1. If \[ a : 2 = b : 5 \] then, what percentage of \(a\) and \(b\) will be equal? (a) 20 (b) 40 (c) 30 (d) 50

2. Four quantities are in continuous proportion. If the first and second quantities are 3 and 2 respectively, then the fourth quantity will be: (a) \(\cfrac{2}{3}\) (b) \(\cfrac{8}{9}\) (c) \(\cfrac{4}{3}\) (d) \(\cfrac{10}{12}\)

3. If \(a, b, c, d\) are in continuous proportion, then what is the value of \(\cfrac{a^2 + b^2 + c^2}{b^2 + c^2 + d^2}\)? (a) \(\cfrac{a}{c}\) (b) \(\cfrac{b}{c}\) (c) \(\cfrac{c}{d}\) (d) \(\cfrac{b}{d}\)

4. If 2a=3b=4c, then what is 2a:5b:7c ? (a) 2:3:4 (b) 4:15:28 (c) 12:20:21 (d) None of these

5. If \(x, y, z\) are in continued proportion, then what is the value of \[ x^2y^2z^2\left(\frac{1}{x^3}+\frac{1}{y^3}+\frac{1}{z^3}\right)? \] (a) \(x+y+z\) (b) \(x^2+y^2+z^2\) (c) \(x^3+y^3+z^3\) (d) None of the above

6. If \(2a = 3b = 4c\), then \(a : b : c\) will be. (a) 2:3:4 (b) 6:4:3 (c) 4:3:2 (d) 3:4:6

7. If \(a + b : \sqrt{ab} = 1 : 1\), then what is the value of \(\sqrt{\cfrac{a}{b}} + \sqrt{\cfrac{b}{a}}\)? (a) 1 (b) 2 (c) 3 (d) 4

8. if a: \(\cfrac{27}{64}=\cfrac{3}{4}\):a then the value of a be- (a) \(\cfrac{81}{256}\) (b) 9 (c) \(\cfrac{9}{16}\) (d) \(\cfrac{16}{9}\)

9. \(a\) is a positive number, and if \(a : \cfrac{27}{64}=\cfrac{3}{4} : a\) , then the value of \(a\) will be ? (a) \(\cfrac{81}{256}\) (b) 9 (c) \(\cfrac{9}{16}\) (d) \(\cfrac{16}{9}\)

10. The fourth proportional of ( ???? 2 − ???? ???? + ???? 2 ) , ( ???? 3 + ???? 3 ) , and ( ???? − ???? ) is (a) \(x^2+y^2\) (b) \(x+y\) (c) \(x^2-y^2\) (d) \(x-y\)

11. If \(a, b, c, d\) are in continued proportion, then what is the value of \(\frac{abc(a + b + c)}{ab + bc + ca}\)? (a) \(\cfrac{1}{a^2}\) (b) \(\cfrac{1}{c^2}\) (c) \(b^2\) (d) \(a^2\)

12. If \( \frac{a}{3} = \frac{b}{4} = \frac{c}{7} \), then what is the value of \( \frac{a + b + c}{c} \)? (a) 1 (b) 3 (c) 4 (d) 2

13. If \( (7a - 5b) : (3a + 4b) = 7 : 11 \), then what is the value of \( (5a - 3b) : (6a + 5b) \)? (a) 777:244 (b) 777:247 (c) 247:778 (d) 247:787

14. If \(x, 2x, 3\), and \(y\) are in continued proportion, then what will be the value of \(y\)? (a) \(4x\) (b) \(6x\) (c) 4 (d) 6

15. If \(2a = 3b = 4c\), the ratio \(a:b:c\) will be – (a) 3:4:6 (b) 4:3:6 (c) 3:6:4 (d) 6:4:3

16. If A:B = 2:3, B:C = 5:8, and C:D = 6:7, then A:D = ____? (a) 2:7 (b) 7:2 (c) 5:8 (d) 5:14

17. Find the fourth proportional of \(a, 2a^2, 3a^3\). (a) \(6a^3\) (b) \(6a^2\) (c) \(6a^4\) (d) \(6a\)

18. If \(\cfrac{p}{q} = \cfrac{5}{7}\) and \(p - q = -2\), then the value of \(p + q\) is – (a) 12 (b) 13 (c) 14 (d) 15

19. The third proportional to 8 and 12 is – (a) 12 (b) 16 (c) 18 (d) 20

20. If the ratio of A to B is 2:3 and the ratio of B to C is 4:3, then what is the ratio A:B:C? (a) 8:12:15 (b) 6:9:8 (c) 8:12:9 (d) 8:16:9

