Matemáticas financieras Manual de soluciones José Luis Villalobos 4ª edición PEARSON

Matemáticas financieras José Luis Villalobos 4ª edición

Manual de soluciones PEARSON

CAPÍTULO 1

Ejercicios 1.2 1. 0 2. 729,421.3

(

)n = a

3. 45(35)5 4. 1

=1

5. 1 6. -1/(58.003)5 7. 8. 9. 10. La raíz cuarta de 10,000 es igual a 10 11. 34 = 81 12. 53 =125 13. 0° no está definido. 14. 4-3 = 1/43 y 43 = 64 15. Se restan los exponentes. 16. 52(-3)2 = 52(3)2 = (5(3))2 = 225 17. (1)(a6)b-4 = a6/b4 18. 43/(4-244) = 43/42 = 43-2 = 4 19.

=

=2

20. (-10)3 – 8 + 5 = -100 = 1 Página 1 de 18



21. = (x + y)21/3 = (x + y)7

22.

23. 5-9/5-8 = 5-1 = 1/5 24. 271/3x-6/3y18/3z6/3w0 = 3x-2y6z2(1) = 3y6z2/x2 25. 2.000000001 26. (33.6323216)1/5 = 2.02 27. -0.625741957 28.

= 1.017289073

29. 7’298,308.346 /(9.98)3/2 = 8.555699854/31.52795572 = 0.271368684

30.

31. 593,782.2553 32. ((1.521)5)7 = (1.521)35 = 2’368,767.806 33. -428.51 34. 7.253216, 7.25322, 7.253, 7.25 35. 0.000100 0.00010 0.000 0.00 36. 725.090091, 725.09009, 725.090, 725.09 37. 16.379154 16.37915 16.379 16.38 Página 2 de 18



38. 3,295.959494, 3,295.95949, 3,295.959, 3,295.96 39. 0.903577 0.90358 0.904 0.90 40. 99 centavos. 41. Cuando su cuenta sea un número entero en los pesos. 42. Es de investigación 43. (-5)4 + 3 - 8 = -5-1 = -1/5 44. 55 = 3,125 ⇒

(a)

45. (x + y – z)4(1/4) = x + y – z 46. 4/5 = 0.8 ⇒

(b)

(d)

(d)

47. (x – 2y)3 + 1/6 = (x – 2y)19/6

(c)

48. 53x-2y4/25x3y-2 = 53y6/52x5 = 5y6/x5 ⇒ 49. 0-2 =

=

(b)

50. 18.475 ⇒

(b)

51. 99 centavos

(b)

(b)

Ejercicios 1.3 6. 4 – 2x = 0, 4 = 2x ⇒ x = 2 7. 5x – 4 = 2 – x/2 ⇒ 10x – 8 = 4 – x ⇒ 11x ⇒ 12 x = 12/11 8. 11 – 0.3x = 0.83 ⇒ x = 10.17/0.3 = 33.9 9. y/5 + 2y – 4/3 = 2 ⇒ 3y + 30y – 20 = 30 ⇒ 33y = 50 ⇒ y = 50/33 Página 3 de 18



10. (1/4)(x – 3) = 5 – x ⇒ x – 3 = 20 – 4x ⇒ 5x = 23 ⇒ x = 23/5 11. 3/(x – 2) = 4 – 1/3 ⇒ 3/(x – 2) = 11/3 ⇒ 9 = 11(x – 2) ⇒ 9 = 11x – 22

x = 31/11

= 42.5 ⇒ 2 – x = (42.5)2 ⇒ x = 2 – 1,806.25 ⇒ x = -1,804.25

12.

13. (1 + x/3)8 = 1.276 ⇒ 1 + x/3= 1.030935119 ⇒ x = 3(0.030935119) = 0.092805357 14. (1 + i/6)6 = 1.083 ⇒ 1 + i/6 =

⇒ i = (0.013377855)6 = 0.08026713

15. (1 + i)3 = 4.82 ⇒ 1 + i = 1.689204953 ⇒ i = 0.689204953 = 2.35 ⇒ x – 3 = (2.35)2 ⇒ x = 5.5225 + 3

16. 17. 2 + x =

x = 8.5225

⇒ 4 + 4x + x2 = 4 + x2 ⇒ 4x = 0 ⇒

x=0

18. x2/(x – 2) = x + 2 ⇒ x2 = x2 – 4 ⇒ 0 = -4 ⇒ no solución. 19. 12,525/11,080 = (1 + i/3)18 ⇒

= 1,006833519

⇒ i/3 = 0.006833519 ⇒ i = 0.020500557 = 2.35 ⇒ 4 – 5x = (2.35)3 ⇒ x = (4 – 12.977875)/5 = -1.795575

20.

