Artículos

7.5: Porcentaje de aumento o disminución


El salario de una persona puede aumentar en un porcentaje. Estos son los tipos de aplicaciones que investigaremos en esta sección.

Incremento porcentual

Ejemplo 1

Agrega texto aquí.

Solución

Dejar X representan el aumento de salario del vendedor. Luego, podemos traducir el problema en palabras y símbolos.

[ begin {array} {ccccc} colorbox {cyan} {Aumento de salario} & text {is} & colorbox {cyan} {5%} & text {of} & colorbox {cyan} {salario original } x & = & 5 \% & cdot & 4000 end {matriz} nonumber ]

Resolver X.

[ begin {alineado} x = 0.05 cdot 4000 ~ & textcolor {rojo} {5 \% = 0.05.} x = 200 ~ & textcolor {rojo} { text {Multiplicar: 0.05 cdot 4000 = 200.}} end {alineado} nonumber ]

Por lo tanto, el aumento de salario es de $ 200. Para calcular el nuevo salario norte, debemos sumar este aumento al salario original.

[ begin {array} {ccccc} colorbox {cyan} {Nuevo salario} & text {is} & colorbox {cyan} {salario original} & text {plus} & colorbox {cyan} {aumento} N & = & 4000 & + & 200 end {matriz} nonumber ]

Por lo tanto, el nuevo salario es N = $ 4, 200 por mes.

Solución alternativa

Si el vendedor va a recibir un aumento del 5% en su salario, entonces su nuevo salario será el 105% de su salario original. Sea N su nuevo salario mensual. Luego,

[ begin {array} {ccccc} colorbox {cyan} {Nuevo salario} & text {is} & colorbox {cyan} {105%} & text {of} & colorbox {cyan} {salario original } N & = & 105 \% & cdot & 4000 end {matriz} nonumber ]

Resolver norte.

[ begin {align} N = 1.05 cdot 4000 ~ & textcolor {red} {105 \% = 1.05.} N = 4200 ~ & textcolor {red} { text {Multiplicar:} 1.05 cdot 4000 = 4200.} end {alineado} nonumber ]

Misma respuesta.

Ejercicio

A un técnico informático se le concede un aumento salarial del 4%. Si el salario actual del vendedor es de $ 2,800 por mes, ¿cuál será su nuevo salario?

Respuesta

$2,912

Ejemplo 2

Un vendedor que gana un salario de $ 4,500 por mes aumenta su salario a $ 5,000 por mes. ¿Cuál es el porcentaje de aumento?

Solución

Para encontrar el aumento de salario, primero reste el salario original del nuevo salario.

[ begin {alineado} text {Aumento de salario} & text {= nuevo salario - salario original} ~ & = 5000 - 4500 ~ & = 500 end {alineado} nonumber ]

Por lo tanto, el vendedor ve un aumento en el salario de $ 500.

A continuación, deja pag representan el porcentaje de aumento salarial del vendedor. Luego, podemos traducir el problema en palabras y símbolos.

[ begin {array} {ccccc} colorbox {cyan} {Aumento de salario} & text {is} & colorbox {cyan} {qué porcentaje} & text {of} & colorbox {cyan} {salario original } 500 & = & p & cdot & 4500 end {matriz} nonumber ]

La propiedad conmutativa de la multiplicación nos permite cambiar el orden de multiplicación en el lado derecho de esta última ecuación.

[500 = 4500p nonumber ]

Resolver pag.

[ begin {align} frac {500} {4500} = frac {4500p} {4500} ~ & textcolor {red} { text {Divide ambos lados entre 4500.}} frac {1} {9} = p ~ & textcolor {red} { text {Reducir dividiendo el numerador y el denominador de 500/4500 por 500.}} end {alineado} nonumber ]

Necesitamos cambiar pag = 1/9 a un porcentaje. Podemos encontrar una respuesta exacta creando una fracción equivalente con un denominador de 100.

[ begin {align} frac {1} {9} = frac {n} {100} ~ & textcolor {red} { text {Hacer una fracción equivalente.}} 9n = 100 ~ & textcolor {red} { text {Multiplica en cruz.}} frac {9n} {9} = frac {100} {9} ~ & textcolor {red} { text {Divide ambos lados entre 9.}} n = 11 frac {1} {9} ~ & textcolor {red} { text {Convertir 100/9 en fracción mixta.}} end {alineado} nonumber ]

Por lo tanto, el aumento porcentual es

[p = frac {1} {9} = frac {11 frac {1} {9}} {100} = 11 frac {1} {9} \%. nonumber ]

Solución alternativa

Un enfoque alternativo es preguntar qué porcentaje del salario original es igual al nuevo salario. En este enfoque, dejemos pag representan el porcentaje del salario original que equivale al nuevo salario.

[ begin {array} {ccccc} colorbox {cyan} {Nuevo salario} & text {is} & colorbox {cyan} {qué porcentaje} & text {of} & colorbox {cyan} {salario original } 5000 & = & p & cdot & 4500 end {matriz} nonumber ]

Resolver pag.

[ begin {align} 5000 = 4500p ~ & textcolor {red} { text {Cambia el orden de multiplicación.}} frac {5000} {4500} = frac {4500p} {4500} ~ & textcolor {red} { text {Divide ambos lados entre 4500.}} frac {10} {9} = p ~ & textcolor {red} { text {Reducir: Divide el numerador y el denominador de 5000/4500 por 500.}} end {alineado} nonumber ]

Necesitamos cambiar 10/9 a un porcentaje. Nuevamente, crea una fracción equivalente con un denominador de 100.

[ begin {align} frac {10} {9} = frac {n} {100} ~ & textcolor {red} { text {Hacer una fracción equivalente.}} 9n = 1000 ~ & textcolor {red} { text {Multiplica en cruz.}} frac {9n} {9} = frac {1000} {9} ~ & textcolor {red} { text {Divide ambos lados entre 9.}} n = 111 frac {1} {9} ~ & textcolor {red} { text {Convierte 1000/9 en una fracción mixta.}} end {alineado} nonumber ]

Por lo tanto,

[p = frac {10} {9} = frac {111 frac {1} {9}} {100} = 111 frac {1} {9} \%. nonumber ]

Por tanto, el nuevo salario es (111 frac {1} {9} \% ) del salario original. Para encontrar el aumento porcentual, reste el 100% de (111 frac {1} {9} \% ).

[111 frac {1} {9} \% - 100 \% = 11 frac {1} {9} \% nonumber ]

Esto representa un aumento (11 frac {1} {9} \% ) en el salario, que es la misma respuesta obtenida con la primera técnica de solución.

Ejercicio

Un estadístico que gana un salario de $ 3,200 por mes aumenta su salario a $ 3,368 por mes. ¿Cuál es el porcentaje de aumento?

Respuesta

(5 frac {1} {4} \% )

Disminución porcentual

Ejemplo 3

Debido al cierre de un molino, la población de Silvertown disminuye en un 8.5%. Si la población original era de 10.200 almas resistentes, ¿cuál es la nueva población?

Solución

Dejar X representan la disminución de la población. Luego, podemos traducir el problema en palabras y símbolos.

[ begin {array} {ccccc} colorbox {cyan} {Disminución de la población} & text {is} & colorbox {cyan} {8.5%} & text {of} & colorbox {cyan} {población original } x & = & 8.5 \% & cdot & 10200 end {matriz} nonumber ]

Resolver X.

[ begin {alineado} x = 0.085 cdot 10200 ~ & textcolor {rojo} {8.5 \% = 0.085.} x = 867 ~ & textcolor {rojo} { text {Multiplicar:} 0.085 cdot 10200 = 867.} end {alineado} nonumber ]

Por lo tanto, la disminución de la población es 867. Para calcular la nueva población P, debemos restar esta disminución de la población original.

[ begin {array} {ccccc} colorbox {cyan} {población nueva} & text {is} & colorbox {cyan} {población original} & text {minus} & colorbox {cyan} {disminución de población } P & = & 10200 & - & 867 end {matriz} nonumber ]

Por lo tanto, la nueva población es P = 9.333 almas resistentes.

Solución alternativa

Reste 8.5% del 100% para obtener

[100 \% - 8.5 \% = 91.5 \%. Nonumber ]

Así, si el 8,5% de la población abandona la ciudad, el 91,5% de la población se queda. Por lo tanto, la nueva población P se calcula a partir de la original de la siguiente manera:

[ begin {array} {ccccc} colorbox {cyan} {población nueva} & text {is} & colorbox {cyan} {91.5%} & text {of} & colorbox {cyan} {población original } P & = & 91.5 \% & cdot & 10200 end {matriz} nonumber ]

Resolver PAG.

