Quinto grado de Secundaria
Editorial
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Sucesiones
144
Denición. Sucesiones grácas. Sucesiones literales. Sucesiones numéricas (sucesión aritmética
líneal o de primer orden, sucesión cuadrática o de segundo orden, progresión geométrica). Series y sumatorias
U3
169
Analogías numéricas. Distribuciones numéricas. Distribuciones gráfcas.
Desigualdades e inecuaciones
178
Desigualdad. Ley de tricotomía. Intervalo.
Logaritmos
187
Denición. Teorema sobre logaritmos. Sistemas
logarítmicos. Cologaritmo. Antilogaritmo. Cerillos
196
Denición previa. Fósforos que se trasladan o
desplazan. Fósforos que se quitan o agregan. Razonamiento geométrico
149
Refuerza practicando.
151
Actividades de razonamiento.
163
Refuerza practicando.
165
Actividades de razonamiento.
172
Refuerza practicando.
174
Actividades de razonamiento.
181
Refuerza practicando.
183
Actividades de razonamiento.
191
Refuerza practicando.
193
Actividades de razonamiento.
200
Refuerza practicando.
202
Actividades de razonamiento.
215
Refuerza practicando.
217
Actividades de razonamiento.
227
Refuerza practicando.
229
155
Serie numérica (serie aritmética, serie geométrica). Sumatorias (propiedades). Principales series y sumas notables. Suma de términos de una serie conociendo su término enésimo. Suma de una serie asociada a una sucesión polinominal usando números combinatorios.
Analogías y distribuciones numéricas
Actividades de razonamiento.
208
Triángulos (propiedades, congruencia, triángu los rectángulos notables). Cuadriláteros (clasi cación). Polígonos (propiedades). Circunferen cia (ángulos en la circunferencia, propiedades).
Perímetros y áreas Perímetros. Áreas de regiones cuadrangulares. Áreas de regiones triangulares. Áreas de regiones circulares.
221
Análisis combinatorio
233
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UNIDAD 1
El tiempo es oro Es una frase que escuchamos muy a menudo, pero es totalmente cierto, el tiempo es oro y en muchas ocasiones es dinero, por ejemplo en todo lo relacionado con los estudios universitarios, cuanto más tiempo tardes en terminarlos más dinero te costará.
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Matemática recreativa Promedio engañoso Pedro y Pablo son dos automovilistas que hacen habitualmente el mismo viaje de ida y vuelta entre dos ciudades, cada uno en su auto. En cierta ocasión hablaron del asunto y Pedro le dijo a Pablo: "El viaje de ida lo hago a 80 km/h y la vuelta a 60 km/h". Pablo contestó a Pedro: "Por las características de mi auto y la carretera, hago el viaje de ida y vuelta a la velocidad constante consta nte de 70 km/h, km/h, que es el promedio de las velocidades que tú me has dicho, de modo que empleamos el mismo tiempo de viaje". ¿El razonamiento de Pablo es correcto? ¿Emplean el mismo tiempo en el viaje?
D iá l l o og o
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Planteo de ecuaciones Impor tante Tanto los números pa res consecutivos como los números impares consecutivos se diferencian en dos unidades.
Para resolver un problema referente a números o relaciones abstractas entre cantidades, basta con traducir dicho problema al idioma algebraico. Ejemplo: Un padre deja una herencia para que sea repartida entre sus hijos de la siguiente manera: al primero S/.100 y la quinta parte del resto, al segundo S/.200 y la quinta parte del resto, al tercero S/.300 y la quinta parte del nuevo resto y así sucesivamente con todos los hijos. Si al final todos recibieron la misma cantidad de dinero, ¿cuánto es la herencia y cuánto son los hijos? Resolución: Al primer hijo: S/.100 y la quinta parte del resto.
Herencia 5x
100
x
4x
1.er Hijo
•
2x + 5 = 3x - 7 Es una ecuación de primer grado con una sola variable.
