Decimals: Notes और MCQ Practice

आसान भाषा में

Decimal असल में Fraction का ही दूसरा रूप है, जहाँ Denominator हमेशा 10 की Powers (10, 100, 1000...) में होता है। RRB NTPC में इससे Direct Simplification, BODMAS और Recurring Decimals (Bar वाले सवाल) खूब पूछे जाते हैं, इसलिए basic rules clear होना बहुत जरूरी है।

Important Rules & Formulas

• Addition & Subtraction: Decimal Points को एक सीधी line (vertically) में align करें, खाली places पर '0' लगाएं और normal add/subtract करें।
• Multiplication: Points को भूलकर normal multiply करें, फिर सभी numbers के Decimal Places का sum करके right side से point लगाएं (e.g., 0.2 × 0.03 = 0.006)।
• Pure Recurring Decimal to Fraction:
• 0.ā = a ÷ 9
• 0.ab̄ = ab ÷ 99 (Example: 0.777... = 7 ÷ 9)
• Mixed Recurring Decimal to Fraction:
• 0.ab̄ = (ab − a) ÷ 90
• Example: 0.16̄ = (16 − 1) ÷ 90 = 15 ÷ 90 = 1 ÷ 6

2 Quick Tricks

1. Zero-Balancing Trick (Division): Division में Decimal हटाना हो, तो point के बाद digits की संख्या zero लगाकर equal कर लें और point उड़ा दें।

• Example: 0.6 ÷ 0.15 → 0.60 ÷ 0.15 = 60 ÷ 15 = 4.

1. Bar Direct Conversion: Numerator = (पूरा number − जिस digit पर bar नहीं है), Denominator = (जितने digits पर bar है उतने '9' और जितने non-bar decimal digits हैं उतने '0')।

2 Common Mistakes

1. Recurring Decimals में 9 की जगह 10 लेना: छात्र अक्सर 0.3̄ को 3 ÷ 10 लिख देते हैं, जबकि सही value 3 ÷ 9 = 1 ÷ 3 होती है।
2. Multiplication में Zero Count: 0.02 × 0.04 को कई बार 0.08 लिख देते हैं, जबकि decimal places 4 होंगे, यानी सही answer 0.0008 होगा।

Decimals: Quantitative Aptitude Study Notes for RRB NTPC

1. Concept

A decimal number represents a fraction whose denominator is a power of 10 (such as 10, 100, 1000, etc.). It consists of two parts separated by a dot known as the decimal point:

• Whole number part: Digits to the left of the decimal point.
• Decimal part: Digits to the right of the decimal point.

Place Value System after the Decimal Point:

• First place to the right: Tenths (1/10)
• Second place to the right: Hundredths (1/100)
• Third place to the right: Thousandths (1/1000)

Types of Decimals:

1. Terminating Decimals: The decimal digits end after a finite number of steps (e.g., 0.25, 0.625).
2. Non-Terminating Recurring Decimals: Digits repeat infinitely in a periodic pattern.

• Pure Recurring: All digits after the decimal point repeat (e.g., 0.333... = 0.3̅, 0.4545... = 0.45̅).
• Mixed Recurring: Some digits do not repeat, followed by digits that do (e.g., 0.1666... = 0.16̅).

1. Non-Terminating Non-Recurring Decimals: Digits neither end nor repeat periodically (Irrational numbers like √2, π).

2. Important Formulas / Rules / Facts

• Converting Terminating Decimals to Fractions: Write the number without the decimal point as the numerator. In the denominator, write 1 followed by as many zeroes as there are decimal places. Simplify to the lowest terms.
• Example: 0.075 = 75 ÷ 1000 = 3 ÷ 40

• Converting Pure Recurring Decimals to Fractions:
• Formula: 0.a̅ = a ÷ 9
• Formula: 0.ab̅ = ab ÷ 99
• Formula: 0.abc̅ = abc ÷ 999

• Converting Mixed Recurring Decimals to Fractions:
• Formula: 0.ab̅ (only b repeats) = (ab − a) ÷ 90
• Formula: 0.abc̅ (only c repeats) = (abc − ab) ÷ 900
• Formula: 0.ab̅c̅ (bc repeats) = (abc − a) ÷ 990
• General Rule: Numerator = (Entire number formed by non-repeating and repeating digits) − (Non-repeating digits). Denominator = As many 9s as repeating digits, followed by as many 0s as non-repeating digits after the decimal point.

• Addition and Subtraction:
• Always line up the numbers vertically by their decimal points before adding or subtracting.
• Add zeroes to the right of shorter decimals to keep place values clear.

• Multiplication:
• Multiply numbers normally, ignoring the decimal points.
• Count the total number of decimal digits in all factors combined.
• Place the decimal point in the final product so that it has the same total count of decimal places.

