Joe drank 1/3 of a glass of milk in one swallow, then drank 4/5 of the remaining milk in another swallow. What fraction of the original milk was left?

This classic fraction puzzle hinges on a common trick: the second action is based on the *remaining* amount, not the original total. After the first swallow removes 1/3 of the milk, the important thing to realize is that 2/3 of the original milk is now left in the glass. This 2/3 becomes the new "whole" amount for the second part of the calculation.

The next swallow consumes 4/5 of that remaining 2/3. A simple way to solve this is to think about what is left behind. If 4/5 was drunk, then 1/5 of the remainder must still be in the glass. To find the final answer, we just need to calculate what 1/5 of the remaining 2/3 is. Multiplying these fractions together (1/5 * 2/3) gives us 2/15, the fraction of the original glass of milk that was left.

This type of problem, involving sequential percentages or fractions, is crucial in many real-world scenarios. It's the same logic used to calculate the final price after a discount is applied to an already on-sale item, or to understand the compounded effects of interest or decay over time. The key takeaway is to always identify what quantity a fraction is actually referring to—the original total or a new, smaller amount.