21. 428/324 = 4 – 3x ⇒ 1.320987654 – 4 = -3x ⇒ 2.679012346/3 = x ⇒ x = 0.893004115 22. 2/x + 3/4 = 5/x ⇒ 8 + 3x = 20 (se multiplicó por 4x) ⇒ x = 12/3 ⇒ x = 4

(a)

= 1 ⇒ 4 + x/2 = 1 ⇒ x = (1 – 4)2

23.

x = -6

(a)

= 2x ⇒ 3 – x = 4x2

24.

4x2 + x – 3 = 0 ⇒ (4x – 3)(x + 1) = 0 ⇒ x1 = +3/4, x2 = -1 ⇒

= 2(3/4)

y

≠ -2 ⇒ solución: x = ¾ ⇒

(d)

25. 2x2/(x + 1) – x + 1 = 0 ⇒ 2x2 = (x – 1)(x + 1) ⇒ 2x2 = x2 – 1 ⇒ x2 = -1 ⇒ no hay solución

(b) Página 4 de 18



26. 63.9 = 21.3(t2 + 2) ⇒ 63.9/21.3 = t2 + 2 ⇒ t2 = 1 ⇒ t = ± 1 ⇒ = 2 ⇒ 3 – 1/d = 64 ⇒ -1/d = 61 ⇒ d = -1/61

27.

+3=0⇒

28.

= -3 ⇒ no solución real

(c) (b)

⇒ x2 = 3 – x/2 ⇒ 2x2 = 6 – x ⇒ 2x2 + x – 6 = 0

29. x =

⇒ (x + 2)(2x – 3) ⇒ x1 = 3/2, 30.

(c)

x2 = -2 no ⇒ x = 3/2

(b)

= x + 3 ⇒ 5 + x2 = x2 + 6x + 9 ⇒ 6x = -4 ⇒ x = -2/3

31. x2/(x – 1) = x + 1 ⇒ x2 = x2 – 1 ⇒ 0 = -1 no

(a)

32. (3 – 2i)4 = 625 ⇒ 3 – 2i = 5 ⇒ i = -1 ⇒

(d)

33. 2/x – 3 = 4 – 5/x ⇒ 2 – 3x = 4x – 5 ⇒ 7x = 7 ⇒ x = 1

(a)

(a)

Ejercicios 1.4 2. 0.153(428) = 65.484 3. 0.962(x) = 4,321 ⇒ x = 4,321/0.962 = 4,491.683992 4. 1.6035(48.5) = 77.76975 5. (x/100)(0.65)1,729 = 400.0906 ⇒ x = 400.0906/1,729(0.65) = 35.60% 6. 1.25(0.596)5,129.3 = 3,821.3285 7. 0.753(x/100)(829) = 42.8 ⇒ x = 42.8(100)/0.753(829) = 6.856370256% 8. 0.35(0.63)(1.3)7,991 = 2,290.62015 9. 70% de 45 = 0.70(45) = 0.45(70) ⇒ 45% 10. 0.253(80.3) = 0.803(25.3) ⇒ 80.3 11. 0.65(4,750) = 3,087.50 12. x(1.0175) = 12.25 ⇒ x = 12.25/1.0175 o bien, $12.0393 13. 0.95(0.18)(34,250) = 5,856.75 Página 5 de 18



14. x = 58.3(1.053) = $61.3899 15. (1 – 0.084)x = 5,000

x = 5,000/0.916 = 5,458.515284

16. 9.73/9.35 = 1.0406417 ⇒ 4.06417% 17. x = 48/52 = 0.923076923 ⇒ 1 – x = 0.076923077 o bien, 7.6923077% 18. 550/680 – 1 = -0.19117647 ⇒ 19.117647% 19. 3,600/27,000 = 0.1

