Proportional distributions: direct and inverse

Another field of proportionality is what we call Proportional Distributions, that is to say, when we want to distribute a quantity in a proportional way, either being direct or inverse, among several parts.

Example

A grandfather decides to distribute $6.000$ € between his three grandchildren, but instead of giving a third to each one he prefers to do it proportionally to the age of every grandchild, whose ages are $7, 12$ and $21$ years old. How much will each of them receive?

To tackle this type of problems it will be necessary to assign an unknown to each of the parts, so that:

The quantity that corresponds to the $7$ -year-old grandchild will be called $x$ , the one corresponding to the $12$ -year-old grandchild will be $y$ , and the one of the $21$ -year-old grandchild will be $z$ .

As the grandfather has decided, for whatever reason, to distribute the money according to age, the youngest grandchild will have $\frac{x}{7}$ parts of the whole, $\frac{y}{12}$ for the middle-aged and $\frac{z}{21}$ for the oldest. The relation can be outlined as follows:

$\frac{x}{7} = \frac{y}{12} = \frac{z}{21}$

Being $x, y, z$ , the quantity that each grandson will receive, the sum of these quantities being the total quantity to distribute, that is to say, $6.000$ €.

In this point it is necessary to cite another important property of the proportions, it is that: $\frac{a}{b} = \frac{c}{d} = \frac{a + c}{b + d}$

So, in a proportion, when we add the numerators and the denominators of the fractions, we obtain a new fraction that is proportional to anyone implied.

Example

Applying this rule to the example mwe have that: $\frac{x}{7} = \frac{y}{12} = \frac{z}{21} = \frac{x + y + z}{7 + 12 + 21} = \frac{C}{40}$

Since the distributed total quantity, $C = 6000$ €, must be the sum of what is assigned to each grandson $x + y + z$ . As the new obtained fraction is equal to any of the others, it is possible to equal it with every unknown, which will let us find its value:

The youngest grandchild will receive:

$\frac{x}{7} = \frac{C}{40} \Rightarrow \frac{x}{7} = \frac{6000}{40} \Rightarrow 40 x = 7 \cdot 6000 \Rightarrow 40 x = 42.000$

$\Rightarrow x = \frac{42.000}{40} = 1050$ €

They middle-aged grandson will receive:

$\frac{y}{12} = \frac{C}{40} \Rightarrow \frac{y}{12} = \frac{6000}{40} \Rightarrow 40 y = 12 \cdot 6000 \Rightarrow 40 y = 72.000$

$\Rightarrow y = \frac{72.000}{40} = 1800$ €

Finally, the oldest grandson:

$\frac{z}{21} = \frac{C}{40} \Rightarrow \frac{z}{21} = \frac{6000}{40} \Rightarrow 40 z = 21 \cdot 6000 \Rightarrow 40 z = 126.000$

$\Rightarrow z = \frac{126.000}{40} = 3150$ €

If the operation has been well done, the sum of the distributed quantities has to be equal to the whole: $1050 + 1800 + 3150 = 6000$ €.

The previous example is a clear case of directly proportional distribution since the grandchildren with more age receive more money, and vice versa. But:

What would have happen if the grandfather had decided to distribute the money in an inversely proportional form regarding the age of the grandchildren?

That is, the more age the less money is received, and vice versa. It is necessary to build a relation that follows this premise.

If we continue with the same unknown for every grandchild, the youngest will receive a quantity inversely proportional to his age, so that if in the direct distribution he received $\frac{x}{7}$ now he will get $\frac{x}{\frac{1}{7}}$ or, what is the same $7 x$ :

$Distribution directly proportional: \frac{x}{7}$

$Distribution inversely proportional: \frac{x}{\frac{1}{7}}$

So, to express the inverse distribution it is necessary to invert the denominator of the fraction corresponding to each grandchild, so that:

$\frac{x}{\frac{1}{7}} = \frac{y}{\frac{1}{12}} = \frac{z}{\frac{1}{21}}$

Now, to find the fraction comparable to these it will be necessary to add the numerators and the denominators:

$\frac{x}{\frac{1}{7}} = \frac{y}{\frac{1}{12}} = \frac{z}{\frac{1}{21}} = \frac{C}{\frac{1}{7} + \frac{1}{12} + \frac{1}{21}}$

If we operate the denominator that contains the sum of fractions we obtain:

$\frac{1}{7} + \frac{1}{12} + \frac{1}{21} = \frac{12 + 7 + 4}{84} = \frac{23}{84}$

So that the relation of the distribution will stay:

$\frac{x}{\frac{1}{7}} = \frac{y}{\frac{1}{12}} = \frac{z}{\frac{1}{21}} = \frac{C}{\frac{23}{84}}$

Or what is the same:

$7 x = 12 y = 21 z = \frac{84 \cdot C}{23}$

At this point the distributions corresponding to every grandson can already be done.

The youngest will get:

$7 x = \frac{84 C}{23} \Rightarrow x = \frac{84 \cdot 6000}{23 \cdot 7} = \frac{504.000}{161} = 3130, 43$ €

The middle-aged will receive:

$12 y = \frac{84 C}{23} \Rightarrow y = \frac{84 \cdot 6000}{23 \cdot 12} = \frac{504.000}{276} = 1826, 09$ €

And the oldest:

$21 z = \frac{84 C}{23} \Rightarrow z = \frac{84 \cdot 6000}{23 \cdot 21} = \frac{504.000}{483} = 1043, 48$ €

It is possible to verify that everything is correct by adding the quantities to see if the result is $6.000$ € total to be distributed:

$3130, 43 + 1826, 09 + 1043, 48 = 6000$ €.

In the problems of proportional distributions, it is common that the total amount to distribute is not known, but in these cases some clues are given find it out.

Example

Anthony, Claire and Albert are three bartenders who always distribute the month’s tips according to the daily hours that each one works. Anthony works $8$ hours a day and this month he has got $124$ €. If Claire works $6$ hours a day and Alberto $4$ hours a day: how much corresponds to them? How much have the total tips been this month?

The first thing that it is necessary to observe is that it is a proportional distribution. The second thing is to realize that from the amount Anthony receives everything else can be found out.

If we call $x$ the part that corresponds to Anthony, $y$ to that of Claire and $z$ that of Albert, the scheme of the distribution will be:

$\frac{x}{8} = \frac{y}{6} = \frac{z}{4}$

But, in fact, the value of $x$ is known, since the statement says that it represents $124$ €, so equaling the above mentioned fraction with both other fractions will allow us to know the information that is missing.

This will correspond to Claire: $\frac{x}{8} = \frac{y}{6} \Rightarrow \frac{124}{8} = \frac{y}{6} \Rightarrow 8 y = 124 \cdot 6 \Rightarrow$

$8 y = 744 \Rightarrow y = \frac{744}{8} = 93$ €

While this corresponds to Albert: $\frac{x}{8} = \frac{z}{4} \Rightarrow \frac{124}{8} = \frac{z}{4} \Rightarrow 8 z = 124 \cdot 4 \Rightarrow$

$8 z = 496 \Rightarrow z = \frac{496}{8} = 62$ €

Now, to know the total amount of tips, the quickest option consists of adding the quantities that every bartender takes: $124 + 93 + 62 = 279$ €