Ratio & rate

A rate compares two different quantities, such as speed or work per day. A ratio compares parts of the same whole. For combined rates, add the rates, not the times: if one pipe fills \(\frac{1}{3}\) tank per hour and another fills \(\frac{1}{6}\) tank per hour, together they fill \(\frac{1}{3}+\frac{1}{6}=\frac{1}{2}\) tank per hour, so they take $2$ hours. For average speed over equal distances, use total distance divided by total time, not the average of the two speeds. To split a quantity by a ratio $a:b$, divide it into $a+b$ equal parts; one part is the total divided by $a+b$. Map scale $1:n$ means $1$ unit on the map equals $n$ units in reality. For inverse proportion, more workers means fewer days: total worker-days stays constant.