Proporciones directas e inversas

Imagina una fábrica en la que el trabajo debe ser realizado por varias personas y hay que cumplir un plazo. Se da un plazo máximo y no deben trabajar demasiados trabajadores, de lo contrario no será rentable.

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    ¿Cómo elegiría el director de la fábrica el número de trabajadores para hacer la tarea en un tiempo determinado, dado que si la hiciera 1 persona, se haría en 6 horas, si la hicieran 3, tardaría 2 horas, y así sucesivamente?

    Esta situación da una idea vaga de cómo se relacionan entre sí dos cantidades diferentes y, utilizando esa información, se pueden abordar adecuadamente muchas situaciones del mundo real.

    Las proporciones de dos cantidades pueden determinarse explícitamente observando la relación entre las cantidades. Dichas cantidades pueden relacionarse entre sí de varias formas, pero aquí, para el ámbito de este artículo, nos centraremos en las más básicas de dichas proporciones: Las proporciones directa e inversa .

    Significado de las proporciones directa e inversa

    Significado de proporción directa

    Sean dos cantidades que dependen entre sí de tal manera que al variar una de ellas, la otra varía en la misma proporción.

    Dos cantidades son directamente proporcionales entre sí si y sólo si dependen linealmente la una de la otra y la razón entre ellas es una constante.

    Supongamos que las dos cantidades son el tamaño de una cama y la otra es el coste de fabricarla. A medida que aumente el tamaño de la cama, el coste de fabricarla también aumentará constantemente.

    Supongamos que el coste de una cama de 4 m2 es de 400 $, de 8 m2 es de 800 $, de 16 m2 es de 1600 $, etc. Se dice que estas dos cantidades son directamente proporcionales.

    Significado de proporción inversa

    Dos cantidades son inversamente proporcionales si aumentamos una de ellas y la otra disminuye proporcionalmente. Esta es la razón exacta por la que se utiliza la palabra "inversa", para denotar la relación inversa entre ellas.

    Dos cantidades son inversamente proporcionales si y sólo si al aumentar una, la otra disminuye y viceversa.

    Un ejemplo típico del mundo real de estas dos cantidades sería el crecimiento de la población de bacterias con respecto al tiempo.

    Otro ejemplo sería el de los trabajadores necesarios para realizar una tarea: cuantos más trabajadores, menos tiempo se necesita.

    Fórmula de las proporciones directa e inversa

    Fórmula de las proporciones directas

    Se dice que dos cantidades son directamente proporcionales si una cantidad aumenta, la otra aumenta, y si una cantidad disminuye, la otra disminuye. En notación matemática, para dos cantidades x e y, la proporcionalidad directa entre x e y se expresa como

    xy

    donde el símbolo indica la proporcionalidad entre las dos cantidades.

    La proporcionalidad denota que las cantidades están relacionadas linealmente. Pero para resolver ecuaciones y las propiedades de estas ecuaciones, necesitamos un signo "=" y sustituir el símbolo de proporcionalidad.

    Por tanto, tenemos la siguiente equivalencia, xy es igual a x=ky{"x":[[233,231,231,231,232,234,236,237,239,247,251,261,265,277,284,288,298,303,308,311,312,313,311,308,304,298,291,283,280,267,265,261,260],[388,388,389,390,390,390,390,388,388,385,383,381,375,371,367,355,346,338,331,328,320,317,313,313,315,319,325,333,338,349,355,361,383,397,410,436,441],[503,506,511,520,526,535,540,544,557,563,568,573,576,577,578],[489,490,493,498,506,510,526,531,546,551,561,566,576,584,591,597,603,605,608],[730,731,731,732,733,734,734,734,733,731,729,727,726,723,721,720,717,717,716,716,716,716,717,719,720],[794,794,794,793,792,790,786,780,771,762,753,744,740,729,726,718,713,711,710,710,711,713,720,724,731,734,743,752,761,766,780,784,796,799,803,808,813,816,817],[860,862,863,864,866,868,868,871,871,873,874,876,877,879,885,887,892,900,903,913,915,924,927,930,935,940,942,946,950,953,954,956,956,955,954,952,949,947,943,941,936,933,930,928,927,927,926,926,925,925,924,924,923,921,920,916,914,910,906,899,897,891,889,888,888]],"y":[[362,357,356,352,351,347,345,342,341,336,335,332,332,332,335,337,344,351,359,368,378,389,399,410,421,431,440,449,453,463,464,464,463],[345,344,343,341,339,338,337,335,334,334,334,334,335,337,338,347,356,365,377,382,399,404,421,425,435,442,446,449,451,452,452,452,451,448,445,439,438],[354,354,355,355,355,354,354,354,354,354,355,356,357,359,361],[416,418,418,418,417,415,412,411,407,406,404,404,402,402,401,401,402,402,404],[260,260,259,259,261,263,275,287,302,317,331,347,354,369,377,384,403,409,425,434,442,447,449,453,454],[334,333,332,331,331,332,335,340,348,357,368,377,381,392,394,403,408,411,412,414,415,415,415,415,415,415,415,415,415,415,417,417,418,418,418,418,418,418,418],[332,331,331,330,332,336,338,351,357,372,376,382,385,387,391,392,393,392,391,387,385,377,373,369,363,355,351,344,338,332,329,327,324,324,325,328,334,337,347,351,367,378,392,399,408,415,436,442,462,468,479,485,489,497,501,506,508,509,509,507,504,495,488,478,468]],"t":[[0,5,11,13,17,28,33,33,44,50,60,66,77,83,94,100,110,116,127,133,144,150,160,166,177,183,194,200,211,233,235,244,250],[444,451,453,458,467,469,477,483,494,500,500,511,517,517,527,541,542,550,560,567,577,583,593,600,610,617,627,633,644,650,655,661,667,676,684,701,708],[1044,1067,1077,1083,1094,1100,1100,1111,1117,1127,1134,1144,1150,1160,1167],[1336,1344,1351,1359,1367,1377,1384,1394,1400,1411,1417,1417,1427,1434,1444,1450,1460,1467,1477],[1748,1755,1759,1769,1777,1784,1794,1800,1811,1817,1827,1834,1844,1850,1851,1861,1867,1877,1884,1894,1900,1910,1917,1927,1934],[2123,2128,2135,2151,2151,2160,2167,2177,2184,2194,2201,2211,2217,2227,2234,2244,2251,2261,2267,2268,2284,2294,2301,2311,2317,2318,2328,2334,2344,2351,2361,2367,2377,2384,2384,2395,2400,2409,2417],[2683,2698,2702,2704,2713,2717,2726,2734,2744,2751,2761,2767,2768,2778,2784,2794,2801,2811,2818,2828,2835,2845,2851,2851,2861,2867,2878,2884,2895,2901,2901,2911,2917,2943,2943,2951,2961,2968,2978,2985,2995,3001,3012,3017,3018,3029,3037,3045,3051,3062,3067,3068,3079,3084,3085,3095,3101,3111,3117,3128,3134,3143,3151,3161,3168]],"version":"2.0.0"}donde k es una constante de valor real distinta de cero.

    En otras palabras, tenemos para una constante de valor real no nula k,

    x yx=ky.

    Fórmula de las proporciones inversas

    Se dice que dos cantidades son inversamente proporcionales, si una cantidad aumenta la otra disminuye y viceversa.

