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Mixed Operations (Combined)

Introduction

Mixed operations में fractions, percentages, roots, exponents और BODMAS rule एक ही problem में combine होते हैं। Competitive exams में ये काफी common हैं और आपकी calculation speed और rules को step-by-step apply करने की clarity दोनों test करते हैं। मुख्य बात यह है कि हर हिस्से को एक-एक करके simplify किया जाए, बिना किसी step को skip किए।

Pattern: Mixed Operations (Combined)

Pattern

मुख्य idea: BODMAS को strictly follow करें और simplification rules (fractions, roots, exponents, percentages) को क्रम में, step-by-step apply करें।

Step-by-Step Example

Question

Simplify: (50% of 240 + √81) ÷ (2/3 of 18)

Options:

  • A) 10.75
  • B) 11
  • C) 9.5
  • D) 12

Solution

  1. Step 1: Percentage निकालें

    50% of 240 → (50/100) × 240 = 120.

  2. Step 2: Root simplify करें

    √81 = 9.

  3. Step 3: Numerator बनाएं

    Numerator = 120 + 9 = 129.

  4. Step 4: Fraction operation करें

    Denominator: 2/3 of 18 = (2/3) × 18 = 12.

  5. Step 5: Final division

    129 ÷ 12 = 10.75.

  6. Final Answer:

    10.75 → Option A.

  7. Quick Check:

    Numerator 129, Denominator 12 → 129/12 = 10.75 ✅

Quick Variations

1. Problems में percentages + roots + fractions हो सकते हैं।

2. कुछ में exponents और surds एक ही expression में आते हैं।

3. Decimals और fractions के साथ addition/subtraction भी test किया जा सकता है।

Trick to Always Use

  • Step 1 → हमेशा percentages और fractions को पहले handle करें।
  • Step 2 → Roots और exponents को next simplify करें।
  • Step 3 → BODMAS order follow करें (brackets → division/multiplication → addition/subtraction).
  • Step 4 → हर step को simplest form में लाते जाएं, कभी direct jump न करें।

Summary

Summary

Mixed operation problems में:

  • Problem को छोटे-छोटे parts में तोड़ें।
  • BODMAS को strictly follow करें।
  • Fractions, percentages और roots को ध्यान से simplify करें, फिर addition/subtraction करें।
  • गलतियों से बचने के लिए हमेशा quick recomputation करें।

Practice

(1/5)
1. Simplify: (25% of 200 + √36) ÷ (1/2 of 16)
easy
A. 7
B. 6
C. 8
D. 9

Solution

  1. Step 1: Compute 25% of 200

    25% = 25/100, so (25/100) × 200 = 0.25 × 200 = 50.

  2. Step 2: Evaluate the square root

    √36 = 6.

  3. Step 3: Form the numerator

    50 + 6 = 56.

  4. Step 4: Compute the denominator

    (1/2) of 16 = 0.5 × 16 = 8.

  5. Step 5: Divide numerator by denominator

    56 ÷ 8 = 7.

  6. Final Answer:

    7 → Option A.

  7. Quick Check:

    7 × 8 = 56 (matches numerator) ✅
Hint: Convert percentages to fraction (x/100) and simple fractions to decimals (1/2 = 0.5) for quick calculation.
Common Mistakes: Forgetting to convert % to fraction or mis-evaluating the square root.
2. Simplify: (10% of 240 + √36) ÷ (1/3 of 18)
easy
A. 4
B. 5
C. 6
D. 7

Solution

  1. Step 1: Compute 10% of 240

    10% of 240 = (10/100) × 240 = 0.1 × 240 = 24.

  2. Step 2: Evaluate the square root

    √36 = 6.

  3. Step 3: Form the numerator

    24 + 6 = 30.

  4. Step 4: Compute denominator

    (1/3) of 18 = (1/3) × 18 = 6.

  5. Step 5: Divide

    30 ÷ 6 = 5.

  6. Final Answer:

    5 → Option B.

  7. Quick Check:

    5 × 6 = 30 (matches numerator) ✅
Hint: Find simple percentages (10% = divide by 10) and small fractional parts (1/3) mentally first.
Common Mistakes: Mixing up numerator and denominator operations or rounding too early.
3. Simplify: (√100 + 30% of 100) ÷ (2/5 of 25)
easy
A. 3
B. 5
C. 4
D. 6

Solution

  1. Step 1: Evaluate the square root

    √100 = 10.

  2. Step 2: Compute 30% of 100

    30% = 30/100, so 30.

  3. Step 3: Form the numerator

    10 + 30 = 40.

  4. Step 4: Compute denominator

    (2/5) × 25 = 10.

  5. Step 5: Divide

    40 ÷ 10 = 4.

  6. Final Answer:

    4 → Option C.

  7. Quick Check:

    4 × 10 = 40 (correct) ✅
Hint: Convert percent to decimal/fraction and simplify fraction multiplications before dividing.
Common Mistakes: Forgetting to multiply the denominator fraction correctly.
4. Simplify: (25% of 240 + √441) ÷ (1/2 of 18)
medium
A. 7
B. 8
C. 6
D. 9

Solution

  1. Step 1: Compute 25% of 240

    25% = 25/100, so 0.25 × 240 = 60.

  2. Step 2: Evaluate √441

    √441 = 21.

  3. Step 3: Form the numerator

    60 + 21 = 81.

  4. Step 4: Compute denominator

    (1/2) × 18 = 9.

  5. Step 5: Divide

    81 ÷ 9 = 9.

  6. Final Answer:

    9 → Option D.

  7. Quick Check:

    9 × 9 = 81 (correct) ✅
Hint: Convert % and fractions first, then handle roots.
Common Mistakes: Mis-evaluating square root like √441.
5. Simplify: (50% of 60 + √324) ÷ (1/3 of 18)
medium
A. 8
B. 7
C. 6
D. 9

Solution

  1. Step 1: Compute 50% of 60

    50% = half, so 30.

  2. Step 2: Evaluate √324

    √324 = 18.

  3. Step 3: Form numerator

    30 + 18 = 48.

  4. Step 4: Compute denominator

    (1/3) × 18 = 6.

  5. Step 5: Divide

    48 ÷ 6 = 8.

  6. Final Answer:

    8 → Option A.

  7. Quick Check:

    8 × 6 = 48 (correct) ✅
Hint: Extract simple percentages and use familiar roots to stay fast.
Common Mistakes: Mis-evaluating √324 or miscomputing fraction part.

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