[ begin {alineado} P = 0.915 cdot 10200 ~ & textcolor {rojo} {91.5 \% = 0.915.} P = 9333 ~ & textcolor {rojo} { text {Multiplicar:} 0.915 cdot 10200 = 9333.} end {alineado} nonumber ]

Misma respuesta.

Ejercicio

Varias tiendas minoristas cierran y la población de Atenas disminuye un 7,2% como resultado. Si la población original era de 12,500, ¿cuál es la nueva población?

Respuesta

11,600

Ejemplo 4

Millertown atraviesa tiempos difíciles y su población disminuye de 11,256 a 10,923 en el espacio de un año. ¿Cuál es la disminución porcentual, redondeada a la centésima de porcentaje más cercana?

Solución

Para encontrar la disminución de la población, primero reste la población actual de la población original.

[ begin {alineado} text {Disminución de la población} & = text {población original - población actual} ~ & = 11256 - 10923 ~ & = 333 end {alineado} nonumber ]

Por lo tanto, la población ha disminuido en 333 personas.

A continuación, deja pag representan el porcentaje de disminución de la población. Luego, podemos traducir el problema en palabras y símbolos.

[ begin {array} {ccccc} colorbox {cyan} {Disminución de la población} & text {is} & colorbox {cyan} {qué porcentaje} & text {of} & colorbox {cyan} {población original } 333 & = & p & cdot & 11256 end {matriz} nonumber ]

Resolver pag.

[ begin {align} frac {333} {11256} = frac {11256p} {11256} ~ & textcolor {red} { text {Divide ambos lados entre 11256.}} 0.02958 approx p ~ & textcolor {rojo} { text {Dividir: 333/11256 approx 0.02958.}} end {alineado} nonumber ]

Cambiar pag a un porcentaje, mueva el punto decimal dos lugares a la derecha y agregue un símbolo de porcentaje.

Se nos pide que redondeemos a la centésima de porcentaje más cercana.

Debido a que el dígito de prueba es mayor o igual a 5, agregue 1 al dígito de redondeo y trunque. Es decir,

[p approx 2.96 \%. nonumber ]

Por lo tanto, la población de Millertown disminuye aproximadamente un 2,96%.

Solución alternativa

Un enfoque alternativo es preguntar qué porcentaje de la población original es igual a la nueva población.

[ begin {array} {ccccc} colorbox {cyan} {población nueva} & text {is} & colorbox {cyan} {qué porcentaje} & text {of} & colorbox {cyan} {población original } 10923 & = & p & cdot & 11256 end {matriz} nonumber ]

Resolver pag.

[ begin {align} 10923 = 11256p ~ & textcolor {red} { text {Cambia el orden de multiplicación.}} frac {10923} {11256} = frac {11256p} {11256} ~ & textcolor {red} { text {Divide ambos lados entre 11256.}} 0.97041 approx p ~ & textcolor {red} { text {Dividir:} 10923/11256 approx 0.97041.} end {alineado} sin número ]

Cambiar pag a un porcentaje, mueva el decimal dos lugares a la derecha y agregue un símbolo de porcentaje.

Se nos pide que redondeemos a la centésima de porcentaje más cercana.

Debido a que el dígito de prueba es menor que 5, deje el dígito de redondeo solo y trunque. Es decir,

[p approx 97.04 \%. nonumber ]

Por lo tanto, permanece el 97,04% de la población de Millertown. Para encontrar el porcentaje de disminución (el porcentaje que se fue), reste 97.04% de 100%.

[100 \% - 97.04 \% = 2.96 \% nonumber ]

Por lo tanto, la población de Millertown disminuye en un 2,96%. Misma respuesta.

Ejercicio

El cierre de una fábrica textil hace que la población de la ciudad adyacente disminuya de 8,956 a 7,800. ¿Cuál es el porcentaje de disminución de la población, redondeado a la décima de porcentaje más cercana?

Respuesta

12.9%

Descuento

Otra aplicación importante del porcentaje es el descuento de bienes.

Ejemplo 5

Un par de esquís tiene un precio de 310 dólares. Sin embargo, un letrero en la tienda indica que los esquís tienen un descuento del 15%. ¿Cuál será el nuevo precio de venta de los esquís?

Solución

Dejar D representan el descuento (en dólares) otorgado por nuestro par de esquís. Luego, en palabras y símbolos:

[ begin {array} {ccccc} colorbox {cyan} {Descuento} & text {is} & colorbox {cyan} {15%} & text {of} & colorbox {cyan} {precio marcado original } D & = & 15 \% & cdot & 310 end {matriz} nonumber ]

Resolver D.

[ begin {align} D = 0.15 cdot 310 ~ & ~ textcolor {red} {15 \% = 0.15.} D = 46.50 ~ & ~ textcolor {red} { text {Multiplicar:} 0.15 cdot 310 = 46.50.} end {alineado} nonumber ]

Por tanto, el descuento es de $ 46,50. Para encontrar el nuevo precio de venta, reste este descuento del precio de venta original.

[ begin {alineado} text {Nuevo precio de venta} & text {= precio de venta original - descuento} ~ & ~ = 310 - 46,50 & ~ = 263,50 end {alineado} nonumber ]

Por tanto, el nuevo precio de venta es $ 263,50.

Solución alternativa

Reste 15% de 100% para obtener

[100 \% - 15 \% = 85 \%. Nonumber ]

Es decir, si un artículo tiene un descuento del 15%, entonces su nuevo precio de venta S es el 85% de su precio original marcado.

[ begin {array} {ccccc} colorbox {cyan} {Nuevo precio de venta} & text {is} & colorbox {cyan} {85%} & text {of} & colorbox {cyan} {original precio marcado} S & = & 85 \% & cdot & 310 end {array} nonumber ]

Resolver S.

[ begin {alineado} S = 0.85 cdot 310 ~ & textcolor {rojo} {~ 85 \% = 0.85.} S = 263.50 ~ & textcolor {rojo} { text {Multiplicar:} 0.85 cdot 310 = 263.50.} end {alineado} nonumber ]

Por tanto, el nuevo precio de venta es $ 263,50. Misma respuesta.

Ejercicio

Un par de botas de montaña tiene un precio de $ 200. Durante una venta, las botas tienen un descuento del 8%. ¿Cuál es el nuevo precio de las botas?

Respuesta

$184

Ejemplo 6

Un par de botas de esquí con un precio de 210 dólares se vende por 180 dólares. Encuentre el porcentaje de descuento, corrija a la décima de porcentaje más cercana.

Solución

Podemos encontrar el descuento (en dólares) restando el precio de venta del precio marcado original.

[ begin {alineado} text {Descuento} & text {= precio marcado original - precio de oferta} ~ & ~ = 210 - 180 ~ & ~ = 30 end {alineado} nonumber ]

Por lo tanto, las botas tienen un descuento de $ 30.

Dejar pag representan el porcentaje de descuento. Luego, en palabras y símbolos:

[ begin {array} {ccccc} colorbox {cyan} {Descuento} & text {is} & colorbox {cyan} {porcentaje de descuento} & text {of} & colorbox {cyan} {precio marcado original } 30 & = & p & cdot & 210 end {matriz} nonumber ]

Resolver pag.

[ begin {align} 30 = 210p ~ & textcolor {red} { text {Cambiar el orden de multiplicación.}} frac {30} {210} = frac {210p} {210} ~ & textcolor {red} { text {Divide ambos lados entre 210.}} frac {1} {7} = p ~ & textcolor {red} { text {Reducir: Divide el numerador y el denominador de 30/210 entre 30.}} p approx ~ & textcolor {rojo} { text {Dividir: 1/7} approx 0.1428.} End {alineado} nonumber ]

Cambiar pag a un porcentaje, mueva el punto decimal dos lugares a la derecha y agregue un símbolo de porcentaje.

Redondea a la décima de porcentaje más cercana.

Debido a que el dígito de prueba es mayor o igual a 5, agregue 1 al dígito de redondeo y trunque. Por lo tanto, con corrección a la décima de porcentaje más cercana, el porcentaje de descuento es

[p approx 14.3 \%. nonumber ]

Solución alternativa

Un enfoque alternativo es preguntar qué porcentaje pag del precio marcado original es igual al precio de venta.

[ begin {array} {ccccc} colorbox {cyan} {Nuevo precio de venta} & text {is} colorbox {cyan} {qué porcentaje} & text {of} & colorbox {cyan} {marcado original precio} 180 & = & p & cdot & 210 end {matriz} nonumber ]

Resolver pag.

