A plot of land is divided among three heirs. The first receives one third of the land. The second receives half of what is left. The third receives the remaining 200 square metres. What is the total area of the land?

Step-by-step solution

1. 1

   Name the whole and read the wording carefully. Let the total area be $x$ square metres. The critical phrase is that the second heir gets half of what remains, not half of the whole plot — fractions in these problems are always taken of whatever quantity the sentence names, so the base changes as you go.

2. 2

   First share and what is left. The first heir takes

   $\frac{1}{3}x,$

   leaving

   $x-\frac{1}{3}x=\frac{3}{3}x-\frac{1}{3}x=\frac{2}{3}x.$

3. 3

   Second share, taken from the remainder. Half of $\frac{2}{3}x$ is

   $\frac{1}{2}\cdot\frac{2}{3}x=\frac{2}{6}x=\frac{1}{3}x.$

   So the second heir also ends up with exactly one third of the whole plot — the halving and the doubling in $\tfrac{2}{3}$ cancel. This coincidence is the trap the problem is built on: "half of the rest" is not half of the plot.

4. 4

   Third share. The first two shares together are $\frac{1}{3}x+\frac{1}{3}x=\frac{2}{3}x$, so the third heir receives

   $x-\frac{2}{3}x=\frac{1}{3}x.$

5. 5

   Solve for the total and check. Setting the third share equal to the given area:

   $\frac{1}{3}x=200\ \Longrightarrow\ x=600\ \text{m}^{2}.$

   Check: the three shares are $200$, $\tfrac12(600-200)=200$ and $200$, which sum to $600\ \text{m}^{2}$. The land splits into three equal parts, even though the problem never says so directly.