• Division:
• Make the divisor a whole number by shifting its decimal point to the right.
• Shift the decimal point of the dividend to the right by the same number of places.
• Divide normally.

3. Solved Examples

Example 1: Converting Mixed Recurring Decimal to Fraction

Problem: Express 0.4777... (0.47̅) as a simplified fraction.

Solution:

• Step 1: Identify repeating and non-repeating parts. Non-repeating digit = 4 (1 digit), Repeating digit = 7 (1 digit).
• Step 2: Use the formula: (Complete number − Non-repeating part) ÷ (9 for repeating digits followed by 0 for non-repeating digits).
• Step 3: Numerator = 47 − 4 = 43.
• Step 4: Denominator = 90 (one 9 for digit '7', one 0 for digit '4').
• Step 5: Fraction = 43 ÷ 90.

Answer: 43 ÷ 90

Example 2: Algebraic Simplification with Decimals

Problem: Evaluate the value of:

(0.87 × 0.87 × 0.87 + 0.13 × 0.13 × 0.13) ÷ (0.87 × 0.87 − 0.87 × 0.13 + 0.13 × 0.13)

Solution:

• Step 1: Let a = 0.87 and b = 0.13.
• Step 2: The expression is in the form: (a³ + b³) ÷ (a² − a×b + b²).
• Step 3: We know that a³ + b³ = (a + b)(a² − a×b + b²).
• Step 4: Therefore, the expression simplifies directly to: a + b.
• Step 5: Substitute the values: 0.87 + 0.13 = 1.00 = 1.

Answer: 1

Example 3: Decimal Shift in Division

Problem: If 12276 ÷ 15.5 = 792, find the value of 122.76 ÷ 155.

Solution:

• Step 1: Express the given equation: 12276 ÷ 15.5 = 792.
• Step 2: Rewrite the target expression: 122.76 ÷ 155.
• Step 3: Compare dividends: 122.76 = 12276 ÷ 100.
• Step 4: Compare divisors: 155 = 15.5 × 10.
• Step 5: Substitute these relationships:

(12276 ÷ 100) ÷ (15.5 × 10) = (12276 ÷ 15.5) ÷ (100 × 10) = 792 ÷ 1000 = 0.792.

Answer: 0.792

Example 4: Addition of Recurring Decimals

Problem: Find the value of 0.3̅ + 0.6̅ + 0.7̅ + 0.8̅.

Solution:

• Step 1: Convert each pure recurring decimal into a fraction:
• 0.3̅ = 3 ÷ 9
• 0.6̅ = 6 ÷ 9
• 0.7̅ = 7 ÷ 9
• 0.8̅ = 8 ÷ 9
• Step 2: Add the fractions together:

Sum = (3 + 6 + 7 + 8) ÷ 9 = 24 ÷ 9

• Step 3: Simplify the fraction: 24 ÷ 9 = 8 ÷ 3
• Step 4: Convert to mixed fraction or decimal: 8 ÷ 3 = 2 + 2/3 = 2.666... = 2.6̅

Answer: 8 ÷ 3 (or 2.6̅)

4. Shortcuts and Tricks

1. Decimal Count Matching in Products:

If an algebraic formula or calculation has equal powers in the numerator and denominator, check the ratio of powers of 10 directly without calculating large numbers.

1. Multiplying by 0.5, 0.25, 0.125:

• Multiplying by 0.5 is the same as dividing by 2.
• Multiplying by 0.25 is the same as dividing by 4.
• Multiplying by 0.125 is the same as dividing by 8.

1. Quick Division by 5, 25, 125:

• N ÷ 0.5 = N × 2
• N ÷ 0.25 = N × 4
• N ÷ 0.125 = N × 8

1. Spotting Recurring Additions:

Do not round off recurring decimals (e.g., using 0.67 for 0.666...). Always convert recurring decimals to simple fractions (using 9s in the denominator) before performing addition, subtraction, or multiplication.

5. Common Mistakes to Avoid

• Misalignment of Decimals: Adding 4.2 + 0.042 as 4.62 instead of aligning place values (4.200 + 0.042 = 4.242).
• Miscounting Zeros in Division: When dividing 0.0036 by 0.06, students often guess 0.6.

Correct method: (36 × 10⁻⁴) ÷ (6 × 10⁻²) = 6 × 10⁻² = 0.06.

• Denominator Rule Confusion in Recurring Numbers:

Writing 0.23̅ (only 3 repeats) as 23 ÷ 99 instead of (23 − 2) ÷ 90 = 21 ÷ 90.

• Ignoring the Whole Number in Conversions:

Writing 2.3̅ as (23 − 2) ÷ 9.

Correct method: Separate whole part: 2 + 0.3̅ = 2 + (3 ÷ 9) = 2 + 1/3 = 7 ÷ 3.