    En notación matemática, para dos cantidades x e y , entonces la proporcionalidad inversa de x e y viene dada por,

    x1y

    Si convertimos la proporcionalidad en una ecuación introduciendo una constante de proporcionalidad, obtenemos

    x=ky

    que puede reordenarse para obtener la expresión

    xy=k

    donde k es una constante de valor real. Observamos que el producto de dos magnitudes de proporción inversa es siempre constante.

    En otras palabras, tenemos para una constante de valor real no nula k,

    x α 1yxy=k.

    Gráficas de proporciones directas e inversas

    Gráfica de proporciones directas

    La proporcionalidad directa puede visualizarse geométricamente trazando la ecuación x=ky de la siguiente manera,

    Gráfico de proporcionalidad, matemáticas puras, estudiar mejorFigura1.- La gráfica de la recta x=ky

    Observamos que la ecuación está representada por una recta que pasa por el origen (ya que ambas interceptas son 0).

    Si reordenamos la ecuación en términos de y , obtenemos y=1kx y la pendiente puede leerse como 1k.

    Por lo tanto, podemos establecer diferentes proporciones utilizando un dato dado y convertirlo en una ecuación utilizando una constante de proporcionalidad. Recuerda que la constante de proporcionalidad puede ser cualquier número real, no es necesario que sea un número positivo.

    Hemos visto que dos cantidades son directamente proporcionales, de modo que si cambiamos una, la otra también cambia en consecuencia y la tasa de cambio es constante. Esa tasa de cambio es k y también se conoce como gradiente de la curva x=ky.

    Gráfica de proporciones inversas

    La proporcionalidad inversa puede visualizarse geométricamente trazando la ecuación xy=kque puede reformularse en

    y=kx=k1x.

    Se trata de una hipérbola rectangular.

    Gráfico de proporcionalidad inversa, matemáticas puras, studysmarterFigura2.- La gráfica de la función xy=k, k >0

    También se puede trazar la gráfica calculando las asíntotas horizontal y vertical, que no son otras que los propios ejes.

    En la gráfica, hemos supuesto que la constante k es positiva. Para k negativa, la gráfica se reflejará a lo largo del eje y de la siguiente manera,

    Gráfico de proporcionalidad inversa, matemáticas puras, studysmarterFigura 3.- Gráfica de la función xy=k, k<0

    Las asíntotas vertical y horizontal siguen siendo las mismas. La propiedad de ambas gráficas sigue siendo la misma: al aumentar una variable, la otra disminuye y al disminuir una variable, la otra aumenta.

    ¿Cómo resolver proporciones Directas e Inversas?

    ¿Cómo resolver proporciones directas?

    Para resolver una proporcionalidad directa a partir de un conjunto de datos dado, tenemos en cuenta los siguientes pasos,

    Paso 1. Convierte la proporcionalidad en una ecuación mediante una constante de proporcionalidad,

    xαyx=ky

    Paso 2. A partir de los datos dados, determina el valor de la constante de proporcionalidad: k=x1y1.

    Paso 3. Vuelve a sustituir k en la ecuación original, y ahora la proporcionalidad está resuelta: x=ky, donde ahora se conoce k.

    ¿Cómo resolver proporciones inversas?

    Para resolver una proporcionalidad inversa a partir de un conjunto de datos dado, tenemos en cuenta los siguientes pasos,

    Paso1 : Convierte la proporcionalidad en una ecuación mediante una constante de proporcionalidad:

    xα1yx=ky

    Paso 2 : A partir de los datos dados, determina el valor de la constante de proporcionalidad: k=x1y1.

    Paso 3 : Vuelve a sustituir k en la ecuación original, y ahora la proporcionalidad está resuelta: xy=k, donde ahora se conoce k .

    Veamos algunos ejemplos de proporciones directas e inversas.

    Ejemplos de proporciones directas e inversas

    Un autobús se desplaza a 40 km/h. Suponemos que la velocidad es constante durante todo el trayecto.

    (i) Halla la distancia recorrida por el autobús en los primeros 30 minutos.

    (ii) Halla el tiempo necesario para que el autobús recorra una distancia de 240 km.

    Solución

    El autobús se desplaza a velocidad constante, lo que implica que la distancia recorrida es uniforme para un intervalo de tiempo dado.

    Por tanto, la distancia recorrida es directamente proporcional al tiempo empleado en recorrer dicha distancia.

    Sea t el tiempo empleado y d la distancia recorrida, por lo que nuestra proporcionalidad es,

    dt

    Convirtiendo esto en una ecuación, obtenemos

    d=kt

    Reordenando la ecuación, obtenemos

    k=dt

    Pero la relación entre la distancia y el tiempo se define como velocidad, por lo que la constante k es la propia velocidad, por lo que

    40=dt

    (i) Por tanto, parahallar la distancia recorrida en 30 min=0,5 hora, sólo tenemos que sustituir t=0,5 hora en la ecuación anterior,

    d=40×0.5=20 km

    Por tanto, la distancia recorrida por el autobús en 30 minutos es de 20 km.

    (ii) Para hallar el tiempo empleado en recorrer 240 km, sustituyendo d=240 km, obtenemos

    t=24040=6 hr

    Por tanto, el tiempo empleado en recorrer 240 km es de 6 h.

    Este problema podría haberse resuelto simplemente utilizando la fórmula de la velocidad, pero aquí lo resolvemos desde lo más básico, utilizando el concepto de proporciones.

    El coste de 2 kilogramos de manzanas es de 4 $. Halla el coste de 7 kilogramos de manzanas.

    Solución

    Sea x el peso de las manzanas e y el coste de las manzanas.

    A medida que aumenta el peso de la manzana, también aumenta su coste. Por tanto, el peso y el coste de las manzanas son linealmente proporcionales, por lo que podemos utilizar la fórmula