13.33%

20. 37,750/37,910 – 1 = -0.004220522 ⇒ 0.4220522% 21. 5,724/5,300 = 1.08 ⇒8% 22. 22.36/21.50 = 1.04 ⇒ 4% 23. x = 81.92/83.15 = 0.985207456 ⇒ 1 – x = 0.014792544 o bien, 1.4792544% 24. 19/13 = 1.461538 ⇒ 46.1538% 25. 1.0425(x) = 6,850 ⇒ x = 6,850/1.0425 = 6,570.7434 26. d = 1 – 3/4 = 0.25 > 24.8 ⇒ 4 por 3 27. 2.513(251.3) = 631.5169

(c)

28. 0.783(x/100)(14,320) = 1,753 ⇒ x = (1,753/14,320)100/0.783 x = 15.634253%

(b)

29. 0.583(3.25)301.48 = 571.229

(d)

30. 0.257(0.032)1.25(A) = 5,9624 ⇒ A = 580 ⇒ 31. 31.5/18 = 1.75

75%

32. 7.56/6.3 = 1.20 ⇒ 7.56(1.2) = 9.072

(a)

(c) (b)

33. 1’006,250/875,000 = 1.15 = r ⇒ 1’006,250(1.15)2 = 1’330.765.62

34. x(1.25) = 113,597 ⇒ x = 90,877.60 millones ⇒

(c)

(a) Página 6 de 18



Ejercicios 1.5 5. A = k(p – q)/ ⇒ A= 12(10 – 2)/ 6. 28 = 12(12 – q)/

30 = k(8 – 3)/

⇒ k = 30(2)/5 = 12

= 12(8)/4 = 24 ⇒ 28 = 4(12 – q) ⇒ q = 12 – 7 ⇒ q = 5

7. x/382 = 85/180 ⇒ x = 180.3 o bien, 180.39 8. C = ka/n 9. C = ka/n

85 = k(22)/48 ⇒ k = 185.4545455

⇒ C = k(21)/56 = 69.54545457 o bien, 69.55 10. 75 = k/10,000 ⇒

k = 750,000

⇒ C = 750,000/10,800 = 69. 11. p = k/p

⇒ $69.44

75 = k/10,000 ⇒ k = 750,000

P = 750,000/15,000 = 50(mínimo) ⇒ no es posible. 12. 63 = 750,000/n ⇒ n = 11,904.7619 o bien, 11,905 13. p = k/

5,000 = k/

⇒

= k/5,000

5(100 – C) = (25,174.39/5,000)2 = 25.34999647 ⇒ C = 94.93 14. 50 + 60 + 83 + 22 = 215 ⇒ 1/215 = 0.004651163 = k a) 50(k)135,500 = $31,511.63 b) 60(k)135,500 = $ 37,813.95 c) 83(k)135,500 = $52,309.30 d) 22(k)135,500 = $13,865.12 15. A = kx2/y3 16. P = k/q(q + r)2 Página 7 de 18



17. A = k(20 + x)/ 18. z = k/(a2 + b2) 19. A = kh/(v – t)2 20. M = C(1 + in) 21. c = kI/in 22. A es proporcional b y a h. 23. V es inversamente proporcional a t. 24. K es proporcional a v y a t. 25. D es proporcional al doble de r (o proporcional a r). 26. A es proporcional a la suma de b y B y a h. 27. Y es proporcional a la suma de a y b, e inversamente proporcional al cubo de la diferencia c – d. 28. D es proporcional a la suma (a + b – 2C). 29. P es proporcional al cubo de (2 + 3a – b) e inversamente proporcional a c. 30. N es proporcional al logaritmo de la suma de y y z. 31. B = k/t

520 = k/3 ⇒ k = 1,560 ⇒ B = 1,560/2 = $780

32. B = kn

325 = k(405) ⇒ k = 325/405 = 0.802469136

B = k(432) = $346.67 33. T = 4 + kt/n

10 = 4 + k(16)/6 ⇒ k = 6(6)/16 = 2.25

⇒ T = 4 + 2.25(13)/4 = 11.3125 semanas. 34. T = 4 + kf/n

10 = 4 + k(16)/6 ⇒ k = 2.25

⇒ 8.5 = 4 + 2.25(12)/n ⇒ 4.5/2.25(12) = 1/n ⇒ n = 6 35. B = kn

756 = k(18)

k = 756/18 ⇒ B = k(15) = $630,00

(b) Página 8 de 18


36. Suma = 135


Carlos: 63,000(35/135) = 16,333.33

Jorge: 63,000(47/135) = 21,933.33 ⇒ 21,933.33 ⇒

Luis: 63,000(53/135) = 24,733.33

(a)