21. If \(4x = 5y = 6z\), what is the value of the ratio \(x : y : z\)? (a) 12:10:15 (b) 10:12:15 (c) 15:12:10 (d) 15:10:12

22. Which number should be added to each of the numbers 6, 7, 15, and 17 so that the resulting sums are in proportion? (a) 2 (b) 3 (c) 4 (d) 5

23. If \( (3x - 2y) : (3x + 2y) = 4 : 5 \), then what is the value of \(x : y\)? (a) 1:6 (b) 1:1 (c) 2:1 (d) 6:1

24. A's amount is \(\cfrac{3}{4}\) of B's amount, and B's amount is \(1\cfrac{1}{5}\) times C's amount. What is the ratio of A's amount to C's amount? (a) 19:20 (b) 10:9 (c) 9:10 (d) 20:19

25. In a partnership business, the capital ratio between A and B is 2:3, and the capital ratio between B and C is 6:5. If A's profit share is ₹400, what will be C's profit share? (a) ₹ 500 (b) ₹ 600 (c) ₹ 700 (d) ₹ 800

26. If \(a : b : c = 2 : 3 : 5\), then find the value of \(\frac{2a + 3b - 3c}{c}\). (a) \(=-\cfrac{2}{5}\) (b) \(=-\cfrac{3}{5}\) (c) \(=\cfrac{2}{5}\) (d) \(=\cfrac{3}{5}\)

27. If \(\frac{a}{2} = \frac{b}{3} = \frac{c}{4} = \frac{3a - 2b + 4c}{p}\), then what is the value of \(p\)? (a) 12 (b) 13 (c) 16 (d) 18

28. The ratio of monthly income of K and Kh is 3 : 8, and the ratio of monthly income of Kh and G is 4 : 9. What will be the ratio of monthly income of K and G? (a) 2:3 (b) 1:6 (c) 1:4 (d) 3:2

29. If \(\cfrac{x-3y}{2y}=\cfrac{6x-5y}{5x}\), then what is the value of \(x:y\)? (a) 5:1 or 2:5 (b) 1:5 or 2:5 (c) 1:5 or 5:2 (d) None of the above

30. The two angles adjacent to side BC are in the ratio 4 : 5 ⇒ \( \frac{4}{9} : \frac{5}{9} \) \(\therefore\) The measures of the two angles = \(180^\circ \times \frac{4}{9},\ 180^\circ \times \frac{5}{9}\) = \(80^\circ,\ 100^\circ\) (a) ₹ 900 (b) ₹ 1000 (c) ₹ 1100 (d) ₹ 1200

31. If \((a^2 + b^2)(x^2 + y^2) = (ax + by)^2\), then what are the values of \(x\) and \(y\)? (a) \(b:a\) (b) \(b^2:a^2\) (c) \(a^2:b^2\) (d) \(a:b\)

32. \(a, b, c, d\) ক্রমিক সমানুপাতী হলে \(\cfrac{abc(a+b+c)}{ab+bc+ca}\) -এর মান কত? - translate in english (a) \(\cfrac{1}{a^2}\) (b) \(\cfrac{1}{c^2}\) (c) \(b^2\) (d) \(a^2\)

33. If \(y\) is the geometric mean between \(x\) and \(z\), then what is the geometric mean between \(x^2 + y^2\) and \(y^2 + z^2\)? (a) (b) (c) (d)

34. If \(x : y = 3 : 4\), then the value of \(\cfrac{x^2 - xy + y^2}{x^2 + xy + y^2}\) will be: (a) 37:13 (b) 13:35 (c) 13:37 (d) 20:13

35. If 7, x, y, 189 are in continued proportion, then the values of x and y respectively will be: (a) 63,21 (b) 21,23 (c) 21,63 (d) 23,21

36. What is the combined ratio of \(a:bc\), \(b:ca\), and \(c:ab\)? (a) \(1:1\) (b) \(1:abc\) (c) \(abc:1\) (d) None of these

37. If ???? 3 = ???? 5 = ???? 8 , then what is the value of 3 ???? − 5 ???? + 2 ???? ???? ?" (a) \(\frac{1}{5}\) (b) 5 (c) 0 (d) 10

38. Here’s the English translation of your question: **"If \( x = \frac{4ab}{a + b} \), then what is the value of \( \frac{x + 2a}{x - 2a} + \frac{x + 2b}{x - 2b} \)?"** Would you like me to solve it step by step? (a) 1 (b) -2 (c) 2 (d) -1