•
Queda
Del gráfico: herencia = 5x + 100 Al segundo hijo: S/.200 y la quinta parte de lo que va quedando.
Queda= 4x 4x
-
200
2
x + 5x - 6 = 0 Es una ecuación de segundo grado con una sola variable. x + y = 10 x - y = 6 Es un sistema de ecuaciones de primer grado con dos variables.
200
4x - 200
4
5
5
b
4x - 200 5
l
2.° Hijo
Y así sucesivamente con todos los hijos, al final todos reciben la misma cantidad de dinero. Entonces de acuerdo a este dato, planteamos que lo recibido por el primer hijo es igual a lo recibido por el segundo hijo. Es decir:
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Problemas 1
resueltos
En una colecta se reunió S/.2300. ¿Cuántas personas colaboraron, si el número de nuevos s oles que dio cada uno excede en 4 al número de personas?
Por condición del problema: 4x- (3x + 10) = 5 4x- 3x - 10 = 5 x= 15 Luego: Tú: 30 Yo: 55 Piden: 55 - 30 = S/.25
Resolución:
De los datos: n.° de personas: n Cantidad de dinero por persona: n + 4 Si se reunió en total S/.2300: n(n + 4) = 2300 2 n + 4n - 2300 = 0 n +50 n -46
4
(n + 50)(n - 46) = 0 n = -50 `
2
0
Mario dice: Ayer tuve la mitad de lo que tengo hoy, y lo que tengo hoy es el triple de lo que tuve anteayer, que es S/.40 menos de lo que tuve ayer. ¿Cuánto tiene Mario? Resolución:
n = 46 De los datos
El número de personas es 46.
Anteayer: 2x Ayer: 3x Hoy: 6x Lo que tuve anteayer es S/.40 menos de lo que tuve ayer: 3x - 2x = 40 x= 40 Mario tiene: 6x = 6(40) = S/.240
En un examen de 30 preguntas, cada respuesta correcta vale 4 puntos, la incorrecta -1 y en blanco 0. Si un estudiante obtuvo 82 puntos y notó que por cada respuesta en blanco tiene 3 correctas, ¿cuántas contestó incorrectamente? Resolución:
Haciendo un esquema: 5
30 Blanco x
Correctas 3x
0 puntos
4 puntos
Incorrectas 30 - 4x -
1 punto
En una fiesta la relación de mujeres y hombres es de 3 a 4. En un momento dado se retiran 6 damas y llegan 3 hombres, con lo que la relación es ahora de 3 a 5. Indica ¿cuántas mujeres deben llegar para que la relación sea de 1 a 1?
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6
En un teatro las entradas de adultos costaban S/.30 y la de los niños S/.10. Cierto día acudieron un total de 752 espectadores y se recaudó S/.18 240. Da como respuesta la suma de cifras del número de adultos que asistieron ese día.
Reemplazando (I) en (II): Hija = 90 +
100Hija = 9000 + 900 + Hija
Resolución:
99Hija= 9900 Hija= 100
Haciendo un esquema:
Luego: Señora
752
`
Adultos x
Niños 752 - x
S/. 30
S/. 10
Por condición del problema: 30x + 10(752 - x) = 18 240
30x + 7520 - 10x = 18 240
20x = 10 720 Piden: 5 + 3 + 6 = 14
900 + Hija 100
&
x = 536
9
=
90 +
100 10
Juntas pesan 200 kg.
Entre 10 personas tenían que pagar una cierta cantidad de dinero, pero resulta que 4 de ellos solo pueden pagar la mitad de lo que les corresponde, obligando de esta manera a que cada uno de los restantes dé 40 soles más. Halla la cantidad de dinero a pagar. Resolución:
7
En un corral se observa que el número de patos excede en 8 al número de pavos. Además, si inclui-
100
=
Sea x la cantidad de dinero.