    x=ky{"x":[[210,209,209,208,207,207,206,206,206,206,206,207,208,209,212,214,217,222,228,231,237,243,250,253,259,265,268,274,277,281,282,287,287,287,287,282,280,278,276,268,265,259,256,253,248,245,239,232],[362,359,357,355,350,348,342,339,336,333,327,324,321,314,308,303,296,292,289,288,287,287,287,289,291,297,300,310,314,323,333,338,343,353,359,364,368],[447,448,451,453,457,460,463,469,472,479,485,491,496,501,503],[416,416,417,419,420,425,432,445,457,461,470],[607,608,608,609,610,610,609,608,607,606,605,604,604,603,603,602,602,602,602,604,605,607],[656,653,651,649,647,643,636,634,625,622,614,611,608,602,600,596,593,591,590,589,589,592,593,599,601,609,612,620,623,632,634,637,642,644,646,648,652,654,656,658,660],[712,712,712,712,713,713,713,713,713,715,718,720,723,727,728,734,736,744,747,755,757,765,770,772,776,778,780,783,784,786,787,788,788,788,787,786,785,781,779,777,776,773,772,771,770,769,767,767,767,767,766,766,766,766,766,767,768,769,769]],"y":[[316,316,315,314,311,310,308,306,303,301,299,295,293,291,287,285,283,279,277,276,275,275,277,278,281,285,287,293,296,303,306,319,323,337,342,357,363,368,373,386,390,397,401,403,407,409,411,411],[286,284,282,281,280,280,280,281,282,283,288,290,293,300,309,319,336,348,359,365,370,381,386,399,403,411,414,419,420,422,422,422,422,420,419,418,416],[306,306,306,306,305,305,305,305,305,304,304,304,304,305,306],[355,356,356,356,356,354,352,350,348,348,347],[218,218,220,222,235,242,265,279,286,301,314,326,332,338,348,362,371,377,380,385,386,386],[271,274,275,277,279,285,294,296,303,306,312,314,316,321,322,326,328,329,330,332,334,336,337,340,340,343,343,344,345,347,347,348,350,350,351,352,353,354,355,357,359],[280,281,282,283,285,288,297,307,317,329,335,340,343,346,347,349,349,348,347,341,338,330,324,321,314,311,308,302,300,295,291,288,287,286,286,286,287,295,303,307,311,327,334,341,355,362,382,394,399,415,420,428,432,439,443,446,447,447,446]],"t":[[0,4,15,27,33,42,49,49,59,66,68,78,82,82,93,99,99,109,115,126,132,143,149,149,160,167,177,182,183,192,200,209,216,226,232,243,249,250,260,267,278,282,283,293,299,299,309,325],[663,669,675,682,693,700,709,716,716,726,732,733,742,749,759,766,779,783,793,799,802,810,817,826,833,842,849,859,866,879,882,883,891,899,899,909,913],[1308,1310,1317,1327,1332,1333,1343,1349,1349,1359,1371,1380,1383,1394,1399],[1635,1642,1658,1666,1666,1682,1687,1699,1709,1717,1726],[2131,2141,2158,2166,2177,2183,2194,2199,2210,2216,2226,2233,2233,2243,2249,2259,2266,2278,2284,2294,2300,2307],[2572,2575,2587,2591,2592,2600,2610,2616,2626,2634,2643,2650,2650,2660,2666,2675,2683,2683,2696,2700,2716,2726,2733,2743,2750,2760,2766,2776,2785,2797,2800,2800,2808,2816,2817,2827,2836,2836,2843,2850,2858],[3215,3219,3242,3250,3250,3260,3266,3277,3283,3292,3300,3308,3318,3326,3333,3343,3351,3360,3366,3377,3383,3396,3400,3411,3416,3417,3427,3433,3433,3444,3450,3461,3468,3477,3484,3496,3500,3518,3525,3526,3535,3544,3550,3551,3560,3567,3577,3583,3597,3600,3612,3617,3617,3627,3634,3644,3650,3660,3667]],"version":"2.0.0"}

    donde kes la constante de proporcionalidad. Introduciendo los valores que nos dan, obtenemos

    2=4k{"x":[[196,196,199,201,202,206,208,211,213,215,220,221,226,228,229,230,230,230,230,230,229,228,226,224,220,219,214,210,207,202,198,195,193,189,188,187,188,193,195,199,205,211,223,231,240,248,253,264,267,277,280,287,288,290,293,295,296],[389,390,392,393,398,400,406,409,416,418,420,425,427,429,431,438,440,445,447,449,452,453,455,459,461],[394,400,404,406,408,416,419,428,431,437,443,446,448,453,460,464,467,469],[559,560,561,561,561,560,559,555,554,552,549,547,546,544,541,539,537,536,536,536,536,537,540,541,547,549,552,556,559,569,575,582,589,592,600,602,607,610,613,614,615,615],[579,580,580,580,580,580,579,578,578,578,578,578,578,578,580,581,582,584,586,587,589],[695,695,695,695,695,695,695,695,695,695,695,695,695,695,695,695,694,694,694,694,694,695,696,697,697,698,700],[742,741,739,737,735,728,725,716,714,705,700,696,694,692,689,688,687,687,687,687,688,691,698,703,708,714,720,723,733,736,742,747,752,755,756,758,760,761,762]],"y":[[275,274,254,253,252,250,250,249,249,249,250,251,255,259,263,266,268,273,276,279,285,288,291,297,303,305,313,319,323,327,332,335,336,340,341,343,343,343,343,343,343,342,341,340,339,338,338,338,338,338,338,339,339,340,341,342,343],[260,259,259,259,258,258,257,257,257,257,257,257,257,257,257,257,257,257,257,257,257,257,258,259,259],[304,304,303,302,302,300,299,298,298,297,297,297,297,297,296,296,296,296],[233,234,235,238,239,245,247,254,257,259,265,267,270,272,277,281,284,287,289,290,292,292,293,293,293,293,292,291,291,289,288,286,285,284,283,282,282,282,282,282,282,283],[253,253,254,256,261,269,283,292,299,303,312,314,322,324,329,330,331,332,333,333,332],[199,200,207,212,217,230,236,247,258,268,278,282,292,300,308,312,321,324,329,331,333,333,332,330,329,326,321],[255,255,256,258,259,264,265,271,273,278,282,285,287,288,291,292,293,294,295,296,296,297,298,299,299,300,302,302,305,306,307,309,312,314,314,316,317,318,319]],"t":[[0,3,7,9,12,26,28,36,37,44,54,61,71,78,88,94,95,104,111,111,121,128,128,138,144,155,161,171,180,192,195,203,211,222,228,238,255,271,278,288,294,305,311,321,328,338,344,355,362,371,378,388,395,395,405,411,419],[1017,1062,1074,1078,1090,1095,1107,1112,1121,1128,1128,1139,1145,1145,1157,1162,1172,1181,1181,1193,1195,1195,1203,1211,1221],[1535,1545,1550,1553,1562,1572,1578,1588,1595,1607,1612,1612,1622,1629,1642,1645,1657,1662],[1963,1968,1972,1987,1995,2005,2012,2022,2028,2029,2039,2045,2045,2056,2062,2072,2078,2091,2095,2095,2106,2112,2122,2128,2139,2147,2149,2157,2162,2172,2182,2193,2195,2204,2212,2222,2228,2239,2245,2246,2256,2262],[2504,2509,2537,2545,2555,2562,2572,2580,2589,2596,2605,2612,2622,2630,2639,2645,2646,2658,2662,2672,2674],[3965,3979,3989,3996,3996,4006,4012,4023,4029,4040,4046,4046,4056,4064,4075,4079,4089,4096,4106,4113,4122,4129,4139,4146,4146,4158,4163],[4339,4355,4363,4373,4379,4389,4396,4409,4413,4423,4429,4441,4446,4446,4459,4463,4463,4471,4479,4480,4490,4496,4506,4513,4526,4530,4540,4547,4559,4563,4573,4579,4589,4596,4606,4613,4622,4630,4639]],"version":"2.0.0"}