[ begin {align} 180 = 210p ~ & textcolor {red} { text {Cambia el orden de multiplicación.}} frac {180} {210} = frac {210p} {210} ~ & textcolor {red} { text {Divide ambos lados entre 210.}} frac {6} {7} = p ~ & textcolor {red} { text {Reducir: Divide el numerador y el denominador de 180/210 por 30.}} p approx ~ & textcolor {red} { text {Dividir:} 6/7 approx 0.8571.} end {alineado} nonumber ]

Para cambiar p a un porcentaje, mueva el punto decimal dos lugares a la derecha y agregue un símbolo de porcentaje.

Redondea a la décima de porcentaje más cercana.

Debido a que el dígito de prueba es menor que 5, no haga nada con el dígito de redondeo y trunque. Por lo tanto, corrija a la décima de porcentaje más cercana,

[p approx 85.7 \%. nonumber ]

Por tanto, el nuevo precio de venta es el 85,7% del precio marcado original. Reste 85,7% de 100%.

[100 \% - 85,7 \% = 14,3 \%. Nonumber ]

Es decir, si el nuevo precio de venta es el 85,7% del precio original, el porcentaje de descuento es el 14,3%. Esta es la misma respuesta encontrada con el primer método.

Ejercicio

Una computadora marcada en $ 2,000 se vende con un descuento de $ 1,850. Encuentre el porcentaje de descuento, corrija a la décima de porcentaje más cercana.

Respuesta

7.5%

Ejercicios

1. Un televisor tiene un precio de $ 447. Sin embargo, un cartel en la tienda indica que el televisor tiene un descuento del 20,5%. ¿Cuál será el nuevo precio de venta del televisor? Redondea tu respuesta al centavo más cercano.

2. Un equipo de música está marcado a $ 380. Sin embargo, un letrero en la tienda indica que el equipo estéreo tiene un descuento del 7,5%. ¿Cuál será el nuevo precio de venta del equipo estéreo? Redondea tu respuesta al centavo más cercano.

3. Debido al cierre de una planta de rodamientos de bolas, Anselm atraviesa tiempos difíciles y su población disminuye de 10.794 a 8.925 en el espacio de un año. ¿Cuál es la disminución porcentual, redondeada a la centésima de porcentaje más cercana?

4. Debido al cierre de una fábrica de tala, Carlytown atraviesa tiempos difíciles y su población disminuye de 12,113 a 10,833 en el espacio de un año. ¿Cuál es la disminución porcentual, redondeada a la centésima de porcentaje más cercana?

5. Un portaequipajes está marcado en $ 500. Sin embargo, un letrero en la tienda indica que el portaequipajes tiene un descuento del 3,5%. ¿Cuál será el nuevo precio de venta del portaequipajes? Redondea tu respuesta al centavo más cercano.

6. Un portaequipajes está marcado en $ 405. Sin embargo, un letrero en la tienda indica que el portaequipajes tiene un descuento del 17,5%. ¿Cuál será el nuevo precio de venta del portaequipajes? Redondea tu respuesta al centavo más cercano.

7. Debido al cierre de una fábrica textil, la población de Silvertown disminuye en un 4,1%. Si la población original era de 14 678 almas resistentes, ¿cuál es la nueva población, según la persona más cercana?

8. Debido al cierre de una tienda por departamentos, la población de Petroria disminuye en un 5.3%. Si la población original era de 14.034 almas resistentes, ¿cuál es la nueva población, según la persona más cercana?

9. A un camarero se le concede un aumento salarial del 4,6%. Si el salario actual del cantinero es de $ 2,500 por mes, encuentre el nuevo salario mensual del cantinero, redondeado al dólar más cercano.

10. A un camarero se le concede un aumento salarial del 5,5%. Si el salario actual del cantinero es de $ 2,900 por mes, encuentre el nuevo salario mensual del cantinero, redondeado al dólar más cercano.

11. Un portaequipajes marcado en $ 358 se vende por $ 292. Encuentre el porcentaje de descuento, corrija a la décima de porcentaje más cercana.

12. Una bicicleta marcada con $ 328 se vende por $ 264. Encuentre el porcentaje de descuento, corrija a la décima de porcentaje más cercana.

13. Debido al cierre de una planta de fabricación de automóviles, Carlytown atraviesa tiempos difíciles y su población disminuye de 14.393 a 12.623 en el espacio de un año. ¿Cuál es la disminución porcentual, redondeada a la centésima de porcentaje más cercana?

14. Debido al cierre de una planta de rodamientos de bolas, Mayville atraviesa tiempos difíciles y su población disminuye de 8.494 a 6.609 en el espacio de un año. ¿Cuál es la disminución porcentual, redondeada a la centésima de porcentaje más cercana?

15. Debido al cierre de una planta de fabricación de automóviles, la población de Silvertown disminuye en un 2,4%. Si la población original era de 8.780 almas resistentes, ¿cuál es la nueva población, según la persona más cercana?

16. Debido al cierre de una fábrica textil, la población de Ghosttown se reduce en un 6,1%. Si la población original era de 14.320 almas resistentes, ¿cuál es la nueva población, correcto a la persona más cercana?

17. Una empleada que gana un salario de $ 2,600 por mes aumenta su salario a $ 2,950 por mes. Encuentre el aumento porcentual correcto a la décima de porcentaje más cercana.

18. Una empleada que gana un salario de 3.600 dólares mensuales aumenta su salario a 4.100 dólares mensuales. Encuentre el aumento porcentual correcto a la décima de porcentaje más cercana.

19. Un camarero que gana un salario de $ 4,200 por mes aumenta su salario a $ 4,300 por mes. Encuentre el aumento porcentual correcto a la décima de porcentaje más cercana.

20. Un camarero que gana un salario de $ 3,200 por mes aumenta su salario a $ 3,550 por mes. Encuentre el aumento porcentual correcto a la décima de porcentaje más cercana.

21. Un jardinero recibe un aumento de sueldo del 5,1%. Si el salario actual del jardinero es de $ 3,200 por mes, encuentre el nuevo salario mensual del jardinero, redondeado al dólar más cercano.

22. A un secretario se le concede un aumento salarial del 2,8%. Si el salario actual de la secretaria es de $ 3,600 por mes, encuentre el nuevo salario mensual de la secretaria, redondeado al dólar más cercano.

23. Un televisor marcado en $ 437 se vende por $ 347. Encuentre el porcentaje de descuento, corrija a la décima de porcentaje más cercana.

24. Una cámara marcada en $ 390 se vende por $ 328. Encuentre el porcentaje de descuento, corrija a la décima de porcentaje más cercana.


25. Suponga que el precio de una lata de salsa de tomate de 8 onzas aumentó de $ 0.20 a $ 0.28.

a) ¿Cuál fue la cantidad de aumento?

b) ¿Cuál fue el porcentaje de aumento?

26. La siguiente tabla resume los precios de la gasolina durante el verano en San Francisco, CA. El precio es la cantidad de dólares necesarios para comprar un galón de gasolina sin plomo. Datos de gasbuddy.com.

[ begin {array} {cc} text {Año} & text {Precio por galón} 2003 & 1.80 2004 & 2.28 2005 & 2.57 2006 & 3.20 2007 & 3.28 2008 y 4.61 2009 y 3.01 end {array} nonumber ]

¿Cuál es el porcentaje de aumento o disminución de 2003 a 2005? Redondea tu respuesta al porcentaje entero más cercano.

27. Consulte la tabla de precios de la gasolina en el ejercicio 26. ¿Cuál es el porcentaje de aumento o disminución de 2005 a 2007? Redondea tu respuesta al porcentaje entero más cercano.

28. ¿Cuál es el porcentaje de aumento o disminución de 2007 a 2009? Redondea tu respuesta al porcentaje entero más cercano.

29. Alza de tarifas. Una familia que paga $ 858 mensuales por cobertura médica se enfrenta a un aumento del 39% en las tarifas. ¿Cuál será su nuevo costo mensual después del aumento? Associated Press-Times-Standard 02/09/10 La secretaria del HHS pide a Anthem Blue Cross que justifique el aumento de tarifas.

30. Multa de estacionamiento. Se esperaba que la Agencia de Transporte Metropolitano de San Francisco considerara aumentar las multas por el uso de etiquetas de estacionamiento para discapacitados falsas, robadas o prestadas de $ 100 a $ 825. ¿Cuál es el porcentaje de aumento de esta multa? Associated Press-Times-Standard 01/06/10 Las multas por etiquetas falsas de estacionamiento para discapacitados pueden aumentar en San Francisco.