37. G = (35/135)63,000 = 16,333.33 38. Suma; 135 + 27 = 162

0.6(G) = $9,800

(c)

162/135 = 1.20

⇒ U = 63,000(1.2) = 75,600 a Luis: (53/162)75,600 = $24,733.33 ⇒

(a)

39. U = kp/c 30 = k(13)/16 k = 36.92307693 ⇒ U = k(18)/21 = $31’648,351.65

(c)

40. U = kp/c k = 36.92307693 ⇒ 34.46 = 36.92307693(p/15) ⇒ p = 13.999375 o bien, 14 41. U = k(25/25) = k = $36’923,076.93

(d)

(b)

42. 18.3(0.88) = 16.104 millones. (3/37)k + (2/40)k + (4/44)k = 1 K(0.081081081 + 0.05 + 0.090909091) = 1 ⇒ k = 1/0.221990172 = 4.504703929 ⇒ (3/37)4.504703929(16.104) = 5.881925844 o bien, $5’881,925.84 ⇒

(c)

Ejercicios 1.6 2. loga(5.4) = 384.5 ⇒ a384.5= 5.4 3. log3(A) = 8.5 ⇒ 38.5 = A 4. log12(P – 3) = 10 ⇒ 1210 = P – 3 5. (1.0283)2 – x = 50.23 ⇒ log1.0283(50.23) = 2 – x Página 9 de 18



6. (A – 2)x = 4 + B ⇒ logA – 2(4 + B) = x 7. logp(5) = 4 ⇒ p4 = 5 8. log7(125) = B ⇒ 7B = 125 9. logx(40.3) = 183.2 ⇒ x183.2 = 40.3 10. log13(C) = 0.383 ⇒ 130.383 = C 11. logx(10/3) = 1.2 ⇒ x1.2 = 10/3 12. log3/5(30C) = 40 ⇒ (3/5)40 = 30C 13. (42.8)10 = R ⇒ log42.8(R) = 10 14. (A – 2B)5 = 3,540 ⇒ logA – 2B(3,540) = 5 15. (73.4)x = 10/3 ⇒ log73.4(10/3) = x 16. (1,258N)4.2 = 100 ⇒ log1,258N(100) = 4.2 17. (253)3x – 2 = 13 ⇒ log253(13) = 3x – 2 18. log10(N) = 1.245 ⇒ 101.245 = N ⇒ N = 17.57923614 19. loga(450) = 5.23 ⇒ a5.23 = 450 ⇒ a =

= 3.215928674

20. log11(1,331) = x ⇒ 11x = 1,331 ⇒ x = 3 porque 113 = 1,331 21. log8(N) = 48.5 ⇒ 848.5= N ⇒ N = 6.307603 X 1043

Página 10 de 18



22. loga(148.5) = 4 ⇒ a4 = 148.5 ⇒ a =

a = 3.490853421

23. log3 – a(25) = 4 ⇒ (3 – a)4 = 25 (c) 24. 5x + 2 = A ⇒ 5x = A – 2 ⇒ log5(A – 2) = x ⇒

(a)

25. 53 + A = B ⇒ 53 = B – A ⇒ log5(B – A) = 3

(a)

26. log4(-3) = 10B, no está definida. ⇒

(c)

27. log3(P) – 5 = 1 ⇒ log3(P) = 6 ⇒ 36 = P

(d)

28. 10x – 4 = y ⇒ 10x = y + 4 ⇒ log10(y + 4) = x ⇒ 29. log7(N) = 1.482347866 ⇒ 71.482347866 = N

(b) N = 17.89490001

(c)

30. log20(3,128) = x ⇒ 20x = 3,128 ⇒ x = 2.68653817 (por tanteos). 31. log15(-4/5) no existe

(d)

32. logx(525) = 3.34 ⇒ x3.34 = 525 ⇒ x =

= 6.5225 ⇒

33. log90(425) = ln(425)/ln(90) = 1.34496559

(c)

(d)

Ejercicios 1.7 3. antiln(ln25.3) = 25.3 y log(antilog4,350) = 4,350 5. log(32.5) = ln(32.5)/ln(10) = 1.51 antilog(32.5) = 1032.5 este es mayor. 6. log(antilog(87.23)) = 87.23 7. ln(e) = 1 antiln(ln (e)) = antiln(1) = e1 = e 8. log(antilog(4x – 2)) = 4x – 2 Página 11 de 18



