The lowest temperature in the night in a city is one-third more than $\frac{1}{2}$ the highest during the day. Sum of the lowest temperature and the highest temperature is 100 degrees. Then what is the lowest temperature?

### Concept & Logic This algebraic word problem requires translating complex chained phrasing into a single equation. The phrase "one-third more than" applied to a base value means multiplying that base value by $\left(1 + \frac{1}{3}\right)$, or $\frac{4}{3}$. Careful translation of the conditions yields a system of two variables that easily collapses into one. ### Step-by-Step Solution * **Given:** Sum of lowest ($L$) and highest ($H$) temperature = 100. $L + H = 100$ * **Calculation:** Translate the relationship for the lowest temperature: "one-third more than $\frac{1}{2}$ the highest" Base value = $\frac{1}{2}H$ Lowest ($L$) = Base + $\frac{1}{3}$(Base) $$ L = \frac{1}{2}H + \frac{1}{3}\left(\frac{1}{2}H\right) $$ $$ L = \frac{1}{2}H + \frac{1}{6}H $$ $$ L = \frac{3}{6}H + \frac{1}{6}H = \frac{4}{6}H = \frac{2}{3}H $$ * Substitute $L$ into the sum equation: $$ \frac{2}{3}H + H = 100 $$ $$ \frac{5}{3}H = 100 $$ $$ H = 100 \times \frac{3}{5} = 60 $$ * Find the lowest temperature $L$: $$ L = 100 - H = 100 - 60 = 40 \text{ degrees} $$ ### Exam Strategy & Shortcut Use the **Options Elimination Method** combined with the $L = \frac{2}{3}H$ ratio. If $L = \frac{2}{3}H$, then the ratio of $L : H$ is $2 : 3$. The total parts are $2 + 3 = 5$ parts. Since the sum is 100, 1 part = 20. The lowest temperature is 2 parts, so $2 \times 20 = 40$. Checking the options immediately yields 40. ### Common Pitfall The most frequent error is miswriting the equation for $L$ as $L = \frac{1}{3} + \frac{1}{2}H$ (adding one-third of a degree) rather than taking one-third *of* the half-value. Precise reading of "one-third more than [value]" is critical. ### Final Answer Therefore, the correct answer is **40 degrees**.