Unit Rates and Complex Ratios

A ratio compares two quantities, but a unit rate is a special kind of ratio where the second quantity is exactly one. For example, instead of saying you ran 10 miles in 2 hours, the unit rate tells us you ran 5 miles in 1 hour ($5$ mph). When we deal with fractions, the concept stays the same: we want to find out how much of the first item exists for every one of the second item. To find this, we simply divide the first number by the second. In math terms, for any ratio $\frac{a}{b}$, the unit rate is the result of $a \div b$.

In the real world, we often compare rates to save money or time. However, it is hard to compare $\frac{2}{3}$ and $\frac{3}{5}$ at a glance. By converting both to unit rates (finding the value for 1 unit), the comparison becomes obvious. The 'better buy' is the item with the lower unit price, while the 'faster' rate is the one with the higher unit value per time period. This process turns complex fractions into simple numbers that our brains can rank easily.

Sometimes we need to use a unit rate to predict future outcomes. Once you have found the constant unit rate, you can multiply it by any amount to find a total. This is the foundation of proportional reasoning. If you know a faucet leaks $\frac{1}{10}$ of a gallon in $\frac{1}{30}$ of an hour, you first find the rate per hour, then you can easily calculate how much water is wasted in an entire day (24 hours).