31. Movimiento industrial. Las regulaciones en California han provocado que los granjeros industriales se muden fuera del estado. La lechería industrial de Idaho pasó de 180 000 vacas en 1990 a 530 000 vacas en 2009. ¿Cuál es el aumento porcentual de las vacas lecheras industriales de Idaho, redondeado al porcentaje más cercano? Associated Press-Times-Standard 02/09/10 Idaho, otros se preparan para el éxodo de las granjas de huevos de California.

32. Grulla chillona. Las poblaciones de grullas chillonas volvieron de forma notable de sólo 15 aves en 1941 a unas 400 aves en todo el mundo en 2010. ¿Cuál es el porcentaje de aumento de la población de grullas chillonas durante los últimos setenta años, redondeado al porcentaje más cercano? Associated Press-Times-Standard 17/03/10 Grulla chillona valiente da esperanza a los expertos en vida silvestre.

33. Submarino. La pérdida del hielo marino del Ártico permitirá un cable de fibra óptica submarino que reducirá el tiempo que lleva enviar un mensaje de Londres a Tokio de 140 milisegundos actuales a 88 milisegundos. Associated Press-Times-Standard 22/01/10 El calentamiento global abre el paso del noroeste para el cable de fibra óptica submarino.

a) ¿Cuál es el porcentaje estimado de disminución en el tiempo de comunicación?

b) A un costo de $ 1.2 mil millones, ¿cuál es el costo por milisegundo de ahorro (redondeado al dólar más cercano)?

34. Salmón Chinook. Durante la temporada 2009 en la cuenca del río Sacramento, se registró un mínimo histórico de 39.500 salmones chinook, muy por debajo de los más de 750.000 contados en 2002. ¿Cuál es el porcentaje de disminución en el recuento de salmones de 2002 a 2009, redondeado al porcentaje más cercano? Associated Press-Times-Standard 02/25/10 Los federales predicen un mejor año para el salmón de California.

35. Horas del dia. En latitudes medias, los días de verano pueden tener hasta 14 horas de luz diurna, mientras que los días de invierno pueden tener hasta 10 horas de luz diurna. ¿Qué porcentaje más de luz diurna hay en verano que en invierno?

36. Expertos cibernéticos. El secretario de Defensa, Robert Gates, dijo que el Pentágono aumentará el número de expertos cibernéticos que puede capacitar cada año de 80 a 250 para 2011. ¿Qué porcentaje de aumento es este? Redondea tu respuesta al porcentaje más cercano. Associated Press Times-Standard 4/19/09

37. Precios de la vivienda. Los datos inmobiliarios del condado de Humboldt, California, se muestran a continuación. Associated Press-Times-Standard 01/17/10 ¿Cómo es nuestro mercado inmobiliario local en comparación con otras regiones?

[ begin {array} {ccc} text {Año} & text {Número de viviendas vendidas} & text {Precio medio de la vivienda} 2000 & 1,358 & $ 152,257 2005 & 1,432 & $ 344,500 2009 & 833 y $ 285,000 end {array} nonumber ]

a) ¿Qué cambio porcentual en el precio promedio de la vivienda ocurrió entre 2000 y 2009?

b) ¿Qué cambio porcentual en las casas vendidas ocurrió entre 2000 y 2009?


Respuestas

1. $355.36

3. 17.32%

5. $482.50

7. 14.076 personas

9. $2,615

11. 18.4%

13. 12.30%

15. 8,569

17. 13.5%

19. 2.4%

21. $3,363

23. 20.6%

25.

a) $ 0.08

b) 40%

27. 28% de aumento

29. $1,193

31. Aumento del 194%

33.

a) 37%

b) Aproximadamente $ 23,076,923 por milisegundo

35. 40% más de luz natural

37.

a) Incremento del 87% en el precio promedio de la vivienda

b) Disminución del 39% en las ventas de viviendas


7.5: Porcentaje de aumento o disminución
















Matemáticas interactivas de Year 8 - Segunda edición


La NRA se abre a las leyes de 'bandera roja'

La NRA se abre a las leyes de 'bandera roja'

Su investigación más reciente analiza cómo las leyes estatales sobre armas afectan la incidencia de homicidios en grandes áreas urbanas.

"Hay muchas conversaciones en este momento sobre las políticas de armas que son efectivas para reducir la violencia", dijo, "y este estudio se suma a un cuerpo de literatura bastante sólido y creciente que muestra que exigir a los posibles compradores de armas de fuego que obtengan una licencia es uno de los mejores políticas para reducir la violencia ".

No está claro por qué un requisito de licencia reduce la tasa de homicidios con armas de fuego. Crifasi dice que puede atribuirse en parte al hecho de que las barreras burocráticas más altas para adquirir armas pueden reducir el número en circulación, lo que significa que hay menos disponibles para ser robadas de hogares y automóviles.

Pero ella dice que las tasas de homicidio más altas en áreas urbanas también están asociadas con la adopción por parte del estado de "leyes de defensa", que permiten una mayor cobertura legal para que las personas se defiendan con armas de fuego.

El bidireccional

Juez rechaza defensa de 'Defiende tu posición' en caso de tiroteo en el teatro de Florida

"En los condados de los estados con leyes de derecho a portar armas y leyes de defensa, vimos aumentos en los homicidios con armas de fuego", dice Crifasi.

Un aspecto de las leyes estatales sobre armas que aún necesita estudio, dice, es el efecto de los diferentes requisitos para obtener un permiso de porte oculto. Algunos estados, como Texas, exigen capacitación y pruebas, mientras que otros, como el estado de Washington, no lo hacen.

"La gente de la política de armas ha tendido a agrupar estos estados con derecho a portar", dice. "Pero los estándares de capacitación son muy, muy diferentes entre los estados. Por lo tanto, nuestro próximo paso en nuestra investigación es analizar si hay elementos de las leyes de porte oculto que son efectivos o quizás protectores contra la violencia con armas de fuego, en comparación con otros".


Burlington Electric propone un aumento de tarifas del 7.5%, primera alza en 12 años

Revista de negocios de Vermont El Departamento de Electricidad de Burlington (BED), una empresa de energía pública sin fines de lucro que presta servicios a la ciudad de Burlington, anunció hoy a la comunidad que propone un aumento del 7.5 por ciento en la tarifa de electricidad que, si es aprobado por la Comisión de Servicios Públicos (PUC), entraría en vigor a finales de este año. El aumento sigue a un período excepcional de 12 años durante el cual BED superó las tendencias de la industria y las presiones inflacionarias a través de una sólida gestión financiera y múltiples esfuerzos de reducción de costos.

Incluso después del aumento de tarifas, las tarifas totales de BED seguirán siendo bajas en comparación con los promedios de Vermont y Nueva Inglaterra:

  • Tarifas residenciales. Se proyecta que la tarifa residencial propuesta se mantendrá aproximadamente un 11.5 por ciento por debajo de la tarifa residencial promedio de Vermont, y más baja que la tarifa residencial promedio de todos los demás estados de Nueva Inglaterra. La factura residencial promedio aumentará en $ 4.92 por mes, y un nuevo programa de asistencia energética BED mitigará el impacto del aumento de tarifas para los clientes de bajos ingresos durante el año fiscal 2022 (FY22).
  • Tarifas comerciales. Para la mayor parte de los clientes comerciales de BED (aquellos con una tarifa de servicio general pequeña), el aumento promedio de la factura será de $ 6.60 por mes. Las tarifas comerciales e industriales de BED serán ligeramente más altas que el promedio de Vermont, pero muy por debajo del promedio de Nueva Inglaterra.
  • Tarifas totales. Si bien las tarifas totales de BED (residencial, comercial e industrial) fueron más altas que el promedio de servicios públicos de Vermont luego de nuestro último caso de tarifas en 2009, al mantener las tarifas estables durante 12 años, las tarifas totales de BED cayeron por debajo del promedio de Vermont y seguirán siendo más bajas que ese promedio tras la implementación de nuestra nueva tarifa propuesta.

El aumento está siendo impulsado principalmente por la pandemia mundial de COVID-19, que ha causado varios impactos financieros negativos en BED. Debido a esos efectos financieros, BED proyecta que comenzará el año fiscal 22 con aproximadamente $ 5 millones menos de efectivo disponible de lo que hubiera sido de otra manera. Los efectos de la pandemia han incluido:

  • Reducir las ventas a los clientes
  • Mayores atrasos debido a que los clientes están atrasados ​​en las facturas y
  • Disminución de las contribuciones de los proyectos de capital de los clientes a medida que se interrumpieron los proyectos de capital y las cadenas de suministro.

