Problems from Proportional distributions: direct and inverse

Development:

The first thing that it is necessary to see is that if the distribution is inversely proportional, the one that has taken less time will take more candies, and vice versa.

If we call $x$ the candies that Patricia gets, $y$ those that Paula deserves and $z$ those that Pablo will take, the relation of the distribution will be as follows:

${\displaystyle \frac{x}{\frac{1}{2}}}={\displaystyle \frac{y}{\frac{1}{3}}}={\displaystyle \frac{z}{\frac{1}{4}}}$

Now we have to find a fraction comparable to the previous ones by the rule of the sum:

${\displaystyle \frac{x}{\frac{1}{2}}}={\displaystyle \frac{y}{\frac{1}{3}}}={\displaystyle \frac{z}{\frac{1}{4}}}={\displaystyle \frac{C}{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}}}$

If we operate the denominator with the sum of fractions we get:

${\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{3}}+{\displaystyle \frac{1}{4}}={\displaystyle \frac{6+4+3}{12}}={\displaystyle \frac{13}{12}}$

With this fact and knowing that, as the statement specifies, the total amount of candies to be distributed $C$ is $65$:

${\displaystyle \frac{x}{\frac{1}{2}}}={\displaystyle \frac{y}{\frac{1}{3}}}={\displaystyle \frac{z}{\frac{1}{4}}}={\displaystyle \frac{65}{\frac{13}{12}}}$

So,

$2x=3y=4z={\displaystyle \frac{12\cdot 65}{13}}\Rightarrow 2x=3y=4z={\displaystyle \frac{780}{13}}\Rightarrow 2x=3y=4z=60$

Therefore, Patricia will have:

$2x=60\Rightarrow x={\displaystyle \frac{60}{2}}=30$ candies

On the other hand, Paula will get:

$3y=60\Rightarrow y={\displaystyle \frac{60}{3}}=20$ candies

And Pablo gets:

$4z=60\Rightarrow z={\displaystyle \frac{60}{4}}=15$ candies

Solution:

$30$ candies to Patricia, $20$ to Paula and $15$ to Pablo.

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