Ratio & Proportion: Detailed Notes, Examples, and Practice Questions

 

Ratio & Proportion: Detailed Notes, Examples, and Practice Questions

---

1. Basic Concepts

A. Ratio

• Definition: A ratio is a quantitative relationship between two numbers showing how many times one value contains or is contained within the other. It is written as $a:b$ or $\frac{a}{b}$12567.

• Key Points:

  • Both quantities must be of the same kind and unit.

  • The first term is the antecedent; the second is the consequent.

  • Ratios can be simplified by dividing both terms by their highest common factor.

  • Multiplying or dividing both terms by the same non-zero number does not change the ratio27.

Example:
If the ratio of apples to oranges is 6:9, simplify by dividing both by 3:
$6:9=2:3$

---

B. Proportion

• Definition: Proportion is an equation stating that two ratios are equal. If $a:b=c:d$, then $a,b,c,d$ are in proportion and written as $a:b::c:d$12567.

• Extremes: $a$ and $d$

• Means: $b$ and $c$

Example:
Are $2:3$ and $4:6$ in proportion?
$\frac{2}{3}=\frac{4}{6}=\frac{2}{3}$ ⇒ Yes, they are in proportion26.

---

2. Types of Proportion

A. Direct Proportion

• When one quantity increases, the other increases in the same ratio (and vice versa).

• Formula: $\frac{a}{b}=\frac{c}{d}$57.

Example:
If 4 pens cost ₹20, how much do 10 pens cost?
Let cost = $x$.
$4:10=20:x$ ⇒ $\frac{4}{10}=\frac{20}{x}$ ⇒ $x=50$

---

B. Inverse Proportion

• When one quantity increases, the other decreases so that the product remains constant.

• Formula: $a\times b=c\times d$57.

Example:
If 8 workers complete a job in 6 days, how many days will 12 workers take?
$8\times 6=12\times x$ ⇒ $x=4$

---

C. Continued Proportion

• If $a:b=b:c$, then $a,b,c$ are in continued proportion7.

Example:
If $2:4=4:8$, then 2, 4, 8 are in continued proportion.

---

3. Partnership Problems

• Definition: When two or more people invest money in a business, the profit (or loss) is divided in the ratio of their investments and the time for which the money was invested.

• Formula:

  • If A invests $P_1$ for $T_1$ months and B invests $P_2$ for $T_2$ months:

    • Profit share of A : B = $P_1\times T_1:P_2\times T_2$15.

Example:
A invests ₹5000 for 12 months, B invests ₹8000 for 6 months.
A:B = $5000\times 12:8000\times 6=60000:48000=5:4$

---

4. Mixtures and Alligation (Applications)

• Definition: Mixture problems involve combining two or more quantities at different ratios to get a mixture of a desired ratio.

• Alligation Rule: Used to find the ratio in which two or more ingredients at different prices must be mixed to obtain a mixture at a given price.

Example:
How much water must be added to 60 L of milk at 30% concentration to reduce it to 20%?
Let water added = $x$ L.
$\frac{60}{60+x}=\frac{20}{30}$ ⇒ $60\times 30=20\times (60+x)$
$1800=1200+20x$ ⇒ $x=30$ L

---

5. Practice Examples

Ratio

• Q: Divide ₹50 between A and B in the ratio 3:2.
  A: $3+2=5$ parts. A gets $\frac{3}{5}\times 50=\text{₹}30$, B gets $\text{₹}20$368.

Proportion

• Q: Are 2:3 and 4:6 in proportion?
  A: $\frac{2}{3}=\frac{4}{6}=\mathrm{0.666...}$ ⇒ Yes26.