    Aislando k obtenemos

    k=24=12

    Esto nos da la ecuación que necesitamos,

    y=2x{"x":[[268,268,268,268,268,268,269,269,269,268,268,268,268,269,271,273,276,280,281,297,305,310,313,318,321,327,328,334,335,339,340,341,342,342,342,340,339,338,336,334,334,331,330,328,327,325,323,321,320,317,313,309,304,301,296,290,285,281,277,275,273,272,272],[440,449,451,453,456,463,469,475,482,484,493,495,500,504,509,510,513,516,517,518,518,519],[449,449,450,452,453,458,460,465,467,470,479,485,492,494,502,508],[608,606,606,606,606,607,609,613,616,621,623,631,633,640,641,645,646,647,647,647,645,644,642,639,636,634,630,628,623,620,617,616,614,614,615,615,618,620,622,626,628,633,639,647,654,657,662,665,668,673,676,680,685,688,689],[723,721,721,721,722,725,728,730,737,739,748,753,756,762,764,767,770,773,775,776,776,778,778,777,775,774,770,768,763,761,757,756,755,754],[821,822,822,822,821,820,819,814,812,804,798,792,789,787,783,781,779,777,774,772,772,773,775,779,790,798,809,820,833,839,859,865]],"y":[[219,218,217,216,215,214,214,217,222,230,238,248,257,265,271,275,278,280,280,276,270,266,263,257,254,246,243,234,231,225,223,219,219,220,226,233,241,251,263,277,283,301,307,325,330,346,353,360,363,368,371,373,374,374,374,371,368,364,358,355,349,342,337],[241,240,240,239,239,238,237,236,235,235,234,234,234,234,234,234,234,234,234,234,235,235],[277,278,278,278,278,277,277,276,275,275,273,272,270,270,269,268],[231,226,225,224,220,219,217,214,212,211,211,210,210,212,213,218,220,226,231,234,240,243,247,253,256,259,265,268,273,277,281,283,286,287,288,289,289,289,289,289,288,288,287,286,286,286,287,287,287,288,288,289,290,290,290],[246,238,234,231,230,225,223,221,218,218,217,216,216,218,219,220,224,228,232,234,237,241,249,255,260,263,269,272,277,278,281,282,283,283],[209,208,206,205,205,204,204,206,208,216,223,231,235,239,248,251,256,260,268,275,281,283,285,288,292,293,294,293,292,292,289,288]],"t":[[0,4,10,16,23,34,51,67,74,84,90,101,108,117,124,134,140,151,157,183,201,207,207,217,224,234,240,251,257,267,273,284,290,317,323,334,340,350,358,367,374,384,390,401,407,417,424,434,440,451,457,467,474,474,484,492,501,507,517,524,534,540,545],[1004,1004,1007,1009,1012,1017,1025,1034,1041,1050,1057,1067,1074,1084,1090,1091,1101,1107,1117,1124,1134,1141],[1390,1400,1407,1417,1424,1434,1441,1449,1451,1457,1467,1474,1484,1491,1504,1507],[1948,1959,1962,1967,1974,1984,1993,2006,2007,2016,2024,2034,2041,2051,2058,2067,2074,2084,2091,2101,2108,2108,2118,2124,2124,2134,2141,2141,2151,2159,2168,2175,2184,2191,2201,2208,2217,2224,2225,2233,2241,2251,2258,2269,2274,2286,2291,2292,2300,2308,2308,2318,2326,2334,2341],[2584,2596,2600,2608,2617,2625,2635,2641,2651,2659,2666,2674,2684,2691,2691,2702,2708,2720,2724,2725,2736,2741,2753,2758,2768,2775,2785,2791,2803,2808,2819,2824,2825,2833],[3027,3035,3041,3052,3059,3068,3076,3085,3092,3100,3108,3118,3125,3125,3135,3141,3141,3152,3158,3168,3174,3175,3185,3191,3202,3209,3218,3226,3235,3243,3252,3259]],"version":"2.0.0"}

    A continuación, para hallar el coste de 7 kilogramos de manzanas, sustituimos x por 7 para obtener ,

    y=2×7=14{"x":[[169,168,167,167,167,167,166,166,166,166,167,170,172,174,177,180,181,187,189,196,198,205,207,210,215,219,221,225,227,229,232,233,234,236,237,238,240,240,241,241,241,241,240,240,239,239,238,238,238,238,237,237,237,236,234,233,231,229,227,222,221,215,211,206,204,201,196,193,188],[339,341,343,345,347,352,358,361,364,370,373,377,380,388,391,396,398,403,407,409,411,414],[341,342,343,344,348,352,358,361,369,372,381,383,385,389,392,394,396,400,402,404,407,408],[466,467,468,468,470,471,474,475,477,480,484,488,490,491,495,499,501,503,504,504,504,503,502,498,497,491,489,485,478,476,473,471,469,467,465,465,464,464,466,467,471,472,478,483,486,489,498,505,511,516,519,522,526,528,530,532],[574,575,575,576,577,580,581,584,585,587,591,592,596,600,604,612,614,620,621,626,629],[623,623,622,622,620,618,613,610,603,598,595,588,586,580,579,577,576,576],[667,667,667,668,669,671,673,676,681,685,691,695,699,703,706,707,710,711,713,714,715,715,715,715,715,715,714,711,709,708,706,704,703,701,700,699,698,698,697,697,697],[684,683,682,682,684,685,691,695,701,706,712,717,720,724,727,729,730,731,732],[776,777,780,784,786,791,794,797,803,812,816,820,823,826,827,828,828],[772,772,773,776,780,786,793,797,808,811,820,826,834,836,838,839,840,841],[907,911,912,913,913,913,911,910,908,907,905,904,901,901,898,897,896,895,895,895,895,896],[951,950,949,949,948,947,946,945,943,942,941,939,938,936,935,933,933,933,934,937,938,939,942,944,946,949,954,960,965,968,973,978,983,986,989,990,993],[965,966,966,966,966,966,966,966,966,965,965,964,963,963,963,963,964,964]],"y":[[199,199,200,203,205,212,216,220,228,237,243,251,256,259,261,263,264,264,264,261,259,253,251,249,244,238,236,230,227,225,220,217,215,211,210,207,204,203,202,204,206,208,213,216,229,233,238,243,255,262,267,279,285,296,307,317,326,334,337,347,349,353,353,353,352,350,346,344,336],[238,238,238,237,237,237,236,236,236,235,235,234,234,233,233,233,233,233,233,233,234,235],[278,278,278,278,278,277,276,276,274,274,273,272,272,271,271,271,270,270,269,269,268,268],[222,216,213,212,210,209,207,206,206,204,203,202,202,202,202,204,206,209,211,218,221,228,231,240,243,252,255,260,269,271,275,277,281,284,286,288,289,290,291,291,291,291,290,289,288,288,286,285,284,284,284,284,284,284,284,284],[210,210,211,212,213,219,222,228,232,235,240,243,248,251,255,261,263,266,267,269,270],[200,201,201,202,207,210,221,225,237,245,249,259,262,271,273,277,281,283],[191,190,189,189,188,188,188,189,189,190,191,192,192,192,192,192,191,191,191,191,191,194,195,196,198,203,209,227,235,242,250,256,262,268,270,276,277,281,281,282,281],[233,232,232,231,231,230,229,228,228,227,226,226,226,225,225,226,226,226,227],[223,223,223,224,224,225,225,225,225,225,224,224,224,224,224,226,227],[251,252,252,252,252,252,251,251,250,249,249,248,248,248,248,249,249,249],[176,176,178,180,182,188,201,209,218,225,233,237,248,251,262,267,272,275,278,280,281,281],[184,192,194,197,202,207,210,212,218,221,224,229,231,236,238,244,246,249,250,251,252,252,252,252,252,252,252,252,251,251,251,251,251,251,251,251,252],[216,217,218,221,223,228,237,245,252,259,266,269,276,279,286,288,293,295]],"t":[[0,1,25,32,42,48,49,59,65,76,84,97,99,107,115,125,132,142,148,159,165,175,182,182,192,198,207,215,216,226,232,232,242,248,249,259,266,275,292,309,315,316,326,332,342,349,349,359,365,366,376,382,383,392,399,409,415,424,433,442,449,459,465,474,482,483,492,498,509],[1023,1027,1035,1041,1042,1049,1059,1065,1066,1075,1082,1082,1092,1100,1107,1115,1116,1126,1133,1142,1149,1159],[1410,1441,1449,1449,1459,1466,1476,1483,1494,1499,1512,1516,1516,1528,1532,1532,1546,1549,1549,1559,1566,1567],[1938,1947,1958,1958,1966,1976,1982,1983,1993,1999,2013,2016,2016,2026,2033,2043,2050,2063,2066,2079,2085,2098,2100,2108,2116,2126,2133,2141,2149,2159,2166,2166,2176,2184,2193,2200,2209,2216,2226,2234,2243,2250,2259,2267,2276,2283,2293,2299,2311,2316,2316,2327,2332,2333,2344,2347],[2702,2725,2733,2733,2743,2750,2760,2766,2766,2780,2783,2783,2794,2800,2812,2826,2833,2843,2850,2860,2866],[3111,3119,3125,3133,3143,3151,3160,3166,3177,3183,3193,3200,3208,3216,3226,3233,3243,3250],[3581,3586,3593,3600,3610,3616,3627,3633,3643,3650,3660,3667,3680,3684,3694,3700,3714,3717,3728,3733,3743,3758,3767,3767,3777,3783,3797,3810,3818,3827,3834,3843,3851,3860,3867,3877,3884,3894,3901,3910,3917],[4120,4120,4125,4133,4158,4167,4177,4183,4194,4201,4210,4217,4227,4233,4244,4250,4250,4261,4268],[4630,4668,4676,4684,4693,4700,4701,4711,4717,4727,4734,4747,4750,4761,4767,4777,4792],[4991,5001,5026,5034,5044,5052,5062,5067,5077,5085,5093,5100,5118,5126,5134,5134,5144,5150],[5433,5443,5449,5452,5461,5467,5478,5485,5494,5502,5511,5517,5530,5534,5544,5550,5565,5567,5578,5584,5601,5605],[5866,5877,5880,5884,5895,5901,5901,5912,5917,5918,5928,5934,5934,5945,5952,5961,5968,5978,5985,5996,6001,6001,6012,6017,6018,6029,6034,6045,6051,6051,6065,6069,6078,6084,6095,6101,6121],[6335,6335,6338,6351,6361,6368,6378,6385,6395,6402,6411,6418,6428,6435,6445,6451,6462,6468]],"version":"2.0.0"}