39. \(x^3y, x^2y^2\), and \(xy^3\) are in continued proportion. True / False

40. The compound ratio of \(ab : c^2\), \(bc : a^2\), and \(ca : b^2\) is 1:1. True / False

41. The continued proportion of \(2ab : c^2, bc : a^2,\) and \(ca : 2b^2\) is 1 : 1. True / False

42. If \(2a = 3b = 4c\), then \(a : b : c = 2 : 3 : 4\). True / False

43. If A : B = 3 : 2 and B : C = 3 : 5, then what is the ratio A : B : C?

44. If 50% of (P + 2Q) equals 30% of (2P + 3Q), then express P in terms of Q.

45. What is the value of the angle \(x^\circ\) formed between the diagonals of a rhombus?

46. If \(3x = 4y = 5z\), then what is the ratio \(x : y : z =\) _____?

47. If \(a, b, c, d\) are consecutive terms in geometric progression, show that \((a^2–b^2) (c^2-d^2)=(b^2–c^2)^2\).

48. If \(a, b, c, d\) are in continuous proportion, then show that \((a^2 - b^2)(c^2 - d^2) = (b^2 - c^2)^2\).

49. If \(x = cy + bz\), \(y = az + cx\), and \(z = bx + ay\), then prove that \[ \frac{x^2}{1 - a^2} = \frac{y^2}{1 - b^2} \]

50. If \(a, b, c\) are in continued geometric progression, prove that \[ a^2b^2c^2\left(\frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3}\right) = a^3 + b^3 + c^3 \]

51. If \[ \frac{3 - 5x}{x} + \frac{3 - 5y}{y} + \frac{3 - 5z}{z} = 0 \] then what is the value of \[ \frac{1}{x} + \frac{1}{y} + \frac{1}{z}? \]

52. Which of the following is greater: (a) The ratio of 8 meters to 10 meters (b) 20% of \(4\frac{2}{5}\)

53. If \((3x - 2y) : (3x + 2y) = 4 : 5\), then what is the value of \((x + y) : (x - y)\)?

54. If \(a, b, c\) are in continued proportion, prove that \((a + b + c)(a - b + c) = a^2 + b^2 + c^2\).

55. If \(a : b = b : c\), then prove that \[ (a + b)^2 : (b + c)^2 = a : c \]

56. If \(a^2 = by + cz\), \(b^2 = cz + ax\), and \(c^2 = ax + by\), then prove that \[ \frac{x}{a + x} + \frac{y}{b + y} + \frac{z}{c + z} = 1 \]

57. If \[ \frac{a}{b + c} = \frac{b}{c + a} = \frac{c}{a + b} \] then prove that each of these ratios is either \(\frac{1}{2}\) or \(-1\).

58. If \((b + c - a)x = (c + a - b)y = (a + b - c)z = 2\), then show that \[ \left(\frac{1}{x} + \frac{1}{y}\right)\left(\frac{1}{y} + \frac{1}{z}\right)\left(\frac{1}{z} + \frac{1}{x}\right) = abc \]

59. If 75% of A equals 40% of B, find the ratio \(A : B\).

60. \(2x+\cfrac{1}{x}=2\) হলে, \(\cfrac{x}{2x^2+x+1}\) -এর মান কত ? - translate in english

61. Given: \(\frac{x + y}{x - y} = \frac{a}{b}\) Prove that: \(\frac{y^2 + xy}{x^2 - xy} = \frac{a^2 - ab}{b^2 + ab}\)

62. If \(a : b = b : c\), then show that \[abc (a + b + c)^3 = (ab + bc + ca)^3\]

63. If \(\cfrac{a + b − c}{a + b} = \cfrac{b + c − a}{b + c} = \cfrac{c + a − b}{c + a}\) and \(a + b + c ≠ 0\), then prove that \(a = b = c\).

64. If \(\cfrac{x}{y+z}=\cfrac{y}{z+x}=\cfrac{z}{x+y}\), then prove that the value of each ratio is equal to either \(\cfrac{1}{2}\) or \(-1\).

65. If \(x : a = y : b = z : c\), then show that \((a^2 + b^2 + c^2)(x^2 + y^2 + z^2) = (ax + by + cz)^2\)

66. If \(x : y = 3 : 4\), then what is the value of \((3y - x) : (2x + y)\)?

67. Write the ratio of 10 days to 2 months.

68. If \(a, b, c\) are in continued geometric progression, prove that \(\cfrac{1}{b} = \cfrac{1}{b - a} + \cfrac{1}{b - c}\).

