Celsius and $F$ is the same temperature in Fahrenheit. This is not of the form $y = kx$ because of the standalone "$+32$": at $C = 0$, $F = 32$, not $0$. A relationship only counts as direct proportion if it passes through the origin — zero maps to zero — and keeps a constant ratio throughout. Temperature conversion fails the first test, so doubling $C$ does not double $F$.

Problem: A car travels at a constant speed of $90$ km/h. Express this speed in m/s, and then find how far the car travels in $10$ seconds at this speed.

Step 1 — Convert the speed using the factor $\dfrac{5}{18}$.

So $90$ km/h is the same physical speed as $25$ m/s.

Step 2 — Check by the unitary method. $90$ km/h means $90{,}000$ m covered in $3600$ s. Speed per second $= \dfrac{90000}{3600} = 25$ m/s. Matches Step 1.

Step 3 — Use the converted speed to find distance in 10 seconds. Distance $=$ speed $\times$ time $= 25 \times 10 = 250$ m.

Answer: $90$ km/h $= 25$ m/s, and the car covers $250$ m in $10$ seconds.

Competitive stretch

Quick contrast — does temperature obey the same doubling rule? Convert $C = 10^\circ$ and $C = 20^\circ$ to Fahrenheit and check whether doubling $C$ doubles $F$: $F(10) = \frac{9}{5}(10) + 32 = 18 + 32 = 50$, and $F(20) = \frac{9}{5}(20) + 32 = 36 + 32 = 68$. Doubling $C$ from $10$ to $20$ changed $F$ from $50$ to $68$, not to $100$ — confirming temperature conversion is not a direct proportion, exactly because of the $+32$.

• Inverting the conversion factor. Multiplying by $\dfrac{18}{5}$ when going from km/h to m/s (or vice versa) is the single most common error — always sanity-check with $18$ km/h $= 5$ m/s.
• Treating $F = \frac{9}{5}C + 32$ as a plain proportion. Setting up a cross-multiplied "$\frac{a}{b} = \frac{c}{d}$" for temperature ignores the $+32$ and gives a wrong answer; temperature conversion is affine, not direct proportion.
• Mixing units before converting. Adding minutes to hours or grams to kilograms without first bringing both to the same unit corrupts the whole calculation.
• Assuming every "conversion" is proportional. Always test whether the relationship passes through zero (no added or subtracted constant) before applying $y = kx$ logic to it.

Ask the tutor: If you are unsure which way to multiply for a unit conversion, or want to see why temperature conversion breaks the direct-proportion rule with more examples, ask the AI tutor to walk through it with you.