    Por tanto, el coste de 7 kilos de manzanas es de 14 $.

    Para un gas ideal a temperatura constante, la presión aplicada sobre él es inversamente proporcional al volumen ocupado por el gas.

    En un experimento, se observa que a una presión de 10 bares, el volumen del gas es de 3 metros cúbicos. ¿Cuál será su volumen cuando el volumen sea de 5 metros cúbicos?

    Solución

    Sean P y V la presión y el volumen del gas respectivamente. Se da que son inversamente proporcionales, lo que da

    PV=k{"x":[[332,333,334,338,339,343,346,348,351,352,356,358,362,364,367,369,373,377,381,385,389,390,393,395,397,400,401,404,406,408,409,414,416,422,425,432,434,437,442,444,445,447,449,451,451,453,454],[534,535,536,537,541,546,558,565,568,575,581,587,593,599,605,609,611,613,614],[543,542,540,538,537,536,535,534,534,536,539,541,549,552,563,567,582,588,593,603,612,616,623,625,628,633,634,636,637],[780,780,780,779,779,779,779,779,779,779,779,779,779,779,778,778,778,778,778,779,779,779,780,780,780,780,781,781,781,781,781,781,781,782],[854,855,856,855,852,850,844,842,839,832,825,818,814,811,806,804,802,797,794,792,789,787,787,787,788,789,793,794,802,804,813,816,820,827,830,837,844,848,851,861,867,872,877,880,881],[178,174,174,174,173,172,171,169,168,167,165,164,162,161,158,157,154,151,149,148,146,145,145,145,145,145,146,147,148,151],[152,151,152,154,162,170,179,190,194,210,215,229,234,246,253,258,260,262,264,264,263,260,257,251,243,233,223,212,206,189,183,166,161]],"y":[[234,235,237,246,251,268,280,293,304,310,330,335,353,360,372,377,387,396,401,406,409,409,409,408,407,401,398,390,378,367,360,339,332,309,301,279,272,265,252,247,242,237,229,225,222,218,214],[290,290,290,290,289,289,289,289,289,289,289,289,289,289,289,289,289,290,291],[338,338,339,339,339,340,341,341,342,342,342,342,341,340,338,338,335,334,333,332,331,331,330,330,330,330,330,330,330],[206,209,214,217,220,224,236,242,251,256,261,273,279,291,304,316,321,338,344,358,363,367,376,384,391,395,397,400,404,406,407,409,410,410],[280,279,279,279,282,283,288,289,292,297,303,309,312,315,320,323,325,330,334,335,339,343,345,346,347,348,349,349,350,350,351,352,352,353,354,356,357,358,359,361,363,365,367,369,370],[193,194,196,198,211,217,239,254,271,280,291,300,320,340,362,372,393,416,435,445,472,479,498,503,513,515,517,518,517,513],[220,205,202,199,190,184,179,176,175,174,174,178,180,187,194,203,207,212,223,235,241,253,259,272,285,297,309,319,324,335,337,343,343]],"t":[[0,34,39,52,62,64,76,81,92,98,108,115,125,131,139,140,151,162,165,176,181,192,198,198,208,215,215,225,232,241,248,257,265,276,281,292,298,298,308,315,315,323,331,332,340,348,358],[843,847,862,868,878,884,899,907,915,925,931,942,948,957,968,976,983,992,998],[1213,1217,1225,1232,1234,1242,1249,1259,1266,1282,1293,1298,1309,1315,1325,1333,1342,1348,1349,1359,1366,1379,1382,1382,1394,1398,1399,1409,1416],[2247,2251,2260,2265,2267,2281,2283,2294,2299,2299,2310,2315,2316,2326,2337,2342,2349,2359,2366,2376,2382,2383,2394,2400,2407,2415,2416,2426,2432,2433,2443,2449,2459,2466],[3199,3203,3211,3233,3243,3250,3260,3266,3266,3277,3282,3293,3299,3299,3310,3316,3316,3327,3332,3343,3349,3358,3366,3376,3396,3400,3410,3417,3426,3432,3445,3449,3449,3460,3466,3477,3483,3483,3493,3499,3510,3517,3528,3533,3541],[8330,8341,8344,8351,8361,8368,8378,8384,8395,8401,8401,8412,8418,8426,8434,8435,8443,8451,8461,8469,8478,8484,8495,8502,8512,8518,8518,8529,8534,8545],[8854,8865,8868,8876,8884,8895,8901,8912,8919,8926,8938,8943,8953,8962,8968,8979,8985,8985,8996,9001,9012,9018,9023,9029,9036,9045,9051,9062,9068,9079,9086,9095,9102]],"version":"2.0.0"}

    donde k es una constante.