69. If \[ (b + c - a) x = (c + a - b) y = (a + b - c) z = 2 \] then prove that \[ \left( \frac{1}{y} + \frac{1}{z} \right) \left( \frac{1}{z} + \frac{1}{x} \right) \left( \frac{1}{x} + \frac{1}{y} \right) = abc \]

70. If \((3x - 2y) : (x + 3y) = 5 : 6\), then what is the value of the ratio \(x : y\)?

71. If \(\cfrac{x}{2} = \cfrac{y}{3} = \cfrac{z}{4}\), then find the value of \(\cfrac{3x + 4y + 8z}{x + 3y}\).

72. If \(a, b, c, d\) are in continued geometric progression, prove that \[(a^2 + b^2 + c^2)(b^2 + c^2 + d^2) = (ab + bc + cd)^2\]

73. If \(\frac{a^2}{b + c} = \frac{b^2}{c + a} = \frac{c^2}{a + b} = 1\), then prove that \[\frac{1}{1 + a} + \frac{1}{1 + b} + \frac{1}{1 + c} = 1\]

74. If \(\cfrac{a}{1-a}+\cfrac{b}{1-b}+\cfrac{c}{1-c} = 1\), then find the value of \(\cfrac{1}{1-a}+\cfrac{1}{1-b}+\cfrac{1}{1-c}\).

75. If \(a : b = b : c\), then prove that \(\cfrac{abc(a+b+c)^3}{(ab+bc+ca)^3} = 1\).

76. The ratio of Chinmoy's age to Ershad's age is 4:5, and the sum of their ages is 99 years. Find each person's age.

77. If \(\cfrac{a}{2} = \cfrac{b}{3} = \cfrac{c}{4} = \cfrac{2a - 3b + 4c}{p}\), then what is the value of \(p\)?

78. If \(\cfrac{x}{y+z}=\cfrac{y}{z+x}=\cfrac{z}{x+y}\), then prove that each ratio is equal to \(\cfrac{1}{2}\) or 1.

79. If \(a, b, c, d\) are consecutive terms in geometric progression, prove that \((a^2 - b^2)(c^2 - d^2) = (b^2 - c^2)^2\).

80. If the sum of two numbers is three times their difference, find the ratio between the two numbers.

81. Find the fourth proportional of \((x^2 - y^2), (x^2y - xy^2), (x + y)\).

82. If \(a : b = c : d\), then prove that \((a^2 + c^2)(b^2 + d^2) = (ab + cd)^2\).

83. If \((3x - 2y) : (x + 3y) = 5 : 6\), then find the ratio \((2x + 5y) : (3x + 4y)\).

84. If \[ \frac{b + c - a}{y + z - x} = \frac{c + a - b}{z + x - y} = \frac{a + b - c}{x + y - z} \] then prove that \[ \frac{a}{x} = \frac{b}{y} = \frac{c}{z} \]

85. If \(x\) is the geometric mean between \((a^2bc)\) and \((4bc)\), then the value of \(x\) is ______.

86. If \(a:b = 3:2\) and \(b:c = 3:2\), then what is \(a+b : b+c\)?

87. If A:B = 2:3 and B:C = 4:5, then what is the ratio A:B:C?

88. In a partnership business, A, B, and C invest capital in the ratio \(\frac{2}{3}:\frac{4}{5}:\frac{3}{4}\). If the total profit is ₹26,600, what is B's share of the profit?

89. If \(x^2 + y^2 - 4x - 6y + 13 = 0\), then what is the value of \((x + y) : (y - x)\)?

90. If \(a, b, c, d\) are consecutive terms in a geometric progression, prove that \((a^2 - b^2)(c^2 - b^2) = (b^2 - c^2)^2\).

91. If \(\cfrac{x}{y+z} = \cfrac{y}{z+x} = \cfrac{z}{x+y}\), then prove that each ratio is equal to \(\cfrac{1}{2}\) or -1.

92. If the fourth of 5 consecutive geometric numbers is 54 and the fifth is 162, then find the first number.

93. If \((a+b) : \sqrt{ab} = 2:1\), then the value of \(a:b\) will be 1:1.

94. Which number should be added to each of 4, 6, and 10 so that the resulting sums are in continued proportion?

95. If \(\frac{x}{y} = \frac{a + 2}{a - 2}\), then find the value of \(\frac{x^2 - y^2}{x^2 + y^2}\).

96. If \(x : a = y : b = z : c\), then show that \(\frac{x^3}{a^3} + \frac{y^3}{b^3} + \frac{z^3}{c^3} = \frac{3xyz}{abc}\).

