1. A fruit has a water content by weight of $m \%$. When left to dry in the sun, it loses $(m-5) \%$ of this water, leaving it with a water content by weight of $50 \%$. What is the value of $m$ ?
SOLUTION
Let the original weight be 100 . Then $100-m$ is pulp and $m$ water. After the reduction, there is $100-m$ pulp and $m\left(1-\frac{m-5}{100}\right)$ water, and these are equal. Hence $100-m=\left(1-\frac{m-5}{100}\right) m$. Multiplying out, we have $10000-100 m=105 m-m^{2}$ and then, rearranging, we obtain the quadratic equation $m^{2}-205 m+10000=0$. This factorises as $(m-125)(m-80)=0$. As $m \leq 100$ it follows that $m=80$.