Razón y proporción

Glosario

Eslora

Imagen no relevante para la comprensión del texto. Pictograma de una embarcación en la que se indica la longitud de la elora

Definición:: Es la longitud de una embarcación desde la proa hasta la popa.
Ejemplo:: La normativa para los buques de una eslora mayor a 24 metros la puedes ver en la web.

Litoral

Imagen no relevante para la comprensión del texto. Fotografía de una playa

Definición:: Costa marítima de un país o territorio.
Ejemplo:: Las playas del litoral gallego se llenaron de turistas.

Zarpar

Imagen no relevante para la comprensión del texto. Fotografía antigua de un puerto en el que está zarpando un barco y en el muelle la gente se despide

Definición:: Cuando un barco abandona el lugar donde estaba atracado.
Ejemplo:: Los familiares agitaban sus manos a modo despedida desde el puerto, mientras el barco zarpaba rumbo a África.

Razón

Pescadora Cuando estamos haciendo referencia a propiedades que podemos medir y las queremos relacionar, escribimos una razón entre ellas. Una razón es un cociente que escribimos como fracción.

En este caso en la fracción no estamos haciendo referencia a las partes de un todo, estamos indicando una relación.

Por ejemplo, la distancia recorrida y el tiempo que tarda en recorrerse. Si decimos, después de 3 horas de navegación el barco estaba a 200 kilómetros de la costa, tenemos la razón \(\displaystyle \frac{200}{3}\).

O podemos relacionar los kilos de algo que se compra con el precio que se paga. Ha pagado 24,15 euros por kilo y medio de xoubas, tenemos la razón \(\displaystyle \frac{24,15}{1,5}\). Fíjate, cuando hablamos de razón en la fracción pueden aparecer decimales.

También podemos establecer una relación entre dos medidas de la misma magnitud. El primer barco que zarpó tenía 27 metros de eslora, mientras que este que va a salir ahora tiene una eslora de 15 metros, la razón es \(\displaystyle \frac{27}{15}\) y como \(\displaystyle \frac{27}{15} = 1,8\) lo que quiere decir es que el primer barco es 1,8 veces el segundo.

Proporción y constante de proporcionalidad

Cuando tenemos dos razones y estas son equivalentes, tenemos una proporción.

Por ejemplo, de las 30 mariscadoras que salieron ayer a trabajar en Aguiño, 20 ya salían a mariscar de niñas con su abuela y su madre. Tenemos la razón \(\displaystyle \frac{20}{30}\). Y de las 75 que salieron ayer a trabajar en Cambados, 50 también iban a mariscar de niñas con su abuela y su madre. En este caso la razón es \(\displaystyle \frac{50}{75}\).

Como se cumple que \(\displaystyle \frac{20}{30} = \frac{50}{75} = \frac{2}{3}\) tenemos que la proporción de mariscadoras que iban a mariscar de pequeñas con su abuela y su madre es la misma en Aguiño y en Cambados, que sería la de dos tercios de las mariscadoras.

La constante de proporcionalidad directa es el resultado de la división entre las dos cantidades. Cuando varias razones tienen la misma constante de proporcionalidad es porque esas razones forman una proporción.

En el caso anterior \(\displaystyle \frac{20}{30} = 0,67\) que como podemos ver es el mismo para las otras razones \(\displaystyle \frac{50}{75} = 0,67, \frac{2}{3} = 0,67\), es por ello que son razones proporcionales.

Una de las propiedades que nos permite ver que dos razones son equivalentes, y por tanto proporcionales, es comprobar que el producto de extremos es igual al producto de medios. Esta propiedad nos servirá para resolver problemas de proporcionalidad.

Como con \(\displaystyle \frac{20}{30}\) y \(\displaystyle \frac{50}{75}\) se cumple que \( \displaystyle 20 \cdot 75 = 50 \cdot 30\), es decir \(1500 = 1500\), las razones son proporcionales, como ya hemos indicado en los párrafos anteriores.

¡Ahora te toca a ti!

Chica pescadora

En esta sección se te proponen una serie de actividades para practicar el concepto de razón y el de proporción.

Relaciona

Relaciona cada expresión con la razón que le corresponde.

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Tarjetas de memoria de proporciones

Pulsa sobre las cartas para descubrir que oculta y encuentra las parejas de razones que forman una proporción.

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Pulsa sobre las cartas para descubrir que oculta y encuentra las parejas de razones que forman una proporción.

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Cálculo de la razón de proporcionalidad

Vamos a practicar la constante de proporcionalidad.

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Calcula la razón de proporcionalidad para \(\frac{{a}}{{b}}\). Redondea a dos cifras decimales.

Su navegador no es compatible con esta herramienta.

Proporciones

Resuelve todas las misiones y completa el desafío.

Resuelve las misiones, une las soluciones y podrás saber los kilómetros que recorrerías si hicieses un paseo por toda la costa de Galicia.

\(\displaystyle \frac{2}{10} = \frac{a}{5} \)

\(\displaystyle \frac{10}{b} = \frac{5}{2} \)

\(\displaystyle \frac{c}{25} = \frac{36}{100} \)

\(\displaystyle \frac{5}{2} = \frac{20}{d} \)

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