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Decimals Simplification

Introduction

Decimal simplification problems test speed and accuracy with decimals - adding, subtracting, multiplying, and dividing quickly. Knowing simple place-value tricks, how to convert between fractions and decimals, and when to shift the decimal point makes these problems fast and error-free.

Pattern: Decimals Simplification

Pattern

Key idea: Align decimal places for addition/subtraction, shift the decimal for multiplication/division (or convert to integers), and simplify by cancelling factors when possible.

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: Count decimal places

    2.5 has 1 decimal, 1.2 has 1 decimal → total 2 decimal places.

  2. Step 2: Remove decimals by converting to integers

    Remove decimals by treating numbers as integers: 2.5 → 25, 1.2 → 12.

  3. Step 3: Multiply the integers

    Multiply integers: 25 × 12 = 300.

  4. Step 4: Restore the decimal point

    Place the decimal back: total 2 decimal places → 300 → 3.00 → final = 3.0.

  5. Final Answer:

    3.0 → Option B.

  6. Quick Check:

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

Quick Variations

1. Addition/Subtraction: align decimal points (e.g., 12.5 + 0.75 = 13.25).

2. Multiplication by powers of 10: shift decimal right (e.g., 3.45 × 10 = 34.5).

3. Division by powers of 10: shift decimal left (e.g., 345 ÷ 100 = 3.45).

4. Multiplication of many decimals: count total decimal places, multiply integers, restore decimals.

Trick to Always Use

  • Step 1 → For ×/÷: convert to integers by shifting decimals, perform operation, then shift decimal back.
  • Step 2 → For +/-: line up decimal points and add/subtract like integers.
  • Step 3 → Convert simple fractions (1/4, 1/5, 1/8) to decimals to speed up answers: 0.25, 0.2, 0.125.

Summary

Summary

  • Count decimal places when multiplying; shift decimal for powers of 10.
  • Align decimal points for addition/subtraction.
  • Convert to integers for multiplication/division to avoid mistakes, then restore decimal places.
  • Use fraction-decimal equivalences for quick approximations and checks.

Example to remember:
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.

Mock Test

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