Exercise 29

A father splits his land among his three sons. To the eldest son he assigns one third of the land, plus 80 hectares. To the second, a fourth, plus 20 hectares, and to the third, a fourth. The shares of the land that correspond to each son, in hectares, are:

  1. 280, 160 and 140

  2. 200, 150 and 150

  3. 280, 170 and 150

  4. 280, 140 and 150

  5. 200, 170 and 150

Try to solve it before checking the answer.

  1. 280, 170 and 150

Let \( x \) be the number of hectares the land has.

  • The eldest son gets:   \( \cfrac{x}{3} + 80 \)

  • The seccond son gets:   \( \cfrac{x}{4} + 20 \)

  • The third son gets:   \(\cfrac{x}{4}\)

The assigned shares must add up to \( x \), the total of the Land. Then:

\[ \begin{aligned} &\left( \frac{x}{3} + 80 \right) + \left( \frac{x}{4} + 20 \right) + \frac{x}{4} = x \\[.5em] &\hspace{3em}\Rightarrow \frac{x}{3} + \frac{x}{4} + \frac{x}{4} + 100 = x \\[.5em] &\hspace{3em}\Rightarrow x = \frac{x}{3} + \frac{x}{4} + \frac{x}{4} + 100 \\[.5em] &\hspace{3em}\Rightarrow x – \frac{x}{3} – \frac{x}{4} – \frac{x}{4} = 100 \\[.5em] &\hspace{3em}\Rightarrow 12x – 4x – 3x – 3x = 1200 \\[.5em] &\hspace{3em}\Rightarrow 2x = 1200 \\[.5em] &\hspace{3em}\Rightarrow x = 600 \end{aligned} \]
\[ \begin{aligned} \left( \frac{x}{3} + 80 \right) + \left( \frac{x}{4} + 20 \right) + \frac{x}{4} = x &\Rightarrow \frac{x}{3} + \frac{x}{4} + \frac{x}{4} + 100 = x \\[.5em] &\Rightarrow x = \frac{x}{3} + \frac{x}{4} + \frac{x}{4} + 100 \\[.5em] &\Rightarrow x – \frac{x}{3} – \frac{x}{4} – \frac{x}{4} = 100 \\[.5em] &\Rightarrow 12x – 4x – 3x – 3x = 1200 \\[.5em] &\Rightarrow 2x = 1200 \\[.5em] &\Rightarrow x = 600 \end{aligned} \]

Hence,

  • The eldest son got:

    \[ \frac{600}{3} + 80 = \boldsymbol{280} \, \text{ hectares} \]

  • The seccond son got:

    \[ \frac{600}{4} + 20 = \boldsymbol{170} \, \text{ hectares} \]

  • The youngest son got:

    \[ \frac{600}{4} = \boldsymbol{150} \, \text{ hectares} \]

More Math Reasoning Exercises