One third of a garden is planted with corn, and half of the remaining garden is planted with soybeans. What fraction of the whole garden is soybeans?

Step-by-step solution

1. 1

   Take the whole garden as 1. Working with the unit whole turns every "fraction of" into a multiplication and avoids needing an actual area.

2. 2

   Find the remainder after the corn.

   $1 - \frac13 = \frac33 - \frac13 = \frac23$

   This $\tfrac23$ — not the original $1$ — is what the word "half" refers to. Reading "half of the remainder" as "half of the garden" is the misstep this question is built to catch.

3. 3

   Take half of the remainder.

   $\frac12 \times \frac23 = \frac{2}{6} = \frac13$

4. 4

   State the answer. The soybeans occupy $\tfrac13$ of the whole garden, which is option D.

5. 5

   Check the whole garden adds up. Corn $\tfrac13$ + soybeans $\tfrac13$ = $\tfrac23$, leaving $\tfrac13$ unplanted. With a concrete garden of $60\ \text{m}^2$: corn takes $20$, leaving $40$; half of that is $20$ for soybeans, which is $\tfrac{20}{60} = \tfrac13$. The coincidence that corn and soybeans occupy equal shares happens precisely because half of $\tfrac23$ equals $\tfrac13$.