“Entendemos que la pandemia global ha sido un desafío para nuestra comunidad, creando dificultades económicas para muchos de nuestros clientes”, dijo Darren Springer, Gerente General de BED. “La pandemia también afectó negativamente a BED, por lo que es necesario presentar un caso de tarifas. Para ayudar a nuestros clientes, actualmente ofrecemos varios programas de ayuda, que incluyen la suspensión de desconexiones por falta de pago, la exención de cargos por pagos atrasados ​​y la provisión de planes de facturación de presupuesto ampliado. BED también continúa abogando por programas de asistencia atrasados ​​para nuestros clientes. Además, para ayudar a compensar el aumento de tarifas para los clientes de bajos ingresos, ofreceremos por primera vez un programa de crédito en la factura mensual para los clientes que están inscritos en el programa estatal de asistencia de combustible. Nuestro equipo de Burlington Electric está increíblemente orgulloso del valor que hemos brindado a nuestra comunidad durante los últimos 12 años sin un aumento de tarifas y espera continuar trabajando duro para mantener bajos los ajustes de tarifas en el futuro, mientras brindamos una atención al cliente excepcional y energía renovable confiable. nuestra comunidad espera y merece ”.

“Durante décadas, BED y nuestra Comisión se han esforzado por reflejar los objetivos de nuestra comunidad al proporcionar energía segura, confiable, asequible y ecológica”, declaró Gabrielle Stebbins, presidenta de la Comisión Eléctrica de Burlington. “Como empresa de energía pública municipal, operamos como una organización sin fines de lucro, siempre esforzándonos por mantener las tarifas lo más bajas posible. Después de una carrera sin precedentes de 12 años sin un aumento de tarifas y luego de los severos impactos de COVID-19, es necesario un cambio en las tarifas para continuar reflejando los objetivos de nuestra comunidad ".

Los impactos adversos de la pandemia y el aumento de los costos de transmisión contribuyen al caso de tarifas
El equipo de BED ha trabajado arduamente para reducir o aplazar los gastos siempre que sea posible y moderar los aumentos de costos controlables durante los últimos años. La tasa de crecimiento promedio de los costos controlables entre el año fiscal 2007 y el año fiscal 2016 fue del 5,84 por ciento, pero luego de la implementación de los pasos de reducción de costos, la tasa de crecimiento promedio se redujo al 3,55 por ciento para el período desde el año fiscal 2016. A pesar de nuestro éxito con los costos controlables, los efectos adversos en BED de la pandemia, el aumento de los costos de transmisión y los proyectos de clientes retrasados ​​que habrían agregado ingresos por ventas han contribuido a la necesidad de un caso de tarifas. Los factores específicos que conducen al caso de tarifas incluyen:

  • Disminución de los ingresos por ventas en $ 2.1 millones desde el inicio de la pandemia como empresa de energía pública, BED no era elegible para ciertos programas de alivio de la pandemia, como el Programa de Protección de Cheques de Pago, que habría reemplazado algunos de los ingresos perdidos de BED.

  • Interrupción de proyectos de capital y cadena de suministro que han impactado las asignaciones de costos laborales y reducido las contribuciones de capital de los clientes en aproximadamente $ 1.5 millones y
  • Aumento significativo tanto en el número total de clientes que se han atrasado en el pago de sus facturas de servicios públicos como en la cantidad de dinero adeudado en esas facturas, superando cualquier asistencia estatal y federal.

12 años de tasas constantes gracias a una sólida gestión financiera y esfuerzos de reducción de costos

BED pudo evitar casos de tarifas durante este extraordinario período de 12 años a través de una sólida gestión financiera y esfuerzos decididos para reducir costos. Estos esfuerzos incluyeron:

  • La reorganización y compra de BED en 2015 redujo el número de empleados de 133 a 120 y se proyectó que reduciría los costos en $ 1.1 millones anuales (hoy BED tiene 118 puestos de tiempo completo)
  • Improvements to BED’s and the City’s credit ratings, with savings of nearly $7 million (net present value as of City fiscal health report 2019) for BED customers through reduced interest rates
  • Five-year contract extension with Sheffield Wind Farm in August 2019, providing a better price and contributing to significantly reduced power supply costs
  • Focus on participating in renewable energy credit markets that have provided between $7-12 million in additional annual revenue during that period to help keep rates low
  • Addition of over 50,000 MWh-equivalents of strategic electrification measures that will offset load declines, reduce rate pressure over the long-term, and displace more than 25,000 tons of carbon dioxide emissions in support of the City’s goal to reach Net Zero Energy by 2030 and
  • Institution of stringent expenditure controls during the pandemic to defer and cut expenses and projects during FY20 and FY21 to mitigate financial impacts.

Including the proposed rate change, BED’s total rates remain well below what they would have been had they just kept pace with the rate of inflation between 2010 and 2021.

Pandemic relief and support for customers

To help our customers handle pandemic-related challenges, BED currently is offering several relief programs and taking steps to secure additional arrearage assistance, including:

  • Suspending disconnections for non-payment
  • Waiving late fees
  • Providing extended budget billing plans
  • Advocating for and implementing State of Vermont financial assistance programs, such as:
    • Vermont COVID-19 Arrearage Assistance Program (VCAAP) – advocating for and implementing an expected second round of funding to help our customers with their electric bill arrearages beyond the nearly $350,000 we helped customers access during the first round and
    • Vermont Emergency Rental Assistance Program (VERAP) – implementing the utility component of this recently created program that helps renters dealing with financial challenges related to the COVID-19 pandemic

    Additionally, for FY22, BED is proposing to help low-income customers through a new energy assistance program designed to mitigate the effects of the proposed rate change. Eligibility requirements include being enrolled in the state fuel assistance program, which supports customers with incomes at or below 185 percent of federal poverty levels. This proposed BED energy assistance program is projected to help more than 1,300 qualified residential BED customers. While the proposed energy assistance program is a temporary relief measure, BED is exploring a permanent low-income rate with an FY23 implementation goal.

    Opportunities for Public Comment

    The Burlington community will have many opportunities to comment on the proposed rate case, including during the following meetings:

    • Wednesday, May 19 – the Burlington Electric Commission will vote on BED’s 7.5 percent rate increase proposal at its meeting scheduled for 5:30pm
    • Thursday, May 20 – Burlington Board of Finance BED Budget Presentation and
    • Monday, May 24 – Burlington Board of Finance and City Council.

    Further, after BED files its rate case with the PUC on or about June 15, members of the Burlington community will have the opportunity to provide comment at public hearings and through written testimony during the PUC’s thorough rate review process, which is expected to span many months. The rates charged to customers of Vermont public utilities must be reviewed and will be approved only if the PUC determines that the proposed rates are just and reasonable. Prior to a final decision from the PUC and in accordance with PUC rules, BED will add a 7.5 percent rate surcharge on customer bills beginning in August 2021. This surcharge will continue until the rate case is finalized, at which time BED will adjust customer bills based on the outcome of the rate case. The public can learn more about PUC rate case procedures from this document.


    Latest Updates

    In the fourth quarter, digital advertising revenue dropped 2 percent from a year earlier, to $90.1 million. Print ad revenue declined 38 percent in the fourth quarter, to $49.1 million. The print ad decline was quickened by the pandemic but was also related to larger trends, The Times said. In the quarter, 65 percent of total ad revenue came from digital, compared with 54 percent a year earlier.

    Fourth-quarter revenue was $509.4 million, a 0.2 percent rise from 2019. Adjusted operating profit rose 1.4 percent from the fourth quarter of 2019, to $97.7 million, and 0.9 percent over the year, to $250.6 million.

    The subscription gains last year put The Times on its way to achieving its stated goal of reaching 10 million by 2025. The increase occurred even as The Times, which started charging for online content in 2011, raised the prices of some digital news subscriptions.

    “With a billion people reading digital news, and an expected 100 million willing to pay for it in English, it’s not hard to imagine that, over time, The Times’s subscriber base could be substantially larger than where we are today,” Ms. Levien, who succeeded Mark Thompson as chief executive in September after seven years at The Times, said in a statement.

    In its report, The Times projected a roughly 15 percent increase in total subscription revenue in the current quarter from the first quarter of 2020, and a jump of 35 to 40 percent in digital subscription revenue. Ad revenue this quarter is expected to fall a little less than 20 percent, the company added.

    The company also announced an increase in its dividend payments to investors, to 7 cents a share from 6 cents, which will cost the company about $46.8 million a year. It was the second time within a year that the board had agreed to raise the dividend, a move that will benefit the Ochs-Sulzberger family that controls The Times.