Direct Proportion

• Q: If 5 books cost ₹100, what is the cost of 8 books?
  A: $\frac{5}{8}=\frac{100}{x}$ ⇒ $x=160$

Inverse Proportion

• Q: If 4 men can finish a job in 10 days, how many days will 5 men take?
  A: $4\times 10=5\times x$ ⇒ $x=8$

Partnership

• Q: A and B invest ₹6000 and ₹4000 for 8 and 12 months. Find profit share.
  A: A:B = $6000\times 8:4000\times 12=48000:48000=1:1$

Mixture

• Q: In what ratio must 30% acid solution be mixed with 50% acid solution to get 40% acid solution?
  A: By alligation:
  $(50-40):(40-30)=10:10=1:1$

---

6. Practice Questions (with a mix of all topics)

Questions 1–20 (with answers)

1. Divide ₹120 in the ratio 2:3.
   Ans: ₹48, ₹72

2. Are 5:7 and 15:21 in proportion?
   Ans: Yes

3. If 6 pens cost ₹60, what is the cost of 10 pens?
   Ans: ₹100

4. If 8 workers finish a job in 9 days, how many days for 12 workers?
   Ans: 6 days

5. A invests ₹4000 for 10 months, B invests ₹6000 for 8 months. Find profit share.
   Ans: 4000×10 : 6000×8 = 40000:48000 = 5:6

6. In what ratio must 25% and 45% alcohol solutions be mixed to get 35%?
   Ans: 1:1

7. Divide 80 kg into two parts in the ratio 5:3.
   Ans: 50 kg, 30 kg

8. Are 4:9 and 12:27 in proportion?
   Ans: Yes

9. If 7 apples cost ₹35, what is the cost of 12 apples?
   Ans: ₹60

10. If 5 men can do a job in 12 days, how many days for 15 men?
    Ans: 4 days

11. A invests ₹5000 for 6 months, B invests ₹3000 for 12 months. Find profit share.
    Ans: 5000×6 : 3000×12 = 30000:36000 = 5:6

12. In what ratio must 20% and 60% sugar solutions be mixed to get 40%?
    Ans: 1:1

13. Divide ₹90 in the ratio 7:8.
    Ans: ₹42, ₹48

14. Are 3:5 and 9:15 in proportion?
    Ans: Yes

15. If 9 chocolates cost ₹45, what is the cost of 15 chocolates?
    Ans: ₹75

16. If 10 workers finish a job in 15 days, how many days for 6 workers?
    Ans: 25 days

17. A invests ₹8000 for 9 months, B invests ₹6000 for 12 months. Find profit share.
    Ans: 8000×9 : 6000×12 = 72000:72000 = 1:1

18. In what ratio must 10% and 30% acid solutions be mixed to get 20%?
    Ans: 1:1

19. Divide 150 kg in the ratio 2:3.
    Ans: 60 kg, 90 kg

20. Are 8:12 and 20:30 in proportion?
    Ans: Yes

---

Questions 21–100 (Practice Only)

