Model the situation with signed numbers and a rate: $18.4 - 2.6 \times 3.5 + 4.75 - 1.2$. Compute: $2.6 \times 3.5 = 9.1$, so $18.4 - 9.1 = 9.3$, $9.3 + 4.75 = 14.05$, and $14.05 - 1.2 = 12.85$. Final temperature: 12.85°F, which makes sense because overall there was more cooling than warming.

Correct proportion: $\frac{1000\ \text{g}}{1\ \text{kg}}=\frac{3750\ \text{g}}{x\ \text{kg}}$. Cross-multiply: $1000\cdot x=1\cdot 3750\Rightarrow x=3.75$ kg. Unit rate: $3750\ \text{g}\times\frac{1\ \text{kg}}{1000\ \text{g}}=3.75\ \text{kg}$; g cancels. Reasonableness: 3750 g is a few thousand grams, so a few kilograms; 3.75 kg makes sense.

A constant rate means the ratio distance/time stays the same. Compute $r = \frac{d}{t}$: $\frac{150}{2} = 75$ and $\frac{375}{5} = 75$. In $d = rt$, $r$ is 75, so the speed is 75 miles/hour.

The constant of proportionality is the consistent ratio $k=\dfrac{y}{x}$. Using (2, 26): $26/2=13$; using (5, 65): $65/5=13$; using (8, 104): $104/8=13$. So $k=13$, the unit rate (slope) of dollars per hour.

Divide total cookies by hours to get per 1 hour: $1260 \div 7 = 180$, so 180 cookies/hour. Unit rates are useful for comparing production speeds between different days or bakeries.

Divide students by buses to get per 1 bus: $252 \div 9 = 28$, so 28 students/bus. Unit rates help compare how crowded different buses are.

Proportion method: $\frac{0.45\ \text{kg}}{1\ \text{lb}}=\frac{x\ \text{kg}}{150\ \text{lb}}$. Cross-multiply: $1\cdot x=0.45\cdot 150\Rightarrow x=67.5$ kg. Unit rate: $150\ \text{lb}\times\frac{0.45\ \text{kg}}{1\ \text{lb}}=67.5\ \text{kg}$; lb cancels. Reasonableness: kilograms should be less than pounds since 1 lb is less than 1 kg; 67.5 < 150.

A constant rate has a consistent ratio $\frac{d}{t}$. Choice A: $\frac{52}{1}=52$, $\frac{104}{2}=52$, $\frac{156}{3}=52$ (constant). B changes (50, 55, 55). C is quadratic ($t^2$) so the rate varies with $t$. D describes speeding up, not a constant rate.

Total flour for 6 is $2\tfrac{2}{3}=\tfrac{8}{3}$ cups. Per person: $\tfrac{8}{3}\div 6=\tfrac{8}{18}=\tfrac{4}{9}$ cup. For 4 people: $4\cdot \tfrac{4}{9}=\tfrac{16}{9}=1\tfrac{7}{9}$ cups. For 9 people: $9\cdot \tfrac{4}{9}=4$ cups. Extra compared to the 4-person amount: $4-\tfrac{16}{9}=\tfrac{20}{9}=2\tfrac{2}{9}$ cups. These values are proportional to servings, so they are reasonable.

In a proportional relationship, $k=\dfrac{y}{x}$. Compute: $12/4=3$, $18/6=3$, and $24/8=3$. The ratio $y/x$ is the same for all pairs, so $k=3$ (the unit rate/slope).