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Decimals Simplification

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Introduction

Decimal simplification वाले questions आपकी decimals के साथ speed और accuracy को test करते हैं - यानी जल्दी add, subtract, multiply और divide करना। Simple place-value tricks, fractions और decimals के बीच conversion, और decimal point कब shift करना है-ये सब जानने से problems तेज़ और error-free हो जाती हैं।

Pattern: Decimals Simplification

Pattern: Decimals Simplification

मुख्य idea: Addition/Subtraction में decimal places को align करें, Multiplication/Division में decimal को shift करें (या numbers को integers बनाएं), और जहाँ possible हो वहाँ factors cancel करके simplify करें।

Step-by-Step Example

Question

Simplify: 2.5 × 1.2

Options:

  • A) 2.7
  • B) 3.0
  • C) 30.0
  • D) 0.3

Solution

  1. Step 1: Decimal places गिनें

    2.5 में 1 decimal place है, 1.2 में भी 1 decimal place है → कुल 2 decimal places।

  2. Step 2: Decimals हटाकर integers में बदलें

    Decimals हटाकर numbers को integers की तरह लें: 2.5 → 25, 1.2 → 12.

  3. Step 3: Integers को multiply करें

    25 × 12 = 300.

  4. Step 4: Decimal point वापस लगाएँ

    कुल 2 decimal places वापस लगाएँ → 300 → 3.00 → final = 3.0.

  5. Final Answer:

    3.0 → Option B.

  6. Quick Check:

    Approximation: 2.5 × 1.2 ≈ 2.5 × (1 + 0.2) = 2.5 + 0.5 = 3.0 ✅

Quick Variations

1. Addition/Subtraction: decimal points align करें (जैसे 12.5 + 0.75 = 13.25).

2. Multiplication by powers of 10: decimal right shift करें (जैसे 3.45 × 10 = 34.5).

3. Division by powers of 10: decimal left shift करें (जैसे 345 ÷ 100 = 3.45).

4. कई decimals multiply करते समय: total decimal places count करें, integers multiply करें, फिर decimal restore करें।

Trick to Always Use

  • Step 1 → ×/÷ के लिए: decimals shift करके integers बनाएं, operation करें, फिर decimal वापस लगाएँ।
  • Step 2 → +/- के लिए: decimal points line up करें और integers की तरह solve करें।
  • Step 3 → Simple fractions (1/4, 1/5, 1/8) के decimal forms याद रखें: 0.25, 0.2, 0.125.

Summary

  • Multiplication में decimal places count करें; powers of 10 के लिए decimal shift करें।
  • Addition/Subtraction में decimal points align करें।
  • Multiplication/Division में integers बनाकर calculation आसान करें, फिर decimal वापस लगाएँ।
  • Fraction-decimal equivalents से quick approximation और checking आसान होती है।

याद रखने वाला example:
2.5 × 1.2 = 3.0

Practice

(1/5)
1. Simplify: 3.4 + 2.56
easy
A. 5.96
B. 6.00
C. 5.90
D. 5.86

Solution

  1. Step 1: Align decimal places

    Write numbers with same decimal places: 3.40 + 2.56.

  2. Step 2: Add as integers

    Add as integers: 340 + 256 = 596.

  3. Step 3: Restore decimal point

    Place decimal back → 5.96.

  4. Final Answer:

    5.96 → Option A.

  5. Quick Check:

    3.4 + 2.5 = 5.9, plus 0.06 → 5.96 ✅
Hint: Align decimal points and add like integers.
Common Mistakes: Not adding trailing zeros (3.40) before adding decimals.
2. Simplify: 12.5 - 7.36
easy
A. 4.94
B. 5.14
C. 5.16
D. 5.12

Solution

  1. Step 1: Equalise decimal places

    Write with same decimals: 12.50 - 7.36.

  2. Step 2: Subtract as integers

    Subtract as integers: 1250 - 736 = 514.

  3. Step 3: Restore decimal point

    Place decimal back → 5.14.

  4. Final Answer:

    5.14 → Option B.

  5. Quick Check:

    12.5 - 7 = 5.5; minus 0.36 → 5.14 ✅
Hint: Add trailing zeros so decimals align, then subtract.
Common Mistakes: Forgetting to align decimal places before subtracting.
3. Simplify: 2.5 × 1.2
easy
A. 2.8
B. 3.2
C. 3.0
D. 3.5

Solution

  1. Step 1: Remove decimals by shifting

    Remove decimals: 2.5 → 25, 1.2 → 12.

  2. Step 2: Multiply integers

    Multiply integers: 25 × 12 = 300.

  3. Step 3: Restore decimal places

    Total decimal places = 2 → place decimal → 3.00 = 3.0.

  4. Final Answer:

    3.0 → Option C.

  5. Quick Check:

    2.5 × (1 + 0.2) = 2.5 + 0.5 = 3.0 ✅
Hint: Multiply as integers, then restore decimal places.
Common Mistakes: Putting decimal in wrong position after multiplication.
4. Simplify: 15.75 ÷ 0.25
medium
A. 62
B. 64
C. 65
D. 63

Solution

  1. Step 1: Make divisor an integer by shifting

    Convert to integers by shifting decimal two places: 15.75 → 1575, 0.25 → 25.

  2. Step 2: Divide the integers

    Divide: 1575 ÷ 25 = 63.

  3. Final Answer:

    63 → Option D.

  4. Quick Check:

    0.25 × 63 = 15.75 ✅
Hint: Make divisor an integer by same decimal shift in numerator and denominator, then divide.
Common Mistakes: Shifting decimal in numerator but not in divisor (or vice versa).
5. Simplify: (3.6 × 2.5) - 4.8
medium
A. 4.2
B. 4.0
C. 3.8
D. 3.6

Solution

  1. Step 1: Multiply by converting to integers

    Multiply first: remove decimals → 36 × 25 = 900.

  2. Step 2: Restore decimal places

    Total decimal places = 2 → 900 → 9.00. So 3.6 × 2.5 = 9.00.

  3. Step 3: Subtract

    Now subtract: 9.00 - 4.80 = 4.20 → 4.2.

  4. Final Answer:

    4.2 → Option A.

  5. Quick Check:

    3.6 × 2.5 ≈ 9; 9 - 4.8 = 4.2 ✅
Hint: Do multiplication first (convert to integers), then subtraction; keep decimal places aligned.
Common Mistakes: Subtracting before multiplying or misplacing decimal after multiplication.