9. log(5x – 3) = 25 ⇒ antilog(log(5x – 3)) = antilog(25) = 5x – 3 10. antilog(2x2 – 3) = 1 ⇒ log(antilog(2x2 – 3)) = log 1 ⇒ 2x2 – 3 = 0 ⇒ 2x2 = 3 ⇒ x2 = 3/2 ⇒ x = ± 11. 32 + x = 27x/3 = (33)x/3 = 3x ⇒ 2 + x = x ⇒ 2 = 0 no hay solución. 12. log(x – 10) = 25 ⇒ 1025 = x – 10 ⇒ x =1025 + 10 13. ln

= x – 3 ⇒ x = 3 + ln

x = 4.745214258

14. (1,025)n – 3 = 321.5 ⇒ n – 3 = ln(321.5)/ln(1.025) = 233.7945215 n = 236.7945215 15. log(10.5) = 3 – 2x ⇒ 2x = 3 – log(10.5) = 1.978810701 ⇒ x = 0.989405351 16. ln(8.5) = ln(x – 4) ⇒ 8.5 = x – 4 ⇒ x = 12.5 17. (52.1)x + 2 = 328 ⇒ (x + 2) = ln(328)/ln(52.1) ⇒ x = -0.534588442 18. log(6 – 3x) = 3 ⇒ 103 = 6 – 3x ⇒ 3x = 6 – 1000 ⇒ x = -994/3 19. log

= 1.3 ⇒ 101.3 =

⇒

= 19.95262315 ⇒ x = 397.1071706

20. log(5 – 4x) = log(x2) ⇒ 5 – 4x = x2 ⇒ x2 + 4x – 5 = 0 (x + 5)(x – 1) = 0

x1 = -5

x2 = 1

21. log(15.35)4.5 = 4.5(ln15.35/ln10) = 4.5(1.18610838) = 5.33748771 22. ln(315.08)3 = (3)ln(315.08) = 3(5.752826575) = 17.25847973 23. ln

= (1/4) ln(3,253.8) = (1/4)(8.087578823) = 2.021894706

24. ln

= (1/5)ln(1,295.3) = (1/5)(7.166497608) = 1.433299522

25. log(4/7)3 = 3log(4/7) = 3(ln(4/7)/ln(10)) = 3(-0.243038049) = -0.729114146 26. ln(1,425)-20 = (-20) ln(1,425) = -20(7.261927093) = -145.2385419 27. log(735) = ln(735)/ln(10) = 2.866287339 Página 12 de 18



28. ln(-42.39) no está definido. 29. log(2x – 3) = 2 ⇒ 102 = 2x – 3 ⇒ 2x = 102 + 3 = 2x ⇒ x = (102 + 3)/2

(d)

30. ln(4 + 2x) = 2.5 ⇒

(d)

= 4 + 2x ⇒ x = (12.18249396 – 4)/2 = 4.09124698

31. 53x – 2 = 12.97 ⇒ 3x – 2 = ln(12.97)/ln(5) = 1.592257135 ⇒ x = 1.197419045 32. log(

)=2⇒

t=±

(d)

33. log(4 – 3x) + 2 = 0 ⇒ 10-2 = 4 – 3x ⇒ 3x = 4 – 0.01 ⇒ x = 1.33

(b)

34. log(10) + ln(e) + log(2x + 5) = 3 ⇒ log(2x + 5) = 1 ⇒ 2x + 5 = 10

x = 5/2

35. antilog(x + 2) = 5 ⇒ log(antilog(x + 2)) = log(5) ⇒ log(5) = x + 2

(c)

36. ln(t2 – 2) = 4 ⇒ e4 = t2 – 2 ⇒ t = ± 37. logx(4 – x) la base debe ser positiva

(b)

(c)

(c) (c)

38. antiln(2x – 3) = e2 ⇒ 2x – 3 = ln(e2) = 2 ⇒ x = 5/2

(c)

39. log(antilog(61.3) = antilog(log(3x – 2)) ⇒ 61.3 = 3x – 2 ⇒ x = 63.3/3 ⇒ x = 21.1

(c)

40. antiln(ln(15)) = 25 + 3x – x2 ⇒ 15 = 25 + 3x – x2 x2 – 3x – 10 = 0 (x – 5)(x + 2) = 0

x1 = 5, x2 = -2

(a)

