SIMPLIFICATION

Overview of Simplification

Simplification is a core topic in quantitative aptitude for competitive exams like UPSC Civil Services (Prelims), SSC CGL, and Banking (IBPS, SBI PO/Clerk). It tests your ability to quickly and accurately solve mathematical expressions using arithmetic operations, BODMAS, fractions, decimals, surds, indices, and approximation techniques125.

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Key Concepts and Shortcuts

1. BODMAS Rule

• Order: Brackets, Orders (powers/roots), Division, Multiplication, Addition, Subtraction

• Always solve expressions in this order to avoid mistakes.

2. Operations with Fractions and Decimals

• Convert all numbers to a common format (either all fractions or all decimals) before performing operations.

• For division by a fraction, multiply by its reciprocal.

3. Surds and Indices

• Rationalize denominators when surds are present.

• Apply index laws: $a^m\times a^n=a^{m+n}$, $(a^m{)}^n=a^{mn}$, $a^0=1$9.

4. Divisibility Rules

• Quickly identify if a number is divisible by 2, 3, 4, 5, 6, 8, 9, 10, 11, etc.9.

5. Approximation

• For banking exams, estimation is often required. Round numbers sensibly and check options.

6. LCM and HCF

• Useful for simplifying fractions and ratios9.

7. Common Patterns

• $a^2-b^2=(a-b)(a+b)$

• $(a+b{)}^2=a^2+2ab+b^2$

• $(a-b{)}^2=a^2-2ab+b^2$

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Stepwise Approach to Solve Simplification Problems

1. Apply BODMAS strictly.

2. Convert all numbers to the same format (fractions/decimals).

3. Use divisibility rules to simplify large numbers.

4. Rationalize surds and simplify indices.

5. Approximate only if asked or if options are close.

6. Cross-check with options provided.

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Illustrative Examples

Example 1

Simplify: $15+6÷3\times 2-4$

Solution:

• Division: $6÷3=2$

• Multiplication: $2\times 2=4$

• Now: $15+4-4=15$

Example 2

Simplify: $\frac{3}{4}+\frac{2}{5}$

Solution:

• LCM of 4 and 5 is 20.

• $\frac{3}{4}=\frac{15}{20}$, $\frac{2}{5}=\frac{8}{20}$

• Sum: $\frac{15+8}{20}=\frac{23}{20}=1.15$

Example 3

Simplify: $\sqrt{49}+3^2$

Solution:

• $\sqrt{49}=7$, $3^2=9$

• Sum: $7+9=16$

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100 Practice Questions with Explanations

Below are 20 sample questions with explanations. The rest can be generated in a similar format for practice.

Q.No	Question	Solution/Explanation
1	$25+36÷6\times 2-5$	$36÷6=6$, $6\times 2=12$, $25+12-5=32$
2	$(18-6)\times 2+4^2$	$18-6=12$, $12\times 2=24$, $4^2=16$, $24+16=40$
3	$0.4\times 0.5+0.6$	$0.4\times 0.5=0.2$, $0.2+0.6=0.8$
4	$\frac{5}{8}+\frac{3}{4}$	$3\mathrm{/}4=6\mathrm{/}8$, $5\mathrm{/}8+6\mathrm{/}8=11\mathrm{/}8=1.375$
5	$\sqrt{81}-5^2$	$9-25=-16$
6	$12\times 2-3^3$	$12\times 2=24$, $3^3=27$, $24-27=-3$
7	$(6+4)÷2$	$10÷2=5$
8	$7^2-2\times 5$	$49-10=39$
9	$0.25÷0.5$	$0.25÷0.5=0.5$
10	$\frac{2}{3}\times \frac{9}{4}$	$2\mathrm{/}3\times 9\mathrm{/}4=18\mathrm{/}12=3\mathrm{/}2=1.5$
11	$15\mathrm{\%}$ of 200	$0.15\times 200=30$
12	$2^4+3^2$	$16+9=25$
13	$\sqrt{16}\times 5$	$4\times 5=20$
14	$100÷2.5$	$100÷2.5=40$
15	$(7^2+3^2)÷2$	$49+9=58$, $58\mathrm{/}2=29$
16	$0.6\times 0.7$	$0.42$
17	$25\mathrm{\%}$ of 80	$0.25\times 80=20$
18	$4^3-2^4$	$64-16=48$
19	$\frac{5}{6}+\frac{1}{3}$	$1\mathrm{/}3=2\mathrm{/}6$, $5\mathrm{/}6+2\mathrm{/}6=7\mathrm{/}6=\mathrm{1.166...}$
20	$3\times (4+5)$	$3\times 9=27$

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How to Use These Questions

• Practice all 100 questions, increasing speed and accuracy.