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Actividades de razonamiento
1. Halla un número cuyo cuadrado, disminuido en
2. Si al numerador de la fracción 3/5 se le suma un
119 es igual a 10 veces el exceso del número con respecto a 8. Da como respuesta el mayor de ellos.
número y al denominador se le resta el mismo número, se obtiene otra fracción equivalente a la inversa de la fracción dada. Calcula el número.
A) 13
A) 4
B) 10
C) 5
D) 15
E) 8
B) 8
C) 10
D) 2
E) 6
3. Martina compró cierto número de correas por
4. Yo tengo el triple de la mitad de lo que tú tienes,
S/.240. Si hubiera comprado 3 correas más con el mismo dinero, cada correa le habría costado S/.4 menos. ¿Cuánto le costó cada correa?
más 10 soles. Si tuvieras el doble de lo que tienes, tendrías 5 soles más de lo que tengo. ¿Cuánto me quedaría, si compro un artículo donde gasto la cuarta parte de lo que no gasto?
A) S/.30
A) S/.30
B) S/.20
C) S/.10
D) S/.35
E) S/.40
B) S/.55
C) S/.44
D) S/.77
E) S/.66
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9. Si dos números suman 49; siendo uno múltiplo de
10. En una festa se observan 15 mujeres sentadas y
5 y el otro de 6, halla el mayor de los dos números.
tantas parejas bailando como el doble de varones sentados. En la siguiente pieza musical se observa que todas las mujeres se encuentran bailando y 3 varones se encuentran sentados ¿Cuántas personas habría en la festa si hubiera el cuádruple de varones y el triple de mujeres?
A) 13
A) 160
B) 25
C) 20
D) 10
E) 15
11. Si un ladrillo pesa m kg; dos ladrillos pesan juntos
(2y - 1) kg y tres ladrillos pesan (m + y) kg, ¿cuánto pesarán cuatro ladrillos juntos? (Los ladrillos son del mismo peso)
A) 6 kg
B) 3 kg
C) 2 kg
D) 5 kg
E) 4 kg
B) 260
C) 460
D) 280
E) 369
12. Para ganar S/. 28 en la rifa de una radio, se hicieron
90 tickets, vendiéndose únicamente 75 y originando así una pérdida de S/17. Halla el valor de la radio.
A) S/.242 D) S/.200
B) S/.312 E) S/.144
C) S/.412
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Refuerza
practicando NIVEL 1 1
5
Entre A y B tienen 1154 soles y el dinero de B es excedido por el de A en 506 soles. Luego: I. Si B recibe S/.253, S/.253, entonces entonces A y B tendrían la misma candad. II. B ene S/.324. III. Si A pierde S/.330, con lo que queda puede comprar un terno de S/.498. Son ciertas: A) Todas D) I y III
B) I y II E) Solo II
Ayer gané 20 soles más que hoy. Si lo que gané hoy son los 5/6 de lo que qu e gané ayer, ayer, ¿cuánto gané hoy? A) S/.80 D) S/.150
6
C) II y III
Halla el producto de dos números cuya suma es 7 veces su diferencia. Además, si le aumentamos una unidad a la suma y a la diferencia, esta diferencia sería la quinta parte de la nueva suma. A) 32
B) 40
C) 44
D) 48
E) 56
7
C) S/.120
Se toma un número impar, se le suma los 3 números pares que le preceden y el cuádruple del número impar que le sigue, obteniéndose en total 199 unidades. El menor de los sumandos es: A) 20
2
B) S/.100 E) S/.140
B) 22
C) 24
D) 25
En una granja se tienen pavos, gallinas, y patos. Sin contar las gallinas tenemos 5 aves, sin contar los pavos tenemos 7 aves,
E) 27
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10
En la tabla:
13
8
b
2
a
-
x
-
y
2a