    Sustituyendo ahora P=10 y V=3, obtenemos

    k=10×3=30{"x":[[85,83,83,83,84,84,83,83,82,82,82,81,81,81,80,79,79,77,77,75,74,73,71,71,71,70,70,70,70,70],[136,123,120,117,113,103,100,91,89,82,80,78,76,74,73,74,75,77,78,82,85,90,96,103,106,116,120,130,133,142,145,151,153,155,159,160,163],[234,238,240,242,247,256,260,270,273,282,285,292,296,298,300,301,302],[225,225,225,227,228,235,238,249,253,265,269,276,283,287,290,296,300,304,306,309,312],[406,406,406,406,405,404,403,401,398,396,395,394,393,393,393,393,393,393,395,396,399,400,403],[471,471,471,470,470,468,466,465,465,465,466,467,470,471,476,480,485,490,493,502,505,511,516,521,528,532,533,534,534,533,528,525,516,512,509,501,496,492,488,481,478],[593,596,597,598,601,602,608,613,617,619,623,626,630,633,637,639,640,641,642,643],[657,658,658,655,654,652,650,645,642,640,634,627,621,612,602,599,597,596,595],[719,725,726,728,733,739,742,746,749,754,757,760,764,765,767,768,769,769,768,766,764,758,755,748,745,738,735,732,732,731,731,732,735,739,744,749,751,756,761,765,767,771,771,771,770,767,762,757,754,751,748,746,740,737,731,728,724,720,717],[843,847,848,851,854,856,860,863,866,872,875,878,883,891,895,897],[833,832,832,834,836,840,843,853,857,869,873,883,889,894,898,899,901],[953,955,957,958,960,961,965,966,969,971,974,976,977,978,978,978,977,976,974,973,972,969,968,966,964,963,963,963,964,968,969,971,975,976,978,980,981,982,982,982,982,978,976,972,964,959,954,948,942,940,934,932,927],[1026,1026,1026,1026,1026,1025,1022,1020,1019,1018,1018,1019,1021,1023,1026,1030,1035,1038,1047,1050,1059,1062,1068,1070,1072,1074,1074,1073,1072,1070,1069,1064,1060,1056,1048,1044,1035,1028]],"y":[[214,210,208,207,211,213,222,226,235,240,252,259,264,276,282,300,306,324,331,347,352,363,376,379,387,389,391,393,397,399],[283,293,295,298,300,308,310,317,319,326,327,329,332,334,336,337,337,338,338,339,339,339,340,340,340,341,341,343,344,347,348,351,352,353,355,356,357],[285,286,286,286,286,285,285,284,284,284,284,284,284,284,285,286,287],[328,329,330,330,330,330,329,327,327,325,324,324,323,323,323,323,323,323,323,323,323],[236,237,238,240,250,254,259,271,283,295,300,311,322,330,338,342,349,355,361,363,366,367,367],[260,256,255,258,260,269,276,284,293,298,311,315,326,328,335,338,339,339,339,335,333,328,322,315,302,293,283,278,265,262,252,250,246,246,246,247,248,250,252,256,258],[245,258,261,265,272,276,285,290,295,298,304,309,314,318,323,326,328,330,331,330],[257,251,250,250,251,253,255,260,263,266,273,282,290,303,317,324,330,332,339],[254,247,246,245,242,239,238,237,236,236,235,235,236,237,238,241,242,246,247,254,256,263,266,273,275,281,285,288,289,291,292,294,296,299,301,303,305,307,311,314,316,321,323,327,329,334,337,340,341,342,343,344,345,345,344,344,341,338,334],[270,270,270,270,269,269,268,268,268,267,267,267,267,267,267,266],[297,299,300,300,300,300,300,298,298,296,296,295,295,294,294,294,294],[250,247,244,243,241,240,239,238,238,238,239,240,242,242,243,245,247,248,251,252,254,257,259,262,266,269,270,275,277,282,284,286,289,291,293,296,298,300,302,308,309,314,315,318,322,324,325,326,327,327,327,326,325],[256,257,258,259,261,263,272,280,289,293,300,307,314,318,321,324,325,325,323,321,315,312,302,298,294,285,277,269,265,262,259,253,250,248,246,245,245,246]],"t":[[0,11,20,28,69,78,85,97,102,102,114,118,121,132,135,145,153,160,168,178,185,194,202,210,218,219,227,231,235,247],[546,557,560,561,569,578,585,595,602,612,618,619,629,635,652,662,668,669,679,685,686,696,703,711,719,729,736,746,752,765,769,781,785,786,797,802,812],[1160,1169,1172,1177,1185,1196,1202,1212,1219,1229,1235,1246,1252,1262,1269,1279,1285],[1452,1456,1471,1479,1485,1496,1503,1512,1519,1529,1535,1544,1552,1552,1563,1569,1579,1585,1586,1596,1603],[1915,1924,1927,1935,1946,1952,1952,1963,1969,1979,1986,1996,2002,2014,2019,2019,2028,2036,2046,2053,2066,2069,2080],[2274,2283,2289,2320,2328,2336,2346,2353,2363,2369,2380,2387,2396,2404,2411,2419,2429,2436,2446,2458,2463,2469,2480,2486,2496,2504,2513,2519,2530,2537,2546,2552,2563,2569,2570,2580,2586,2586,2597,2603,2607],[2980,2983,2989,2990,2995,3003,3013,3019,3030,3036,3047,3053,3063,3069,3080,3087,3097,3104,3114,3130],[3280,3291,3294,3316,3319,3320,3328,3336,3336,3344,3353,3363,3370,3380,3395,3403,3413,3421,3430],[3749,3760,3763,3765,3770,3780,3786,3787,3797,3803,3803,3814,3820,3820,3831,3836,3837,3847,3853,3864,3870,3880,3887,3897,3904,3917,3920,3928,3936,3937,3947,3953,3964,3970,3981,3986,3987,3997,4003,4014,4020,4030,4037,4047,4053,4064,4070,4080,4086,4087,4096,4098,4103,4114,4120,4121,4131,4137,4147],[4511,4521,4524,4529,4537,4548,4553,4554,4565,4570,4570,4578,4587,4597,4604,4614],[4869,4879,4882,4895,4904,4914,4921,4931,4937,4947,4955,4965,4973,4981,4987,4998,5003],[5333,5333,5337,5338,5346,5354,5364,5370,5381,5388,5398,5404,5412,5420,5421,5431,5437,5448,5454,5454,5465,5470,5471,5482,5487,5498,5504,5514,5521,5531,5537,5537,5548,5554,5554,5562,5570,5571,5581,5587,5598,5604,5612,5620,5631,5638,5648,5654,5665,5671,5682,5687,5698],[5973,5984,5988,5996,5996,6004,6015,6020,6029,6037,6048,6054,6065,6072,6081,6087,6099,6104,6115,6121,6132,6138,6148,6154,6154,6165,6171,6179,6187,6188,6198,6204,6204,6216,6221,6221,6232,6237]],"version":"2.0.0"},