    Modeling Results

    The results from the increased farm labor supply scenario conform to basic economic principles when the supply of one factor or input of production—such as labor, land, or machinery—becomes more plentiful. Greater availability of temporary nonimmigrant foreign-born farmworkers leads to their increased employment at lower wages. This, in turn, results in longrun increases in agricultural output and exports, above and beyond the growth projected by the base forecast. The increases in output and exports are generally larger in labor-intensive sectors such as fruit, tree nuts, vegetables, and nursery products. By Year 15 of the scenario, these four sectors experience a 1.1- to 2.0-percent increase in output and a 1.7- to 3.2-percent increase in exports, relative to the base forecast. Less labor-intensive sectors, such as grains, oilseeds, and livestock production, tend to have smaller increases, ranging from 0.1 to 1.5 percent for output and from 0.2 to 2.6 percent for exports.

    Accompanying this additional growth in agricultural output and employment, however, would be a relative decrease of about 5.7 percent in the number of U.S.-born and other permanent residents employed as farmworkers and a 3.4-percent relative decrease in their wage rate. In the model, U.S.-born and foreign-born permanent resident workers are assumed to compete with foreign-born temporary nonimmigrant workers in the labor market. A 3.4-percent relative decrease in the wage rate does not mean that the wage rate is projected to fall by 3.4 percent over the 15-year period of the projection. Instead, it means that the wage rate in Year 15 is projected to be 3.4 percent lower in the increased farm labor supply scenario than in the base forecast.

    The longrun results from the decreased unauthorized labor supply scenario show a reduction in the labor supply to agriculture with effects on agricultural output and exports that are opposite in sign from the increased farm labor supply scenario and larger in magnitude. Fruit, tree nuts, vegetables, and nursery production are again among the most affected sectors but with longrun relative declines of 2.0 to 5.4 percent in output and 2.5 to 9.3 percent in exports. These effects tend to be smaller in other, less labor-intensive, parts of agriculture—a 1.6- to 4.9-percent decrease in output and a 0.3- to 7.4-percent decrease in exports.


    7.2: Double Number Lines (15 minutes)

    Actividad

    The purpose of this activity is for students to use double number line diagrams to represent situations of percent increase and decrease. Additionally, students identify the original and new amount in the double number lines to reinforce that the original amount pertains to 100%.

    As students work on the task, monitor for students who created various equations for the last question.

    Launch

    Arrange students in groups of 2. Give 5–8 minutes of quiet work time. After 5 minutes allow students to work with a partner or to continue to work alone.

    For each problem, complete the double number line diagram to show the percentages that correspond to the original amount and to the new amount.

    The gas tank in dad’s car holds 12 gallons. The gas tank in mom’s truck holds 50% more than that. How much gas does the truck’s tank hold?

    Expandir imagen

    At a movie theater, the size of popcorn bags decreased 20%. If the old bags held 15 cups of popcorn, how much do the new bags hold?

    Expandir imagen

    Description: <p>A double number line for “popcorn in cups” with 7 evenly spaced tick marks. On the top number line the number zero is on the first tick mark and the remaining tick marks are not labeled. The bottom number line, starting with the first tick mark, zero percent, 20 percent, 40 percent, 60 percent, 80 percent, 100 percent, and 120 percent are labeled.</p>

    A school had 1,200 students last year and only 1,080 students this year. What was the percentage decrease in the number of students?

    Expandir imagen

    Description: <p>Double number line, top line number of people, evenly spaced tick marks labeled 0, 120, 240, 360, 480, 600, 720, 840, 960, 1080, 1200. Bottom line, corresponding tick marks, first labeled 0% and the rest blank.</p>

    One week gas was $ 1.25 per gallon. The next week gas was $ 1.50 per gallon. By what percentage did the price increase?

    Expandir imagen

    Description: <p>A double number line for “price of gas in dollars” with 9 evenly spaced tick marks. On the top number line starting with the first tick mark zero, zero point 2 5, zero point 5, zero point 7 5, one, one point 2 5, one point 5, one point 7 5 and 2 are labeled. On the bottom number line, zero percent is on the first tick mark and the remaining tick marks are not labeled.</p>

    After a 25% discount, the price of a T-shirt was $ 12. What was the price before the discount?

    Expandir imagen

    Description: <p>A double number line for “price of shirts in dollars” with 6 evenly spaced tick marks. On the top number line, the number zero is on the first tick mark and the remaining tick marks are not labeled. On the bottom number line, starting with the first tick mark, zero percent, 25 percent, 50 percent, 75 percent, 100 percent and 125 percent are labeled.</p>

    Compared to last year, the population of Boom Town has increased 25%.The population is now 6,600. What was the population last year?

    Expandir imagen

    Description: <p>A double number line for "number of people" with 6 evenly spaced tick marks. For the top number line, the number 0 is on the first tick mark and the remaining tick marks are blank. For the bottom number line, starting with the first tick mark, the percentages 0%, 25%, 50%, 75%, 100%, and 125% are labeled.</p>

    Respuesta del estudiante

    Para acceder, consulte a uno de nuestros socios certificados de mensajería instantánea.

    Anticipated Misconceptions

    Students may continue to struggle to recognize the original amount and new amount with the proper percentages on the double number line. Remind them that the original amount always corresponds to 100%.

    Síntesis de actividades

    Select students to share the values they identified as original amount and the new amount for a few problems. Discuss how 100% always corresponds to the original value and when there is an increase in the value the new value corresponds to a percentage greater than the original 100%.

    Select students to share the different equations they came up with. Discuss how the distributive property is useful for finding the percentage that corresponds with the new value instead of the percentage of the change.

    Discuss how solving problems about percent change may require either multiplying or dividing numbers. It can be confusing, but it helps to first express the relationship as an equation and then think about how you can find the unknown number. Looking at the examples below, the first two require multiplication, but the others require division.

    Using the structure A% of B is C:

    • ((1.5) oldcdot 12 = c)
    • ((0.80) oldcdot 15 = c)
    • (a oldcdot (1,!200) = 1,!080)
    • (a oldcdot (1.25) = 1.50)
    • ((0.75) oldcdot b = 12)
    • ((1.25) oldcdot b = 6,!600)

    7.5: Percent Increase or Decrease

    If you do not like the background color, you can change it by highlighting the color you prefer in the scroll box below.

    Should a business increase or reduce the number of units it is producing? Should it rely more or less heavily on borrowed money? The answer depends upon how a change would affect risk and return. Operating leverage is the name given to the impact on operating income of a change in the level of output. Financial leverage is the name given to the impact on returns of a change in the extent to which the firm’s assets are financed with borrowed money. Despite the fact that both operating leverage and financial leverage are concepts that have been discussed and analyzed for decades, there is substantial disparity in how they are defined and measured by academics and practitioners.

    In their 1969 college textbook, Weston and Brigham told some of today’s businessmen and women that, "High fixed costs and low variable costs provide the greater percentage change in profits both upward and downward." [Weston, 86]

    Today, in his 1995 textbook, Brigham says that, "If a high percentage of a firm’s costs are fixed, and hence do not decline when demand decreases, this increases he company’s business risk. This factor is called operating leverage." [Brigham, 425] "If a high percentage of a firm’s total costs are fixed, the firm is said to have a high degree of operating leverage ."

    In their 1970 textbook, Grunewald and Nemmers told them that, "When fixed costs are very large and variable costs consume only a small percentage of each dollar of revenue, even a slight change in revenue will have a large effect on reported profits." [Grunewald, 76]

    In his 1970 textbook, Cherry said that, "Operating leverage, then, refers to the magnified effect on operating earnings (EBIT) of any given change in sales. And the more important, proportionally, are fixed costs in the total cost structure, the more marked is the effect on EBIT." [Cherry, 254]

    In his 1971 textbook, Van Horne said that, "one of the most dramatic examples of operating leverage is in the airline industry, where a large portion of total costs are fixed." [Van Horne, 680]

    Archer and D’Ambrosio in their 1972 textbook said that, "The higher the proportion of fixed costs to total costs the higher the operating leverage of the firm. " [Archer, 421]

    In their 1972 textbook, Schultz and Shultz, said that, "Since a fixed expense is being compared to an amount which is a function of a fluctuating base (sales), profit-and-loss results will not bear a proportionate relationship to that base. These results in fact will be subject to magnification, the degree of which depends on the relative size of fixed costs vis-a-vis the potential range of sales volume. This entire subject is referred to as operating leverage." [Schultz, 86]

    q = quantity

    p = price per unit

    v = variable cost per unit

    f = total fixed costs

    Block and Hirt in their 1997 textbook say that operating leverage measures the effect of fixed costs on the firm, and that the degree of operating leverage (DOL) equals:

    Degree of operating leverage =

    In their 1997 article, Buccino and McKinley define operating leverage as the impact of a change in revenue on profit or cash flow. It arises, they say, whenever a firm can increase its revenues without a proportionate increase in operating expenses. Cash allocated to increasing revenue, such as marketing and business development expenditures, are quickly. "consumed by high fixed expenses." (This is certainly a different definition!)

