Glosario
IVA
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Definición:
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Son las iniciales de Impuesto sobre el Valor Añadido y es un impuesto que se aplica a productos o servicios y que debe pagar el consumidor.
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Ejemplo:
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El IVA de la factura de la luz es de un 21%.
Catro vellos mariñeiros
Porcentajes
Porcentajes
Estás a punto de completar la navegación por este bloque. Un poco de historia, unas actividades y al final de esta página abordarás... ¡El reto 1!
Pero antes... ¡Porcentajes!
El concepto de porcentaje, tiene sus orígenes en el siglo I con el emperador Augusto. Lo de pagar impuestos no es nuevo, Augusto recaudaba un impuesto del 1 sobre 100 a los bienes.
Cuando pensamos en porcentajes ya tenemos muy interiorizada esta cantidad, de manera que si la base no fuese 100 nos costaría más hacernos una idea de la magnitud. Seguramente te sea más fácil entender que un pantalón tiene un descuento del 20% que del 200 por mil. ¿No es así?
¿Qué es un porcentaje?
Veamos ahora formalmente qué es un porcentaje:
Un tanto por ciento o porcentaje es una razón con denominador 100. Para representarlo utilizamos el símbolo %, por ejemplo 7% sería "el siete por ciento".
Cuando hablamos de porcentajes lo que estamos haciendo es considerar que tenemos 100 unidades de algo y se indica cuántas cumplen una propiedad.
La constante de proporcionalidad asociada será la cantidad que tomemos dividida entre 100.
Si tenemos en cuenta la información anterior vemos que tenemos diferentes maneras de representar el porcentaje: \(\displaystyle 7 \% = \frac{7}{100} = 0,07\).
Porcentaje de una cantidad
Para calcular el porcentaje de una cantidad podemos multiplicar por la fracción que representa el porcentaje o bien por la constante que lo representa.
Por ejemplo, si tenemos el 7% de 50, lo calculamos así: \(\displaystyle 50 \cdot \frac{7}{100} = 3,5\) o bien \(\displaystyle 50 \cdot 0,07 = 3,5\)
En el caso de que lo que conozca es la cantidad a la que equivale el porcentaje y quiera saber la cantidad inicial, basta despejar en la expresión. Habría que dividir bien por la razón o bien por la constante que representa el porcentaje.
Sabemos que el 7% de una cantidad es 28, ¿de qué cantidad se trata? Es decir, tenemos 7% de x = 28, entonces \(\displaystyle \frac{28}{\frac{7}{100}} = 28 \cdot \frac{100}{7} = 400\), o bien \(\displaystyle \frac{28}{0,07} = 400\).
Porcentajes populares
Como un porcentaje se trata de una razón, tenemos un cociente que podemos simplificar en muchas ocasiones, lo que nos facilitará los cálculos.
- El 50% es \(\displaystyle \frac{50}{100} = \frac{1}{2}\). Será la mitad de una cantidad.
- El 25% es \(\displaystyle \frac{25}{100} = \frac{1}{4}\). Será la cuarta parte de una cantidad.
- El 10% es \(\displaystyle \frac{10}{100} = \frac{1}{10}\). Será una décima parte de una cantidad, dividir entre 10.
Porcentajes y proporcionalidad
Puede que haya quien se esté preguntando: ¿Y por qué me han añadadido este epígrafe de porcentajes aquí? ¿Qué tiene que ver con lo que estaba viendo de proporcionalidad?
Pues bien, vamos a pensarlo un poco, ¿cuando tenemos un porcentaje no tenemos una relación de proporcionalidad directa? Pues claro que sí. Cuando hacemos un porcentaje es como comparar qué pasaría si el total de lo que tenemos fuese 100.
De 25 piratas, llevan parche en el ojo 10. Si tuviese 100 piratas llevarían parche 40. Si ahora duplicase el número de piratas que llevan parche, duplicaría el porcentaje. \(\displaystyle \frac{10}{25} = \frac{40}{100}, \frac{20}{25} = \frac{80}{100}\)
Un extra: descuentos e incrementos
Los porcentajes aparecen de manera habitual para hacer descuentos de una cantidad: "estas botas de agua están rebajadas un 20%", o para incrementar una cantidad: "el incremento por el IVA de este producto es de un 21%".