    lo que nos da la ecuación,

    PV=30{"x":[[266,268,269,270,271,272,275,276,277,279,280,280,281,282,283,284,286,287,289,290,290,292,294,295,297,298,300,301,304,305,308,311,314,316,318,319,323,327,332,339,344,349,354,360,363,371,374,383,386,388,392,394,399,400,401,401,401],[517,523,529,533,537,541,550,561,571,581,586,600,604,616,619,626,629,630,631,631,629,626,625],[528,526,525,527,530,532,537,539,542,551,555,564,574,583,587,599,602,610,613,615],[762,761,761,761,761,763,765,766,770,772,779,782,789,792,795,800,803,805,807,812,814,815,816,815,810,806,800,797,788,785,777,774,769,768,767,767,768,771,773,775,780,787,790,796,799,803,811,813,816,817,817,817,813,811,801,797,784,774,763,759,749,740,731,725,720,720],[888,889,889,889,888,885,884,879,878,875,875,875,875,876,877,878,881,883,890,893,895,898,904,908,916,923,931,939,945,948,957,959,963,964,964,964,964,963,961,959,956,954,946,942,938,929,923,919,909,905,897,892,889,888,887],[103,102,101,101,101,101,101,101,100,100,99,98,98,96,94,93,91,90,89,87,86,85,85,85,86,86,87,88,89,90,91,91,91,92,92,92,92,93,93,93,94,94,94,94,95,95,95,95,95,95,95,95,95,95,95,97,100,103,106,108,115,118,128,131,142,146,157,164,170,173,175,177,181,183,184,186,187,187,187,185,182,178,176,166,163,151,147,134,129,125,115,110,106,101,91,85,81,73,69,62,59,57,57]],"y":[[190,189,189,190,191,193,201,205,210,221,228,234,246,253,259,267,282,298,316,324,333,349,371,384,395,401,412,414,418,418,415,410,402,396,390,385,372,357,341,316,297,280,263,245,237,214,207,189,184,179,171,168,162,159,158,159,160],[229,229,229,229,229,229,229,228,228,227,227,227,227,227,227,228,229,231,231,233,235,238,239],[298,298,298,298,296,296,294,293,292,290,289,286,284,282,281,279,279,278,278,278],[220,214,212,210,208,205,201,199,193,191,186,184,181,181,180,180,181,182,183,186,190,199,202,209,221,230,239,244,257,261,273,277,287,293,298,301,303,307,310,312,316,320,323,327,330,332,340,342,347,350,355,357,365,367,373,375,381,383,385,385,386,385,383,379,370,367],[239,236,239,245,253,263,270,290,297,316,322,340,346,360,364,368,374,376,381,381,382,382,379,377,372,366,357,348,338,332,315,309,291,284,273,267,261,255,245,240,235,231,223,220,217,212,210,210,210,211,214,219,224,226,232],[444,445,445,444,442,439,437,432,430,420,411,402,397,379,365,358,341,331,323,297,288,272,264,257,244,238,230,222,217,213,209,208,204,203,200,199,198,196,195,194,192,190,189,188,187,186,185,184,183,181,178,176,173,166,163,155,145,140,136,133,127,125,119,116,112,111,109,109,111,112,114,116,122,125,128,135,143,151,155,164,175,186,192,211,217,235,240,255,260,264,272,275,278,281,284,286,286,287,287,285,283,279,273]],"t":[[0,0,7,15,15,23,33,40,40,50,56,57,67,73,73,84,90,100,106,107,117,123,133,141,150,157,167,174,183,191,199,206,217,223,224,234,240,250,256,269,274,284,291,303,307,317,323,335,340,340,351,356,367,373,384,400,403],[981,988,992,993,998,999,1007,1017,1025,1034,1040,1051,1057,1067,1073,1084,1090,1101,1107,1117,1123,1134,1136],[1655,1667,1674,1699,1707,1717,1724,1724,1734,1740,1741,1751,1757,1768,1775,1784,1791,1801,1808,1812],[2334,2339,2343,2349,2351,2357,2366,2374,2384,2392,2401,2408,2418,2424,2424,2435,2441,2441,2452,2457,2468,2474,2485,2491,2501,2507,2518,2525,2535,2541,2552,2557,2568,2574,2585,2591,2591,2602,2607,2608,2618,2624,2635,2641,2641,2652,2657,2668,2674,2675,2685,2691,2702,2707,2718,2725,2735,2741,2749,2757,2766,2774,2785,2791,2801,2809],[3207,3213,3241,3251,3258,3268,3276,3284,3291,3301,3308,3318,3325,3336,3341,3343,3352,3358,3368,3374,3375,3385,3391,3391,3404,3408,3419,3425,3435,3441,3454,3458,3468,3474,3483,3483,3491,3491,3502,3508,3508,3519,3524,3525,3536,3541,3542,3553,3558,3558,3569,3574,3585,3592,3599],[8551,8561,8568,8627,8643,8652,8660,8672,8677,8688,8693,8704,8710,8721,8726,8739,8743,8743,8754,8760,8772,8776,8777,8790,8793,8794,8804,8810,8821,8826,8839,8843,8854,8860,8872,8876,8877,8887,8893,8894,8907,8910,8921,8937,8954,8985,8993,9004,9176,9186,9193,9202,9207,9210,9221,9227,9237,9244,9254,9260,9271,9277,9287,9294,9304,9310,9324,9328,9338,9343,9344,9355,9360,9361,9371,9377,9388,9393,9394,9405,9410,9421,9427,9437,9445,9454,9460,9471,9476,9480,9485,9493,9494,9504,9510,9511,9522,9527,9528,9538,9544,9554,9560]],"version":"2.0.0"}

    A continuación, se nos pide que hallemos la presión cuando el volumen es de 5m3, por lo que tenemos,