    In his 1997 article, Rushmore says that positive operating leverage occurs at the point at which revenue exceeds the total amount of fixed costs.

    There seems to be more uniformity in the definition of financial leverage. "Financial leverage," say Block and Hirt, reflects the amount of debt used in the capital structure of the firm. Because debt carries a fixed obligation of interest payments, we have the opportunity to greatly magnify our results at various levels of operations. [Block, 116]

    According to Weston and Brigham back in 1969, the degree of financial leverage is computed as the percentage change in earnings available to common stockholders associated with a given percentage change in earnings before interest and taxes.

    According to Brigham in 1995, "The degree of financial leverage (DFL) is defined as the percentage change in earnings per share [EPS] that results from a given percentage change in earnings before interest and taxes (EBIT), and it is calculated as follows:"

    This calculation produces an index number which if, for example, it is 1.43, this means that a 100 percent increase in EBIT would result in a 143 percent increase in earnings per share. (It makes no difference mathematically if return is calculated on a per share basis or on total equity, as in the solution of the equation EPS cancels out.)

    Clarity in regard to operating and financial leverage is important because these concepts are important to businesses. As Conrad Lortie observes in an article, small and medium-sized business often have difficulty using the highly sophisticated quantitative methods large companies use. Fortunately, he observes, the simple break-even graph is simple and easy to interpret yet it can provide a significant amount of information. The algebra necessary to compute operating and financial leverage, too, is not very complex. Unfortunately, it comes in a several guises not all equally easy to understand or equally useful.

    To make it readily apparent something that is wrong with the typical description of operating leverage, a very simple example is used in Tables 1 and 2. Assumed is that Widget Works, Inc. has fixed costs of $5,000 and variable costs per unit of $1.00. Bridget Brothers, on the other hand, has fixed costs of $2,000 and variable costs per unit of $1.60. Both firms’ selling price is $2.00 per unit. Shown in Tables 1 and 2 (below) are their revenues and costs for the production of up to 25,000 units of output.


    Pennsylvania COVID-19 Early Warning Monitoring Dashboard Update for Feb. 5-11: Case Increases at 19,621 Percent Positivity at 8.0% and 51 Counties with Substantial Transmission

    Governor Tom Wolf and Acting Secretary of Health Alison Beam today released a weekly status update detailing the state’s mitigation efforts based on the COVID-19 Early Warning Monitoring System dashboard, highlighting a seven-day case increase of 19,621, a statewide percent positivity of 8.0% and 51 counties with substantial transmission status.

    The update includes the following:

    • Level of community transmission as a basis for the recommendations for Pre-K to 12 schools to determine instructional models.
    • Data on cases among 5-18-year-olds.
    • Cases that reported visiting a business among potential locations where exposures may have occurred.
    • Actualizado travel recommendations.

    The dashboard is designed to provide early warning signs of factors that affect the state’s mitigation efforts. The data available on the Early Warning Monitoring dashboard includes week-over-week case differences, incidence rates, test percent-positivity, and rates of hospitalizations, ventilations and emergency room visits tied to COVID-19. This week’s update compares the period of February 5 – February 11 to the previous seven days, January 29 – February 4.

    “We are reporting another week of lowered statewide positivity, which is an encouraging sign of as we continue our COVID-19 response,” Gov. Wolf said. “Across the commonwealth, we are getting vaccine into arms to ensure our residents are protected against COVID-19 exposure, but there is still community spread ongoing in our communities, so we cannot get inpatient. We need to stay the course to best fight this virus.”

    As of Thursday, February 11, the state has seen a seven-day case increase of 19,621 cases the previous seven-day increase was 20,815 cases, indicating 1,194 fewer new cases across the state over the past week compared to the previous week.

    The statewide percent-positivity went down to 8.0% from 8.6% last week. We now are seeing three counties who have a positivity rate lower than five percent. There are no counties reporting over 20 percent positivity rate.

    “The Department of Health is laser-focused on working with trusted local providers to get vaccine in arms as quickly as it is available from the federal government as we continue our COVID-19 response,” Acting Sec. of Health Beam said. “The importance of the proven health practices and mitigation efforts remain important. While we await more vaccine, we encourage people to prevent the spread of COVID-19 by wearing a mask, washing hands and maintaining social distance.”

    Community Transmission

    As of Friday’s data, 51 counties were in the substantial level of community transmission, the highest level of transmission.

    There are three levels of transmission: low, moderate and substantial. For the week ending February 12, there was one county in the low level of transmission, 15 counties in the moderate level of transmission, and 51 counties were in the substantial level of transmission.

    • Low – Cameron County
    • Moderate – Armstrong, Bedford, Butler, Cambria, Clarion, Elk, Fayette, Indiana, Jefferson, Somerset, Tioga, Venango, Warren, Washington, Westmoreland
    • Substantial – Adams, Allegheny, Beaver, Berks, Blair, Bradford, Bucks, Carbon, Centre, Chester, Clearfield, Clinton, Columbia, Crawford, Cumberland, Dauphin, Delaware, Erie, Forest, Franklin, Fulton, Greene, Huntingdon, Juniata, Lackawanna, Lancaster, Lawrence, Lebanon, Lehigh, Luzerne, Lycoming, McKean, Mercer, Mifflin, Monroe, Montgomery, Montour, Northampton, Northumberland, Perry, Philadelphia, Pike, Potter, Schuylkill, Snyder, Sullivan, Susquehanna, Union, Wayne, Wyoming, York

    Cases Among 5-18-Year-Olds

    The Department of Health is providing weekly data on the number of statewide cases of COVID-19 among 5 to 18-year-olds.

    Throughout the pandemic, there have been 78,475 total cases of COVID-19 among 5 to 18-year-olds. Of that total, 3,380 occurred between February 5 – February 12. For the week of January 29 – February 4, there were 3,210 cases of COVID-19 among 5 to 18-year-olds.
    Cases by demographic group is available on the DOH website.

    Business Visits

    The Department of Health provides weekly data on the number of individuals who responded to case investigators that they spent time at business establishments (restaurants, bars, gym/fitness centers, salon/barbershops) and at mass gatherings 14 days prior to the onset of COVID-19 symptoms.

    It is important to note that due to the number of cases, the department is prioritizing case investigations to prevent outbreak. In addition to the need for people to answer the call, the significant number of cases helps contribute to the low percentages in case investigation data. All of this reinforces the need for Pennsylvanians to take steps to prevent the spread of COVID-19.

    Of the 24,884 cases reported between January 31 – February 6 and excluding Philadelphia County residents and those who answered the digital case investigation, 11.9 percent (3,067) provided an answer to the question as to whether they spent time at a business establishment.
    Of those who did provide an answer, 1.3 percent, or 333, answered yes, they visited a business establishment 14 days prior to onset of symptoms:

    • 35 percent (116) of those who said yes reported going to a restaurant
    • 37 percent (123) of those who said yes reported going to some other business establishment
    • 11 percent (37) of those who said yes reported going to a bar
    • 18 percent (61) of those who said yes reported going to a gym/fitness center and
    • 8 percent (26) of those who said yes reported going to a salon/barbershop.

    Of the 25,884 cases, 11.9 percent (3,088) answered the question as to whether they attended a mass gathering or other large event. Of the 9 percent of cases, 7.5 percent (230) answered yes to whether they attended a mass gathering or other large event 14 days prior to onset of symptoms.

    Compared to data reported on February 8, this week’s data saw an increase for people who reported going to some other business (37 percent vs. 34 percent last week), going to the gym (18 percent vs. 15 percent last week). The data saw a decrease for people who reported going to a restaurant (35 percent vs. 38 percent last week) and those going to the bar (11 percent vs. 12 percent last week). The number of those going to a salon/barbershop remained the same at 8 percent compared to last week. The number of those who attended a mass gathering or other large event increased from 7.3 to 7.5 percent compared to last week.

    In addition to the traditional case investigation, the Department of Health launched the new Connect & Protect Form as a means to complete a digital case investigation. During the week of January 30 – February 6, there were 1,101 forms completed and returned. Of the 99 percent, or 1,086 people, who answered whether they spent time at any business establishment two days before symptom onset or positive collection date if asymptomatic, 14.5 percent, or 160 individuals, individuals answered yes:

    • 54 percent (87) of those who said yes reported going to a restaurant
    • 4 percent (7) of those who said yes reported going to some other business establishment
    • 16 percent (26) of those who said yes reported going to a bar
    • 22 percent (35) of those who said yes reported going to a gym/fitness center and
    • 8 percent (12) of those who said yes reported going to a salon/barbershop.