Ejercicios 1.8 1. (1/6)375,000 = 62,500, 3/6(375,000) = 187,500 (2/6)(375,000) = 125,000 ⇒ 62,500, 187,500, 125,000 2. 2’800000(0.8) = 2’240,000 Página 13 de 18



8(17) + 6(13) + 5(8) + 3(2) = 260 ⇒ 2’240,000/260 = 8,615,3846 = k 17k = $146,461.54 13k = 112,000

8k = 68,923.08

2k = 17,230.77

⇒ $146,461.54, $112.000, $68,923.08, $17,230.77 3. k/3 + k/4 + k/2 + k/2 = 13,200 ⇒ k(1,58 ) = 13,200 ⇒ k = 8,336.842109 $2,778.95, $2,084.21, $4,168.42, $4,168.42 = 20,000 ⇒ Cn = 175,000 – 20,000(6) = $55,000

4.

5. 450(12) + (450 + x)18 = 990(30) ⇒ x = (29,700 – 13,500)/18 = $900.00 6. (15(3,525) + 15(3,525 + x))/30 = 8,000 105,750 + 15x = 240,000 ⇒ x = 134,250/15 = 8,950 7. u = 0.25(100,000) – 0.75(20,000) = $10,000 8. x(0.09815) + (180,000 – x)(0.08465) = 180,000(0.093) 0.0135x + 15,237 = 16,740 ⇒ x = 1,503/0.0135 = 111,333.33 180,000 – x = 68,666.67 ⇒ $111,333.33 y $68,666.67 9. (270,000)(1.075)x = 540,000 ⇒ x = ln(2)/ln(1.075) = 9.584358957 o bien, 10 ⇒ en el 2020 10. a1 = 12.5

a5 = a1(1 + 0.053)4 = a1(1.229457398) = $15’368,217.48

11. I = 7,200 + 0.0015(x) ⇒ I = 7,200 + 0.0015(8’500,000) = $19,950.00 12. a) 0.15(x) = 850,000 ⇒ x = 850,000/0.15 = $5’666,666.67 b) 0.283(850,000) = $240,550.00 c) 0.07(850,000 – 240,550) = $42,661.50 d) 0.485(42,661.50) = $20,690.83 e) 0.132(850,000 – 240,550) = $80,447.40 Página 14 de 18



13. 38,925.31/38,542.16 = 1.009941062

⇒ 0.9941062%

14. 132’380,000/130’746,000 = 1.012497514 ⇒ 1.2497514% 15. 4.548987/4.530129 = 1.004162795 ⇒ 0.4162795% 16. 4.12/4.01 = 1.027431421 ⇒ 2.7431421% 17. C = kp/r, 90 = k(20/6)

k = 27

18. C = k(10/3) = 90

C = 27(15/5) = 81

(b)

(b)

19. T = k(n/c) 23,000 = k(1/4) ⇒ k = 92,000 ⇒ T = k(1.122)/5 = 20,645 20. T = 92,000(1.122/4) = 25.806

(e)

(a)

21. 3.8 = k(153)(25) ⇒ k = 0.000993464 ⇒ P = k(150)(30) = 4.470588237

(b)

22. P = k(160)27 = 4.291764707 23. i = (10/6)(0.045) = 0.075 o bien, 7.5%

(b)

Problemas propuestos para exámenes 1. Falso (5 – x)2 = 25 – 10x + x2 2. 3x + 2 = 3x32

Falso

3. Verdadero, 0° no existe. 4.

+3=6⇒

Falso

5. log(5) = P ⇒ 10p = 5

Falso

6. 13.2930501 = 13.29306

Falso

7. 4x - 2 = 1/1024

4-3-2 = 4-5 = 2-10 = 1/210 = 1/1024

8. Debe ser ≠ 0

Falso

Verdadero

Página 15 de 18



9. Es una expresión Falso 10.

= x ⇒ 2 + x = x2 ⇒ x2 – x – 2 = 0

(x – 2)(x + 1) = 0

x1 = 2, x2 = -1

-1 no cumple ⇒ 2 11. log15(30) = z ⇒ 15z = 30 12.