    P=30V=305{"x":[[297,297,298,298,303,305,307,313,318,320,325,328,330,334,339,342,343,346,349,351,352,353],[294,294,295,296,299,304,310,317,320,329,333,341,344,350,352,354,358,361,362,363],[422,422,422,423,424,427,428,434,437,442,448,452,455,457,459,461,463,463,463,462,459,457,452,451,446,444,442,441,440,440,441,442,445,448,449,453,456,460,462,466,467,468,466,465,463,462,458,453,448,444,438,433,429,424],[509,509,508,508,507,507,506,506,505,505,504,504,504,505,507,509,511,516,518,526,530,534,537,540,542,544,544,544,542,538,535,530,525,520,517],[426,426,428,429,430,433,435,440,444,451,458,471,476,488,493,506,510,513,520,527,530,536,538,540,544,547,549,550,551,551,550,547,546],[452,452,452,452,452,454,454,456,457,460,461,462,464,466,468,471,472,474,476,478,481,483,484,487,488,491,492,494,494,495,496,497,497,498,499,500,501,503,505,508,511,515,518,522,523,528,529,533,533,536,537,538,538,538,538],[638,646,648,653,655,658,666,668,676,680,684,685,689,690,691,692,693,692],[630,629,629,631,633,634,636,640,647,653,659,662,671,674,681,683,687],[775,775,775,776,777,777,778,779,780,781,783,785,787,790,793,795,796,798,799,800,800,800,799,799,798,797,794,793,792,789,788,787,785,784,783,782,782,782,784,784,785,787,788,791,793,794,794,795,795,795,795,795,793,792,791,788,786,785,781,779,774,770,766,764,760,758,755,754,754,754],[828,827,827,827,826,825,825,824,824,824,824,825,826,828,831,833,836,837,842,844,846,849,851,854,855,856,856,857,857,855,854,853,851,847,844,842,837,834,832],[761,762,764,766,771,773,784,788,797,801,810,819,828,836,843,847,853,856,859,865,866,868,869,870,870,869],[842,842,842,842,840,839,838,836,834,831,830,828,823,820,817,814,813,811,809,808,806,805,804,804,804,804,804,804,805,805,806,806,806,806,806,806,805,805,804,804,804,803,803,803,803,804,805,806,808,810,812,814,815,818,820,821,826,828,830,833,834,836,838,839,840,840,842,842,843,843,843,842,841,839,836,834,829,827,818,815,813,807,804,801,796,793,791,789,782],[163,163,162,162,161,160,160,159,159,158,158,157,157,157,157,157,158,158,159,161,162,163,164,166,168,168,170,170,170,170,170,170,170,170,169,169,168,167,166,165,164,163,161,160,158,158,156,155,155,155,155,155,156,157,160,161,165,170,175,181,188,191,199,202,210,212,214,219,222,225,226,227,227,227,226,220,215,211,201,196,183,179,171,167,163,156,147]],"y":[[202,201,201,200,199,198,197,196,195,195,194,194,194,194,194,193,193,193,193,194,194,194],[237,238,238,238,237,236,234,232,231,229,229,227,227,226,226,226,226,226,226,226],[171,169,168,168,167,165,164,162,161,160,159,159,159,159,159,161,165,168,172,174,180,183,189,191,197,201,203,206,208,209,210,211,212,212,212,213,213,213,214,215,216,218,221,222,223,224,226,229,231,233,234,235,236,235],[171,173,173,174,177,179,183,188,193,200,205,210,215,218,221,223,224,224,224,220,216,212,207,201,196,191,185,181,177,174,173,171,170,170,169],[288,289,289,289,289,288,288,287,287,286,285,283,283,282,282,282,282,282,282,282,282,282,282,282,283,283,284,284,285,286,287,289,289],[328,327,326,325,324,325,326,330,332,339,342,345,349,355,363,371,375,383,392,400,408,416,420,430,433,440,442,447,448,448,449,449,448,447,445,441,435,429,421,412,403,389,380,373,369,360,357,349,347,342,339,337,336,335,334],[229,229,229,229,229,229,228,228,227,227,226,226,226,225,225,225,225,225],[255,256,257,257,257,257,257,256,255,253,252,251,250,249,248,248,248],[176,174,173,172,171,170,168,167,167,166,165,164,163,162,162,162,162,164,165,168,169,170,173,175,177,179,183,185,187,191,192,194,197,199,202,204,207,208,210,211,212,214,215,217,219,220,221,222,223,224,225,226,227,228,229,230,231,231,232,233,234,234,234,234,234,233,230,229,228,226],[179,184,185,187,191,193,195,199,203,205,208,210,211,214,216,217,217,217,215,214,213,210,209,204,202,200,196,194,192,189,188,186,185,183,183,182,181,181,181],[294,294,294,294,294,294,294,294,294,294,293,293,293,293,293,293,294,294,294,295,295,296,296,296,297,297],[328,330,331,332,333,333,334,334,334,335,335,335,335,335,335,335,335,336,336,336,336,336,336,337,338,339,341,342,344,346,349,352,355,356,362,364,369,370,372,375,376,377,379,381,382,382,382,382,381,379,377,376,375,373,372,372,370,370,370,370,371,372,374,375,377,379,385,387,391,393,395,401,406,410,414,416,422,423,428,429,429,430,430,430,429,427,425,423,415],[329,328,326,324,321,316,313,302,298,294,284,279,273,262,249,236,224,218,206,194,183,177,165,155,145,141,128,125,116,113,104,102,101,100,99,98,98,98,98,99,100,101,102,103,103,104,104,104,103,102,101,99,98,97,94,93,90,88,87,86,86,87,91,93,101,104,107,113,120,127,134,137,145,149,154,167,177,182,196,201,214,218,225,228,230,234,236]],"t":[[0,8,16,23,33,39,39,49,56,66,72,73,83,89,99,106,106,115,123,133,139,144],[412,419,426,433,440,450,457,466,473,483,490,500,506,517,523,523,536,539,550,553],[918,928,931,935,941,950,956,967,973,983,989,1000,1006,1007,1017,1023,1033,1041,1051,1056,1067,1073,1087,1090,1100,1106,1119,1123,1134,1139,1150,1156,1165,1173,1173,1184,1191,1200,1207,1216,1223,1233,1249,1256,1260,1267,1273,1284,1290,1300,1306,1317,1323,1333],[156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    P=6 bar

    Por tanto, la presión del gas que ocupa 5m3 de volumen es de 6 bar.

    En una fábrica de fabricación de teléfonos inteligentes, 10 trabajadores pueden montar un teléfono en 6 horas.

    ¿Cuántos trabajadores serán necesarios para montar el teléfono en 4 horas?

    Solución

    Intuitivamente, podemos deducir que a medida que aumente el número de trabajadores, disminuirá el tiempo necesario para montar el teléfono. Por tanto, será una proporcionalidad inversa.

    Sea w el número de trabajadores necesarios para montar 1 teléfono y t el tiempo que tardan en montarlo, entonces la proporcionalidad inversa es,

    w1t.

    Convirtiendo la proporcionalidad en una ecuación, obtenemos

    w=kt

    donde k es la constante de proporcionalidad.

    Se nos da que 10 trabajadores tardaron 6 horas en realizar la tarea, por lo que tenemos, 10=k6

    Supongamos que se necesitaron w trabajadores para completarla en 4 horas, por tanto

    w=k4

    Tomando el cociente de las dos ecuaciones anteriores, se elimina k y la única incógnita que queda es w, tenemos

    w10=64

    w=15

    Por tanto, se necesitarán 15 trabajadores para montar el teléfono en 4 horas.

    Proporciones directas e inversas - Puntos clave

    • Para dos cantidades cualesquiera, si están relacionadas entre sí de forma explícita, se dice que son proporcionales entre sí.
    • Las proporciones directas e inversas son dos tipos principales de proporcionalidades.
    • Dos cantidades son directamente proporcionales entre sí si y sólo si dependen linealmente la una de la otra y la razón entre ellas es una constante. La proporcionalidad directa entre dos cantidades ax e y se denota por xy.
    • Dos cantidades son inversamente proporcionales si y sólo si el producto de ambas es siempre constante y a medida que una aumenta, la otra disminuye,. Su proporcionalidad inversa se expresa como x1y.
    Proporciones directas e inversas Proporciones directas e inversas
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    Preguntas frecuentes sobre Proporciones directas e inversas
    ¿Qué son las proporciones directas?
    Las proporciones directas ocurren cuando una cantidad aumenta o disminuye en proporción directa a otra.
    ¿Qué son las proporciones inversas?
    Las proporciones inversas ocurren cuando una cantidad aumenta mientras la otra disminuye en proporción correspondiente.
    ¿Cómo se identifica una proporción directa?
    Identificamos una proporción directa cuando el cociente entre dos cantidades es constante.
    ¿Cómo se identifica una proporción inversa?
    Se identifica una proporción inversa cuando el producto de las dos cantidades es constante.

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