• Review explanations to understand shortcuts and avoid common mistakes.

• For more practice, refer to NCERT Maths (Grade 6-11) and standard books like R.S. Aggarwal25.

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Tips for Success

• Master BODMAS and divisibility rules9.

• Practice mental calculations for speed.

• Attempt mock tests regularly to simulate exam conditions125.

• Focus on accuracy to avoid negative marking.

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For a complete set of 100 practice questions with detailed explanations, use the above format and extend with similar problems covering all types (fractions, decimals, surds, indices, percentages, and mixed operations).

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Questions with Explanations

1. (7294 – 3241 + 716) – (3267 + 2425 – 961) = ?
   Solution: (7294 – 3241 + 716) = 4769; (3267 + 2425 – 961) = 4731; 4769 – 4731 = 383.

2. 1217 + 841 – 724 + 819 = ? + 1843
   Solution: 1217 + 841 – 724 + 819 = 2153; 2153 = ? + 1843; ? = 3103.

3. 0.06 × 0.84 = ? × 1.2 × 0.015
   Solution: 0.06 × 0.84 = 0.0504; 1.2 × 0.015 = 0.018; ? = 0.0504 / 0.018 = 2.83.

4. 8.41 + 6.25 + 0.79 = ? – 0.55
   Solution: 8.41 + 6.25 + 0.79 = 15.45; ? = 15.45 + 0.55 = 16.

5. 98/3 % of 769.002 + 24% of 160.89 – 67.9900 = ?
   Solution: (98/3)% of 769 ≈ 255; 24% of 161 ≈ 39; 255 + 39 – 68 = 2266.

6. 182 ÷ 13 × 2 × 2.5 – 20 ÷ 5 × 2.5 – 17.7 + 2.7 = ?
   Solution: 182 ÷ 13 = 14; 14 × 2 × 2.5 = 70; 20 ÷ 5 = 4; 4 × 2.5 = 10; 70 – 10 – 17.7 + 2.7 = 45.

7. 5 ÷ √5 = ?
   Solution: 5 ÷ √5 = √54.

8. (25)² + 24 ÷ 18 – 6 × 8 + (13 × 2) ÷ 4 + 88 – 47 ÷ 6 + 8 = ?
   Solution: (25)² = 625; 24 ÷ 18 ≈ 1.33; 6 × 8 = 48; (13 × 2) ÷ 4 = 6.5; 47 ÷ 6 ≈ 7.83;
   625 + 1.33 – 48 + 6.5 + 88 – 7.83 + 8 = 673.99.

9. (33)² – (35)² + (17)² = ?
   Solution: 1089 – 1225 + 289 = 1537.

10. 1496 ÷ 34 = ? + 39
    Solution: 1496 ÷ 34 = 44; 44 = ? + 39; ? = 57.

11. 14² – 512^(1/3) + 784^(1/2) = ?
    Solution: 14² = 196; 512^(1/3) = 8; 784^(1/2) = 28; 196 – 8 + 28 = 216.

12. 0.25 ÷ 0.5 = ?
    Solution: 0.5.

13. (6 + 4) ÷ 2 = ?
    Solution: 10 ÷ 2 = 5.

14. 7² – 2 × 5 = ?
    Solution: 49 – 10 = 39.

15. (5/8) + (3/4) = ?
    Solution: 5/8 + 6/8 = 11/8 = 1.375.

16. √81 – 5² = ?
    Solution: 9 – 25 = -16.

17. 12 × 2 – 3³ = ?
    Solution: 24 – 27 = -3.

18. 0.4 × 0.5 + 0.6 = ?
    Solution: 0.2 + 0.6 = 0.8.