¿Qué fecha será, en un año bisiesto, cuando la séptima parte del número de días transcurridos sea igual a la quinta parte de los días que faltan transcurrir, disminuido en 6? A) 11 de julio C) 15 de julio E) 14 de julio
b
-
La suma de los 3 términos en cada fla, columna y diagonal es la misma. Calcula: x - y
B) 13 de julio D) 16 de julio
UNI 2008-I
A) a - b
B) b - a
C) a + b
D) 4
E) 0
14
NIVEL 2 11
A 10 parejas de jóvenes le van a entregar 2 canarios por pareja. En el momento de la entrega se escapan algunos canarios y luego se ordenan traer tantos
Un alambre se divide en 5 partes iguales, con 4 partes se forma un cuadrado y con la última un triángulo equilátero, resultando el área del cuadrado numéricamente igual a un lado del triángulo. ¿Cuál es la longitud del alambre? A) 5/9
B) 15/7
C) 5/3
D) 10/3
E) 10/7
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17
El largo de un buque, que es 800 cm, excede en 744 cm a los 8/9 del ancho. Halla la medida del ancho del buque. A) 63 cm D) 65 cm
B) 70 cm E) 62 cm
C) 80 cm 21
18
Un cajero automático debe entregar 740 soles, empleando billetes de las siguientes denominaciones:
El día de mi cumpleaños nos reunimos 65 personas. Después de la reunión me di cuenta que el primer amigo bailó con una amiga, el segundo con tres, el tercero con seis, el cuarto con diez y así sucesivamente, hasta que yo fui el último y bailé
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24
Un granjero cría patos, gallinas y conejos. La cantidad de gallinas duplica al número de patos, así como los conejos son tantos como los patos y gallinas juntos. Si el granjero vende 5 patos y 10 gallinas, el número de conejos es el doble del número de patos y gallinas que quedan. ¿Cuántos conejos tiene el granjero? UNI 2003-II
A) 40 D) 35
B) 25 E) 20
C) 30
27
Se tienen 3 montones de clavos donde las cantidades son proporcionales a 6, 7 y 11. Si del montón que tiene más clavos se sacan 12 para redistribuir entre los demás, al final se tendrían los tres montones con igual número de clavos. ¿Cuántos clavos hay en total? A) 74 D) 94
B) 84 E) 96
C) 90
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Edades Ejemplo: Jenny le dice a Jason: “Actualmente yo soy mayor que tú en 20 años. ¿Cuál sería ser ía nuestra diferencia diferen cia de edades si tú hubieras nacido 10 años antes y yo 10 años después?" Resolución: Si Jenny hubiera nacido 10 años después, tendría 10 años menos y Jason hubiera nacido 10 años antes, tendría 10 años más, esto quiere que la diferencia se reduciría en 20 años. Por lo tanto ambos tendrían la misma edad.
CLASIFICACIÓN
Recue rda • Si una persona hubiera nacido n años después tendría n años menos. • Si una persona hubiera nacido m años antes tendría m años más.
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Problemas resueltos 1
Las edades edad es de 3 jóvenes están está n en progresión prog resión aritmética creciente creciente cuya suma es 39. Si la suma de sus cuadrados es 515; la edad del menor es: Resolución:
Sean las edades: x - r, x, x + r Según el enunciado: x - r + x + x + r = 39 3x = 39 & x = 13 También: (x - r)2 + x2 + (x + r)2 = 515 (13 - r)2 + 132 + (13 + r)2 = 515 2
2
Resolución:
Según los datos, en 1976: Año de Nacimiento: 19ab Edad actual: 1 + 9 + a + b Sabemos que: Año de nacimiento + Edad actual = Año actual En el problema: 19ab + (1 + 9 + a + b) = 1976 1900+ 10a + b + 10 + a + b = 1976 11a+ 2b = 66 .
.
