Q.At what temperature do the Celsius and Fahrenheit temperature scales give exactly the same numerical reading? Show your working using the relation $\dfrac{C}{5} = \dfrac{F-32}{9}$.

Let the common reading be $x$, so $C = x$ and $F = x$. Substituting into $\dfrac{C}{5} = \dfrac{F-32}{9}$:

Cross-multiplying: $9x = 5(x - 32) = 5x - 160$.

$9x - 5x = -160 \;\Rightarrow\; 4x = -160 \;\Rightarrow\; x = -40$.

Set $C = F = x$ in the conversion relation and solve the resulting linear equation for $x$.

• Assuming the two scales must agree at $0$ (they do not; $0\,^\circ\text{C} = 32\,^\circ\text{F}$).
• Sign errors when cross-multiplying $9x = 5(x-32)$.