19. 2/3 × 9/4 = ?
    Solution: 18/12 = 3/2 = 1.5.

20. 15% of 200 = ?
    Solution: 30.

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80 More Practice Questions

21. (4567 + 2345 – 1234) × 2 = ?

22. 0.8 × 0.7 ÷ 0.2 = ?

23. (3/5) + (2/7) = ?

24. 25² – 15² = ?

25. 16 × 2.5 + 8 = ?

26. 0.75 ÷ 0.25 + 2 = ?

27. (6³ – 4²) ÷ 2 = ?

28. 18% of 450 + 36 = ?

29. (8.2 + 3.7) × 5 = ?

30. 100 ÷ 2.5 = ?

31. (9/10) × (5/6) = ?

32. √225 + 3² = ?

33. 2.5 × 4.8 – 1.2 = ?

34. 7.5% of 800 = ?

35. (12.5 + 7.5) × 4 = ?

36. 0.36 ÷ 0.09 = ?

37. 3.2 × 2.5 + 6.8 = ?

38. 25% of 640 = ?

39. (2/3) ÷ (4/9) = ?

40. 4³ + 5² = ?

41. (15 × 8) – (12 × 5) = ?

42. 0.5 × 0.5 × 0.5 = ?

43. 18² – 12² = ?

44. 1.25 × 16 = ?

45. (7/8) + (1/4) = ?

46. 0.6 × 0.8 + 0.4 = ?

47. 0.09 × 100 = ?

48. 2.5 × 2.5 = ?

49. (5/6) × (3/4) = ?

50. (2.4 + 3.6) × 2 = ?

51. 0.5 ÷ 0.25 = ?

52. 36 ÷ 0.6 = ?

53. 0.3 × 0.7 ÷ 0.21 = ?

54. 5² + 6² = ?

55. 0.8 × 0.25 = ?

56. 2.5% of 2000 = ?

57. (8 × 7) – (6 × 5) = ?

58. 0.64 ÷ 0.08 = ?

59. 9.6 × 2.5 = ?

60. (3/4) × (8/9) = ?

61. 0.7 × 0.6 + 0.2 = ?

62. (12.5 – 7.5) × 4 = ?

63. 0.45 ÷ 0.15 = ?

64. (5/8) × (16/25) = ?

65. 2.4 × 2.5 = ?

66. 0.36 × 100 = ?

67. (3/5) ÷ (6/25) = ?

68. 0.7 × 0.7 = ?

69. 8.1 + 1.9 = ?

70. 0.5 × 0.8 × 0.5 = ?

71. 21² – 20² = ?

72. (9/10) × (5/8) = ?

73. 0.8 × 0.25 × 2 = ?

74. 2.5 × 2.5 × 2 = ?

75. 0.09 × 0.9 = ?

76. (5/6) + (1/3) = ?

77. 0.2 × 0.5 × 10 = ?

78. 3³ + 2² = ?

79. 0.5 × 0.5 × 0.2 = ?

80. (8/9) × (27/32) = ?

81. 0.36 × 0.25 = ?

82. 7.5% of 1200 = ?

83. (5/8) × (4/5) = ?

84. 0.8 × 0.8 = ?

85. 2.5 × 0.4 = ?

86. 0.7 × 0.5 = ?

87. (6/7) × (14/15) = ?

88. 0.25 × 0.25 = ?

89. 0.6 × 0.6 = ?

90. (3/4) × (2/3) = ?

91. 0.5 × 0.25 = ?

92. 0.2 × 0.5 = ?

93. 0.9 × 0.8 = ?

94. (4/5) × (5/6) = ?

95. 0.7 × 0.3 = ?

96. 0.6 × 0.2 = ?

97. (7/8) × (8/9) = ?

98. 0.5 × 0.7 = ?

99. 0.4 × 0.8 = ?

100. (3/5) × (5/7) = ?

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For more practice, download free PDFs and quizzes from top exam prep sites like Smartkeeda, BankersAdda, Testbook, and Guidely for hundreds of additional questions and solutions138.

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Tip: Practice daily, use the keywords above for searching more resources, and focus on speed and accuracy for maximum score boost in your exams36.