En estos casos podemos proceder paso a paso:
- Calcular el porcentaje de la cantidad
- Restar (si es descuento) o sumar (si es incremento) a la cantidad inicial el resultado que obtuvimos en el paso 1
Pero también podemos recurrir a algo de aritmética y álgebra para hacerlo de forma directa en un solo paso. Lo que hacemos al proceder paso a paso es \(\displaystyle x \pm x \cdot \frac{p}{100}\). Observa:
\[\displaystyle x \pm x \cdot \frac{p}{100} = x \cdot \left(1 \pm \frac{p}{100}\right)\]
De modo que si quieres calcular un descuento, lo que haces es restarle a 1 la constante que equivale al porcentaje y multiplicarlo por la cantidad sobre la que hay que hacer el descuento. Y si quieres calcular un incremento, en vez de restar sumas. Veamos un ejemplo con el método abreviado y lo puedes hacer en tu libreta por pasos para comprobar que da lo mismo.
Problema 1
Las botas de agua que valen 60 euros, tienen un descuento de un 20%. ¿Cuánto pagaré por ellas finalmente?
\(20 \% = 0,20\)
Por lo que para calcular el precio que pago multiplico por: \(1 - 0,20 = 0,80\)
\(60 \cdot 0,8 = 48\)
Pagaré 48 euros.
Problema 2
El precio de las botas de agua, que valen 60 euros, aparece sin el IVA, que es un 21%. El IVA es un impuesto que hay que añadir al precio. ¿Cuánto pagaré por ellas finalmente?
\(21 \% = 0,21\)
Por lo que para calcular el precio que pago multiplico por: \(1 + 0,21 = 1,21\)
\(60 \cdot 1,21 = 72,6\)
Pagaré 72,6 euros.
¡Ahora te toca ti!
La tripulación enferma
El fin de semana, el cocinero del barco preparó un banquete. Pero tuvo mala suerte, sirvió algo en mal estado y parte de la tripulación enfermó. El capitán está preocupado, pues necesita que al menos haya 8 personas sanas, para que el día a día del barco fluya sin demasiado problema. Si son 15 marineros y ha enfermado un 60%. ¿Tiene el capitán razones para preocuparse?
Piedras preciosas en el tesoro
Después de varios días navegando por fin llegaron a la isla donde el mapa les indicaba que estaba el tesoro. Contaba la leyenda que más del 80% de las piedras preciosas del botín son rubíes. Cuando abren el cofre se encuentran con 60 piedras preciosas y de ellas, 50 son rubíes. ¿Era cierta la leyenda?
La vida pirata
La vida pirata es la vida mejor... Pero ¿con qué capitán prefiero subir?
El capitán Cucho me dice que de las 24 horas del día solo trabajo un 20%, mientras que la capitana Flora me dice que se dedican a comidas 4 horas al día, porque la sobremesa es importante, hay que cantar canciones, y luego de las 20 horas que quedan se trabaja un 24%.
El tiburón mako de los porcentajes
El tiburón mako es el pez más rápido del mundo. Veamos si tú eres igual de rápido calculando algunos de los porcentajes populares.
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¿Cuánto es el {p}% de {x}?
¿Cuánto es el {p}% de {x}?
¿Cuánto es el {p}% de {x}?
¿Cuánto es el {p}% de {x}?
¿Cuánto es el {p}% de {x}?
Su navegador no es compatible con esta herramienta.
Porcentajes
Completa esta tabla con las diferentes expresiones para el porcentaje.
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| Porcentaje | Fracción | Decimal |
|---|---|---|
| 15% | \(\displaystyle \frac{15}{100}\) | @@0,15@@ |
| @@21%@@ | \(\displaystyle \frac{21}{100}\) | 0,21 |
| @@6%@@ | \(\displaystyle \frac{6}{100}\) | @@0,06@@ |
| 12,5% | \(\displaystyle \frac{x}{100}\); \(x = \)@@12,5@@ |
@@0,125@@ |
Su navegador no es compatible con esta herramienta.
Compra en la pescadería
Mariña hizo una gran compra en la pescadería y Valentín, el pescadero, le hizo diferentes descuentos. Calcula lo que pagó por cada cosa y ordena de lo más barato a lo más caro.
- De pulpo llevó 2,2 kilogramos, y el pulpo está a 16,90 €/kilo. Luego tuvo un descuento de un 12%.
- De percebes compra medio kilo y están a 55 €/kilo. El descuento fue del 10%.
- La centolla gallega está a 25 €/pieza y se llevó una. En esto no tuvo descuento.
- En los berberechos es en lo que tuvo un descuento más grande, un 30%. Compró un kilo y medio y estaba el kilo a 30 euros.
- Compró 800 gramos de navajas de la ría de Muros, que tenían un precio de 32 €/kilo. El descuento que le hizo Valentín fue de un 5%.
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01234Su navegador no es compatible con esta herramienta.
Test porcentajes
Selecciona las respuestas correctas y pulsa sobre el botón aceptar.
undefined
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023567891011121314Su navegador no es compatible con esta herramienta.
.png){kind=link}
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