6
0
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Resolución:
De los datos: Año de nacimiento + Edad actual = Año actual E2 + E = 2070 E(E + 1) = 45# 46 E = 45 Luego Evelyn nació en el 2025. ` En el 2030 tuvo: 2030 - 2025 = 5 años Andy y Ángela tienen edades cuya suma es 5 veces
Resolución:
Según los datos: Hace (n - S) años
Edad actual
Dentro de (n + S) años
Carmen
nx
nx + (n - S)
nx + (n - S) + (n + S)
María
x
x + (n - S)
x + (n - S) + (n + S)
Por condición del problema:
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9
Camila nació 6 años antes que Valentina. Hace “2n” años sus edades eran como 7 es a 4 y hace
10
Las edades de Aldolfo, Julio y José están en la relación de 3, 5 y 4; luego de 12 años dichas edades
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Actividades de razonamiento 1. Un alumno de la academia nació en 19ba y en 1980
tuvo (a + b) años. ¿En qué año cumplió (2a años?
+
b)
2. Jhon le dice a Elvis: “Tengo el triple de la edad que
tú tenías, cuando yo tenía la mitad de la edad que tienes y cuando tengas la edad que tengo yo, yo
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9. Rosa tenía en 1978 la mitad de años de los que tenía
en 1992. ¿Cuántos años tendrá en 1999?
10. La edad actual de una persona es el doble de la otra
persona; hace 7 años la suma de sus edades era igual al promedio de sus edades actuales, disminuido en 0,5. Halla la edad del menor menor..
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Refuerza
practicando NIVEL 1 1
Andrea tuvo su primer hijo a los 30 años, 2 años
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10
Ricardo dice: “Si al año en que cumplí 18 años le sumas el año en que cumplí 20 años y luego
14
Una persona de 40 años le preguntó a Carlos y
Juan sobre sus edades, a lo que Carlos contestó:
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o c i t á m e t a M o t n e i m a n o z
R AZONAMIENTO MATEMÁTICO QUINTO GRADO DE SECUNDARIA COLECCIÓN INTELECTUM EVOLUCIÓN
©
Ediciones Lexicom S. A. C. - Editor RUC 20545774519 Jr. Dávalos Lissón 135, Cercado de Lima Teléfonos: 331-1535 / 331-0968 / 332-3664 Fax: 330 - 2405 E-mail:
[email protected] www.editorialsanmarcos.com
Responsable de edición: Yisela Rojas Tacuri Equipo de redacción y corrección: Eder Gamarra Tiburcio / Jhonatan Peceros Tinco Josué Dueñas Leyva / Saby Camacho Martinez Óscar Díaz Huamán Diseño de portada: Miguel Mendoza Cruzado / Cristian Cabezudo Vicente Retoque fotográco:
Luis Armestar Miranda Composición de interiores: Miguel Lancho / Carol Clapés Hurtado / Melissa Chau / Cristian Cabezudo / Vilma Jacquelín Añazco / Lourdes Zambrano Ibarra Grácos e Ilustraciones:
Juan Manuel Oblitas / Ivan Mendoza Cruzado Primera edición: 2013 Tiraje: 9000 Hecho el depósito legal en la Biblioteca Nacional del Perú N.° 2013-18811 ISBN: 978-612-313-116-6 Registro de Proyecto Editorial N.° 31501001300694 Prohibida la reproducción total o parcial de esta obra, sin previa autorización escrita del editor. Impreso en Perú / Printed in Peru
Pedidos: Av. Garcilaso de la Vega Av. Vega 978 - Lima. Teléfonos 331-1535 / 331-0968 / 332-3664 E-mail :
[email protected] Impresión: Industria Gráca Cimagraf S.A.C.
La COLECCIÓN INTELECTUM EVOLUCIÓN para Secundaria ha sido concebida a partir de los lineamientos pedagógicos establecidos en el Diseño Curricular Nacional de la Educación Básica Regular, además se alinea a los patrones y estándares de calidad aprobados en la Resolución Ministerial N.º 0304-2012-ED. La divulgación de la COLECCIÓN INTELECTUM EVOLUCIÓN se adecúa a lo dispuesto en la Ley 29694, modicada por la Ley
N.º 29839, norma que protege a los usuarios de prácticas ilícitas en la adquisición de material escolar. El docente y el padre de familia orientarán al estudiante en el debido uso de la obra.
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