21. Divide ₹250 in the ratio 3:2.

22. Are 6:11 and 12:22 in proportion?

23. If 5 kg rice costs ₹200, what is the cost of 8 kg?

24. If 15 men complete a job in 10 days, how many days for 25 men?

25. A invests ₹3000 for 7 months, B invests ₹5000 for 5 months. Find profit share.

26. In what ratio must 40% and 70% milk solutions be mixed to get 50%?

27. Divide 120 liters in the ratio 7:5.

28. Are 9:14 and 18:28 in proportion?

29. If 8 apples cost ₹56, what is the cost of 14 apples?

30. If 12 men finish a job in 18 days, how many days for 9 men?

31. A invests ₹7000 for 10 months, B invests ₹5000 for 12 months. Find profit share.

32. In what ratio must 15% and 25% salt solutions be mixed to get 18%?

33. Divide ₹180 in the ratio 4:5.

34. Are 2:7 and 10:35 in proportion?

35. If 4 books cost ₹80, what is the cost of 9 books?

36. If 20 workers finish a job in 5 days, how many days for 8 workers?

37. A invests ₹9000 for 8 months, B invests ₹6000 for 10 months. Find profit share.

38. In what ratio must 12% and 36% acid solutions be mixed to get 24%?

39. Divide 200 kg in the ratio 3:5.

40. Are 5:8 and 25:40 in proportion?

41. Divide ₹350 in the ratio 5:2.

42. Are 7:9 and 21:27 in proportion?

43. If 6 kg wheat costs ₹90, what is the cost of 11 kg?

44. If 18 men complete a job in 16 days, how many days for 24 men?

45. A invests ₹6000 for 11 months, B invests ₹9000 for 7 months. Find profit share.

46. In what ratio must 35% and 65% milk solutions be mixed to get 50%?

47. Divide 180 liters in the ratio 13:17.

48. Are 8:13 and 16:26 in proportion?

49. If 10 oranges cost ₹80, what is the cost of 15 oranges?

50. If 14 men finish a job in 21 days, how many days for 7 men?

51. A invests ₹4000 for 15 months, B invests ₹6000 for 10 months. Find profit share.

52. In what ratio must 22% and 38% acid solutions be mixed to get 30%?

53. Divide ₹240 in the ratio 5:7.

54. Are 11:15 and 22:30 in proportion?

55. If 3 pens cost ₹45, what is the cost of 7 pens?

56. If 16 men finish a job in 20 days, how many days for 32 men?

57. A invests ₹10000 for 6 months, B invests ₹8000 for 9 months. Find profit share.

58. In what ratio must 18% and 42% salt solutions be mixed to get 30%?

59. Divide 300 kg in the ratio 2:7.

60. Are 9:13 and 27:39 in proportion?

61. Divide ₹420 in the ratio 4:3.

62. Are 6:8 and 18:24 in proportion?

63. If 12 kg rice costs ₹96, what is the cost of 20 kg?

64. If 10 men complete a job in 30 days, how many days for 25 men?

65. A invests ₹8000 for 12 months, B invests ₹6000 for 16 months. Find profit share.

66. In what ratio must 28% and 56% milk solutions be mixed to get 35%?

67. Divide 250 liters in the ratio 9:11.

68. Are 7:12 and 21:36 in proportion?

69. If 5 apples cost ₹25, what is the cost of 12 apples?

70. If 18 men finish a job in 15 days, how many days for 30 men?

71. A invests ₹5000 for 9 months, B invests ₹7000 for 6 months. Find profit share.

72. In what ratio must 16% and 40% acid solutions be mixed to get 24%?

73. Divide ₹360 in the ratio 11:7.

74. Are 8:11 and 24:33 in proportion?

75. If 7 books cost ₹56, what is the cost of 10 books?

76. If 24 men finish a job in 8 days, how many days for 16 men?

77. A invests ₹9000 for 7 months, B invests ₹7000 for 9 months. Find profit share.

78. In what ratio must 10% and 30% salt solutions be mixed to get 18%?

79. Divide 400 kg in the ratio 5:3.

80. Are 12:19 and 36:57 in proportion?

81. Divide ₹500 in the ratio 7:3.

82. Are 4:9 and 8:18 in proportion?

83. If 8 kg wheat costs ₹64, what is the cost of 15 kg?

84. If 20 men complete a job in 12 days, how many days for 15 men?

85. A invests ₹12000 for 5 months, B invests ₹8000 for 8 months. Find profit share.

86. In what ratio must 24% and 36% milk solutions be mixed to get 30%?

87. Divide 180 liters in the ratio 2:5.

88. Are 5:7 and 15:21 in proportion?

89. If 9 oranges cost ₹54, what is the cost of 20 oranges?

90. If 16 men finish a job in 18 days, how many days for 24 men?

91. A invests ₹7000 for 8 months, B invests ₹9000 for 6 months. Find profit share.

92. In what ratio must 20% and 60% acid solutions be mixed to get 36%?

93. Divide ₹600 in the ratio 3:5.

94. Are 11:17 and 22:34 in proportion?

95. If 10 pens cost ₹80, what is the cost of 16 pens?

96. If 25 men finish a job in 10 days, how many days for 20 men?

97. A invests ₹6000 for 12 months, B invests ₹9000 for 8 months. Find profit share.

98. In what ratio must 12% and 48% salt solutions be mixed to get 24%?

99. Divide 360 kg in the ratio 7:5.

100. Are 6:13 and 18:39 in proportion?

---

Use these notes and questions for a thorough grasp of Ratio & Proportion, including all major applications in exams.
Practice regularly for speed and accuracy.