97. If \(3x = 4y = 5z\), then \(x:y:z =\) _____.

98. If \(\frac{ay - bx}{c} = \frac{cx - az}{b} = \frac{bz - cy}{a}\), then prove that \(\frac{x}{a} = \frac{y}{b} = \frac{z}{c}\).

99. If \(\frac{2}{3}\) of A = 75% of B = 0.6 of C, then find the ratio A : B : C.

100. If 50% of A = 60% of B = \(\frac{4}{5}\) of C, then find the ratio A : B : C.

101. If \(\cfrac{y+z-x}{b+c-a} = \cfrac{z+x-y}{c+a-b} = \cfrac{x+y-z}{a+b-c}\), then show that \(\cfrac{x}{a} = \cfrac{y}{b} = \cfrac{z}{c}\).

102. If \(\cfrac{x}{y+z}=\cfrac{y}{z+x}=\cfrac{z}{x+y}\), then prove that each ratio is equal to \(\cfrac{1}{2}\) or (-1).

103. \[ (a+b) : \sqrt{ab} = 2:1, \quad \text{Find the value of } a:b. \]

104. 75% of A = 50% of B; Find the ratio A : B.

105. Find the geometric mean between \(6a^3b\) and \(24ab^3\).

106. In a business, the ratio of Lalu and Bhulu's capital is 3:2. If Bhulu's investment period is 4 months, for how many months should Lalu invest so that the profit ratio becomes 9:4?

107. If \(a, b, c\) are in continued geometric progression (i.e., in successive proportion), then prove that \[ \cfrac{1}{b^2} = \cfrac{1}{b^2 - a^2} + \cfrac{1}{b^2 - c^2} \]

108. If \(\cfrac{a}{b + c} = \cfrac{b}{c + a} = \cfrac{c}{a + b}\) and \(a + b + c \ne 0\), then prove that \(a = b = c\).

109. If \[ \cfrac{a^3+3ab^2}{b^3+3a^2b}=\cfrac{63}{62} \] determine the ratio \( a:b \).

110. If \[ (a+b+c)x = (b+c-a)y = (c+a-b)z = (a+b-c)w \] prove that \[ \cfrac{1}{y}+\cfrac{1}{z}+\cfrac{1}{w}=\cfrac{1}{x} \]

111. If \(a, b, c, d\) are in continuous proportion, prove that \((a^2 - b^2)(c^2 - d^2) = (b^2 - c^2)^2\).

112. Let \(a\) be a positive number, and given: \[ a : \frac{27}{64} = \frac{3}{4} : a \] Find the value of \(a\).

113. If \(a : b = 3 : 4\) and \(x : y = 5 : 7\), then find the value of \((3ax - by) : (4by - 7ax)\).

114. If \[ \frac{x^2 - yz}{a} = \frac{y^2 - zx}{b} = \frac{z^2 - xy}{c} \] then prove that \[ (a + b + c)(x + y + z) = ax + by + cz \]

115. If \(bcx = cay = abz\), then prove that \[ \frac{ax + by}{a^2 + b^2} = \frac{by + cz}{b^2 + c^2} = \frac{cz + ax}{c^2 + a^2} \]

116. If \[ \frac{2x}{3} = \frac{4y}{5} = \frac{7z}{9} \] then find the value of \[ \frac{4x + 12y - 21z}{3y} \]

117. If \(\cfrac{x}{y+z} = \cfrac{y}{z+x} = \cfrac{z}{x+y}\), then prove that each ratio is equal to \(\cfrac{1}{2}\) or (-1).

118. Given: \[ \frac{x}{b + c} = \frac{y}{c + a} = \frac{z}{a + b} \] Prove that: \[ \frac{a}{y + z - x} = \frac{b}{z + x - y} = \frac{c}{x + y - z} \]

119. If \(3a = 4b = Sc\), then \(a:b:c\) will be _____.

120. 1 + (tan A / tan B) = 1 + (tan(90° − B) / tan B) = 1 + (cot B / cot B) = 1 + cot² B = cosec² B (Proved)

121. If \[ \frac{by + cz}{b^2 + c^2} = \frac{cz + ax}{c^2 + a^2} = \frac{ax + by}{a^2 + b^2} \] then prove that \[ \frac{x}{a} = \frac{y}{b} = \frac{z}{c} \]

122. If \( a:b = 3:2 \) and \( b:c = 3:2 \), find the value of \( a+b : b+c \).