    The numbers above highlight business settings and mass gatherings as possible sites for transmission. It is important to note that currently Pennsylvania is experiencing widespread community transmission. With less than 10 percent of those asked about what types of businesses they visited or if they attended a mass gathering responding to the question, the department is again reminding Pennsylvanians that it is essential that they answer the phone when case investigators call and to provide full and complete information to these clinical professionals or on the Connect & Protect form the case investigator provides.

    Travel Recommendations

    The Department of Health, along with the Centers for Disease Control and Prevention, continue to discourage any nonessential travel, even for people who are fully vaccinated.

    In November, the Department of Health provided an updated travel order requiring anyone over the age of 11 who visits from another state to provide evidence of a negative COVID-19 test or place themselves in a travel quarantine for 14 days upon entering Pennsylvania. Travel quarantine guidance was changed to 10 days on Dec. 5 based on new CDC guidance.

    This order does not apply to people who commute to and from another state for work or medical treatment, those who left the state for less than 24 hours, or those complying with a court order, including child custody.

    It is important that people understand that this Order is in place to prevent the spread of COVID-19 in Pennsylvania. A concerning number of cases have been linked to travel, and if people are going to travel, we need them to take steps to protect themselves, their loved ones and their community, and that involves having either a negative test, or placing themselves in a quarantine.

    Gov. Wolf continues to prioritize the health and safety of all Pennsylvanians through the COVID-19 pandemic. Pennsylvanians should continue to take actions to prevent the spread of COVID-19, regardless of in what county they live. This includes wearing a mask or face covering any time they are in public. COVID-19 has been shown to spread easily in the air and contagious carriers can be asymptomatic. Pennsylvanians are encouraged to wash their hands, social distance, avoid gatherings and download COVID Alert PA.


    Voter Turnout in New York City Jumped 7.5% from 2016 to 2020

    Despite concerns about how the coronavirus pandemic may affect voter turnout, more than 2.97 million New York City voters cast a ballot in the 2020 general election, an increase of about 7.5% or 210,000 more votes than were cast in the city in the last presidential election, in 2016.

    By election night, more than 2.25 million New York City residents had voted in person, 1.1 million of whom cast a ballot in the nine days of early voting before Election Day. As expected, absentee voting swelled, with more than 720,000 absentee ballots returned to the New York CIty Board of Elections by the November 10 deadline. Counting those ballots just began and is expected to take weeks.

    With 5.56 million registered voters in the city as of November 1, the turnout rate was roughly 53.4%. The rate was slightly lower than the 56.3% in 2016 there were fewer registered voters in the city that year – 2.76 million out of 4.9 million registered voters cast a ballot. There are roughly 13.56 million registered voters statewide in New York.

    President-elect Joe Biden easily won New York State and, as expected, is winning New York City by an even larger margin. The city is home to 3.74 million registered Democrats, roughly 568,000 Republicans, and more than 1 million party-unaffiliated voters, as well as many thousands of voters registered with smaller political parties. But President Donald Trump received more votes in the city this year than four years ago – even without absentee ballots, he had garnered 586,900 votes compared to a total of 494,549 in 2016.

    An analysis by Steven Romalewski, director of the mapping service at the Center for Urban Research, CUNY Graduate Center, shows that from 2016 to 2020 turnout in the city increased the most on Staten Island, large portions of Queens, and parts of Brooklyn. In Manhattan, turnout only increased slightly while in large parts of the Bronx, fewer voters cast ballots this year than four years ago. It’s unclear what role the coronavirus pandemic had in boosting or suppressing voter turnout.

    “I think it reflects the intensity of the race and the heightened attention to this particular election. So, as far as civic engagement in New York City, that's a good thing,” Romalewski said in a phone interview.

    Recognizing the challenges posed by the pandemic, state lawmakers and Governor Andrew Cuomo took several steps to make voting easier in the election. These came on top of the fact that early voting had been passed at the state level last year and was being implemented in a presidential election for the first time.

    The governor signed several bills into law on August 20 that immediately allowed all voters to request an absentee ballot by citing the risk of contracting COVID-19 as an excuse allowing voters to postmark their ballots as late as Election Day and mandating that absentee ballots received the day after the election without a postmark are counted.

    New York State still does not allow no-excuse absentee ballots, which requires an amendment of the state constitution. The state Legislature has already voted to pass that amendment once and must do so again before presenting it to voters for their approval as a ballot question, which is also true for same-day voter registration which would allow new voters to register and vote on the same day.

    “Anything that helps increase turnout, makes it easier for people to vote is good,” Romalewski said. “It doesn't seem to make sense to have artificial restrictions on when you can submit an absentee ballot, whether you can vote by mail or not. And I think the early voting definitely helped in terms of making it easier for people to vote in a less crowded indoor environment.”

    Among the neighborhoods that saw the highest increase in votes – between 5,001 to 10,995 more votes – were the Upper East Side and the Upper West Side in Manhattan Greenpoint, South Williamsburg, Cobble Hill, Carroll Gardens, Gowanus, Windsor Terrace, Kensington, Borough Park, Midwood, Sheepshead Bay and Brighton Beach in Brooklyn and large swathes of western, central and northeastern Queens including Sunnyside, Ridgewood, Long Island City, Queensbridge, Ravenswood, Astoria, Woodside, Maspeth, Ditmars Steinway, Forest Hills, Rego Park, Jamaica Hills, Jamaica Estates, Forest Hills, Middle Village, Kew Gardens, Fresh Meadows, Bay Terrace, Auburndale, Clearview, Douglaston and Little Neck.

    Notably, according to Romalewski’s analysis, two Assembly districts – District 80 in the Bronx and District 56 in Brooklyn – saw a marked decrease in turnout, of between 5,001 and 6,888 fewer votes. District 56 had one of the few competitive Assembly races in the city, at least in the Democratic primary, since longtime Democratic Assemblymember Tremaine Wright did not run for reelection while unsuccessfully seeking a State Senate seat. Democrat Stefani Zinerman won Wright’s seat, running uncontested in the general election. In District 80, first-term Democratic Assemblymember Nathalia Fernandez handily won reelection, defeating nominal general election opposition in the heavily Democratic district.

    There were parts of the city that saw a smaller drop in turnout, up to 2,500 fewer votes. That includes Brooklyn’s Assembly District 60, which covers New Lots, East New York and Starrett City and is represented by Assemblymember Charles Barron, who was reelected unopposed and Manhattan’s Assembly District 71, which is represented by Assemblymember Al Taylor and includes Hudson Heights, Sugar Hill and Hamilton Heights. Taylor was also reelected and did not have an opponent in the general election.

    Large swathes of the Bronx saw a similar decrease in voting including Fordham Heights, Fordham Manor, West Bronx, Mt. Hope, Crotona, Belmont, Claremont Village, Morrisania, Concourse Village, Charlotte Gardens, Highbridge, Mt. Eden, Soundview, Clason Point, Longwood, and Hunts Point.

    While the presidential election has shown additional encouraging signs of participation in a city that has long been known for voter apathy, but has seen marked increases in 2018 and now 2020, the turnout percentages are still relatively low, and a real test is coming in the municipal elections next year. The future of the city will be on the ballot via June primaries and the fall general election as the city holds elections for mayor, comptroller, public advocate, five borough presidents, and all 51 members of the City Council. Most of those offices, including mayor, will see a new person elected given most of city government is leaving their current offices due to term limits.

    The last situation like this, in the presidential election of 2012 and the city elections of 2013, when most of city government including the mayor was turning over, general election turnout in the city went from 53% in 2012 to 23.9% in 2013. But, given the city’s overwhelming Democratic enrollment imbalance, what often matters most is voter involvement in the party primaries -- in 2013, that stood at 22% of registered Democrats, about 40% of whom advanced Bill de Blasio to the general election and mayoralty. In 2016 and 2017, with the Hillary Clinton-Donald Trump presidential election followed by de Blasio cruising to reelection, the dropoff was even more severe.

    “If there is a fall off again in turnout. it's really unfortunate,” Romalewski said, “because in some ways, the person we elect as mayor and the people we elect as City Council representatives have a more direct impact on our everyday lives than people at the federal level. So you would hope that there would be even higher turnout for municipal elections, but that hasn't been the case.”

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    by Samar Khurshid, senior reporter, Gotham Gazette
    @samarkhurshid @GothamGazette