1510.42 – Farmer Fox

Farmer Fox has left his herd of cows to this four daughters. According to the will, the oldest gets half of them, the next youngest gets $\dfrac{1}{4}$ , the third gets $\dfrac{1}{5}$ and the youngest gets the remaining 7 cows. How large is the herd?

Solution

Solution. Let $x$ be the size of the herd. Then

$\begin{align*}x &= \dfrac{x}{2} + \dfrac{x}{4} + \dfrac{x}{5} + 7 \\[1em]&= \left(\dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{5}\right)x + 7 \\[1em]&= (10 + 5 + 4)\dfrac{x}{20} + 7 \\[1em]&=19 \cdot \dfrac{x}{20} + 7\end{align*}$

So $\dfrac{x}{20} = 7$ and $x = 140$ .