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Situación de Aprendizaje - Construyendo un cine en el centro comercial de la ciudad

Proporciones y porcentajes en la vida diaria

4. Problemas de repartos proporcionales

4. Problemas de repartos proporcionales

Problemas de repartos proporcionales

En los problemas de reparto proporcionales, una cantidad total se distribuye en varias partes siguiendo una determinada relación entre ellas.

Esa relación puede ser:

  • Directa: cuando una parte mayor corresponde a un valor mayor del criterio.
  • Inversa: cuando una parte mayor corresponde a un valor menor del criterio.
\( \displaystyle \text{Reparto directo: } x:y:z = m:n:k \qquad\qquad \text{Reparto inverso: } x:y:z = \frac{1}{m}:\frac{1}{n}:\frac{1}{k} \)
La clave del problema es identificar si la cantidad debe repartirse en el mismo sentido del criterio o en sentido contrario.

Repartos directamente proporcionales

Cuando una cantidad se reparte directamente en proporción a varios valores, quien tiene el doble del criterio recibe el doble, quien tiene el triple recibe el triple, etc.

Ejemplo. Tres personas comparten un gasto de \(60\) € de forma directamente proporcional a los días \(2\), \(5\) y \(3\).
Días 2 5 3
Coste (€) 12 30 18
\( \displaystyle 2+5+3=10,\qquad \frac{60}{10}=6 \)
\( \displaystyle 2\cdot 6=12,\qquad 5\cdot 6=30,\qquad 3\cdot 6=18 \)
Comprobación: \[ \frac{12}{2}=\frac{30}{5}=\frac{18}{3}=6 \] El cociente es constante, por eso el reparto es directamente proporcional.

Método general del reparto directo

Para repartir una cantidad \(C\) en partes directamente proporcionales a \(m\), \(n\) y \(k\), se sigue este procedimiento:

  1. Se suman los términos de la proporción.
    \( \displaystyle m+n+k \)
  2. Se divide la cantidad total entre esa suma para hallar el valor de una unidad proporcional.
    \( \displaystyle p=\frac{C}{m+n+k} \)
  3. Se multiplica ese valor por cada término de la proporción para obtener las partes.
    \( \displaystyle \frac{1}{p}=\frac{m}{x}=\frac{n}{y}=\frac{k}{z} \qquad \iff \qquad x=m\,p,\qquad y=n\,p,\qquad z=k\,p \)
Ejemplo: Si se reparten \(84\) € en proporción \(2:3:7\), entonces:
\( \displaystyle 2+3+7=12 \)
\( \displaystyle p=\frac{84}{12}=7 \)
\( \displaystyle 2\cdot 7=14,\qquad 3\cdot 7=21,\qquad 7\cdot 7=49 \)

Repartos inversamente proporcionales

Cuando la cantidad debe repartirse de forma inversamente proporcional a varios números, primero se transforman esos números en sus inversos.

\( \displaystyle \text{Si el reparto es inverso a } m,\ n,\ k, \text{ entonces se reparte en proporción } \frac1m,\ \frac1n,\ \frac1k \)
Ejemplo. Repartir \(190\) en partes inversamente proporcionales a \(2\), \(4\) y \(5\).

Primero se escriben los inversos:

\( \displaystyle \frac12,\ \frac14,\ \frac15 \)

Después se suman:

\( \displaystyle \frac12+\frac14+\frac15 = \frac{10+5+4}{20} = \frac{19}{20} \)

Se calcula la unidad proporcional:

\( \displaystyle p=\frac{190}{19/20}=200 \)

Finalmente, se obtienen las partes:

\( \displaystyle \frac12\cdot 200=100,\qquad \frac14\cdot 200=50,\qquad \frac15\cdot 200=40 \)
Las partes del reparto son \(100\), \(50\) y \(40\).

Actividades

Resuelve estos problemas aplicando lo aprendido sobre repartos proporcionales.

  1. Reparte \(72\) € en partes directamente proporcionales a \(2\), \(4\) y \(6\).
  2. Reparte \(180\) en partes inversamente proporcionales a \(2\), \(3\) y \(6\).
  3. Tres personas aportan dinero en proporción \(3:5:2\) para pagar una factura de \(200\) €. ¿Cuánto debe aportar cada una?
Ver soluciones

1) \[ 2+4+6=12,\qquad p=\frac{72}{12}=6 \] Partes: \[ 2\cdot 6=12,\qquad 4\cdot 6=24,\qquad 6\cdot 6=36 \]

2) Inversos: \[ \frac12,\ \frac13,\ \frac16 \] Suma: \[ \frac12+\frac13+\frac16=1 \] Entonces: \[ p=\frac{180}{1}=180 \] Partes: \[ \frac12\cdot 180=90,\qquad \frac13\cdot 180=60,\qquad \frac16\cdot 180=30 \]

3) \[ 3+5+2=10,\qquad p=\frac{200}{10}=20 \] Aportaciones: \[ 3\cdot 20=60,\qquad 5\cdot 20=100,\qquad 2\cdot 20=40 \]

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