= ±4

4 o bien, -4

13. (x/100)(0.30)(4,800) = 324 ⇒ x = 324(100)/0.3(4,800) = 22.5

22.5%

14. 10.83x = 42 ⇒ log10.8(42) = 3x 15. (3 – x)4 = 2,401 ⇒ 3 – x =

=7 ⇒ x = -4

x = -4

16. y = kx3/(z – x)2 103 = 1 – 3t ⇒ 999 = -3t ⇒ t = -333

17. log(1 – 3t) = 3

t = -333

18. a5 = a1(1 + 0.035)4 = 70,500(1.147523001) = 80,900 19. a17 = a1 r16 ⇒ 70,500 = a1(1,035)16 ⇒ a1 = 40,657.77

40,658

20. a7 = a1(1.0022) = 37,650(1.0022) = 37,733 21. (

)2 = 721.3

22. 56(5-5) = 56-5 = 5 23. 47/163 = 47/(42)3 = 47/46 = 4 24. ln(65.3)3/2 = (3/2)ln(65.3) = 6.268488055 25. (30.2)1/2 26. x3

= 30.21/2 + 1/2 = 30.2

/

x6 = x3(x + 3)/(x + 3)2x6 = 1/(x + 3)x3

27. (7a)4(7a)5 = (7a)9 28. (x – y)2/(x – y)-3 = (x – y)2 + 3 = (x – y)5 29. 3x – 3 = 9 = 32 ⇒ x – 3 = 2 ⇒ x = 5 Página 16 de 18



30. log3(x + 2) = 4 ⇒ 34 = x + 2 ⇒ x = 81 – 2 o bien, x = 79 31.

4 – 3x = 4 ⇒ x = 0

=2

32. 300 = 325(1 – 0.85d) ⇒ 300/325 – 1 = -0.85d ⇒ d = 0.090497738 33. x/2 + 3/4 = 4 – x/3 ⇒ 6x + 9 = 48 – 4x ⇒ 10x = 39

x = 39/10

34. (x + 3)3 = 27 ⇒ x + 3 = 3 ⇒ x = 0 35. 36.

= 1.293 ⇒ 1 + i = 1.29310 ⇒ i = 12.06126524 = x ⇒ 4 – 3x = x2 ⇒ x2 + 3x – 4 = 0 ⇒ (x + 4)(x – 1) = 0

x1 = -4 ⇒ x2 = 1 ⇒ x = 1 (-4 no cumple). 37.

= 7.962541478

38. log7(429.5) = ln(429.5)/ln(7) = 3.115571263 39. ln(70.98)3 = (3)ln(70.98) = 3(4.262398147) = 12.78719444 40. (5,293)3/4 = 620.5494864 41. (21.7)-2 = 0.002123638 42.

=

= 2.89848191

43. log(70) – log(10) + lne = log(70) – 1 + 1 = log(70) = 44. 45.

=

= 1.84509804

= 0.874032049

= 35,000 ⇒ C = 35,000(7) + 60,300 = 305,300

46. u = 0.40(53,200) – 0.60(19,500) = $9,580 47. a4 = 925,000(1.036)3 = 1’028,539.557 o bien, 1’028,540 48. a86 = a1(1.036)85 = a1(20.21060014) ⇒ a1 = 925,000/20.21060014 ⇒ a1 = 45,768 hab. Página 17 de 18



49. 81.5/89.3 – 1 = 0.912653975 – 1 = .0.087346025

8.7346025%

50. 98/125 = 0.784 ⇒ 21.6% 51. 6250 + x(256,500) = 16,510 ⇒ x = 0.04 ⇒ 4% 52.

= x ⇒ N = xk ⇒

53. A = k/x

(b)

(a)

54. t/5 = 4 – 2x (4 – t/5)/2 = x ⇒ x = 2 – t/10 55. 3,624.95/3,500 = 1.0357 ⇒ 3.57%

(d)

56. (x5)x -2/x3 = x5-2/x3 = x3/x3 = 1 ⇒

(d)

57. 4 – x < 0 ⇒ x > 4 58. 59.

(d)

= 7.134563757 ⇒ 7.1346 =5⇒

(c)

4 + x = 25

(b)

x – 21 = 0

(c)

60. loga(54) – 10 = 0 ⇒ loga(54) = 10 ⇒ a10 = 54 ⇒ 61. 1.25(Y/100)5,000 = 4,875 ⇒ Y = 78% 62. A = logx(P – Q) ⇒ XA = P – Q ⇒

(c) (a)

(a)

Página 18 de 18

Matemáticas financieras Manual de soluciones José Luis Villalobos 4ª edición PEARSON

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