Fractions: Notes और MCQ Practice

आसान भाषा में

Fraction का मतलब होता है किसी Whole (पूरी चीज़) का एक Part (हिस्सा), जिसे हम Numerator / Denominator (p/q, जहाँ q ≠ 0) के रूप में लिखते हैं। RRB NTPC में Fraction Comparison, Recurring Decimals और Simplification से सीधे सवाल पूछे जाते हैं। Fractions पर अच्छी पकड़ आपकी calculation speed को double कर देती है।

Important Rules & Formulas

• Proper Fraction: Numerator < Denominator (Value हमेशा 1 से छोटी, जैसे 3/5)
• Improper Fraction: Numerator > Denominator (Value हमेशा 1 से बड़ी, जैसे 7/4)
• Addition / Subtraction: a/b ± c/d = (a×d ± b×c) ÷ (b×d)
• Division Rule: (a/b) ÷ (c/d) = (a/b) × (d/c)
• Recurring Decimals to Fraction:
• 0.a̅ = a ÷ 9
• 0.ab̅ = (ab − a) ÷ 90
• 0.abc̅ = (abc − ab) ÷ 900

2 Quick Tricks

1. Cross Multiplication Trick (Comparison के लिए):

अगर compare करना हो कि 4/7 और 5/9 में कौन बड़ा है, तो cross-multiply करें: 4 × 9 = 36 और 7 × 5 = 35। चूंकि 36 > 35, इसलिए 4/7 > 5/9। LCM निकालने का time बच जाएगा!

1. Equal Difference Trick:

अगर Proper Fractions में Numerator और Denominator का difference समान हो, तो जिसका Numerator बड़ा होगा, वह Fraction भी सबसे बड़ा होगा।

उदाहरण: 3/5, 5/7, 7/9 (सब में difference = 2 है)। यहाँ सबसे बड़ा 7/9 और सबसे छोटा 3/5 है।

2 Common Mistakes

1. Mixed Fraction Multiplication में गलती:

2(1/2) × 3(1/2) को सीधे 6(1/4) मत लिखना! हमेशा पहले Improper Fraction में बदलें: (5/2) × (7/2) = 35/4 = 8(3/4)।

1. Cancellation Mistake:

(a + b) ÷ a में 'a' से 'a' cancel नहीं होता! Cancellation सिर्फ Multiplication में ही allowed है, Addition या Subtraction में नहीं।

Fractions: Comprehensive Study Notes

1. Concept

A fraction represents a numerical value that is a part of a whole. It is written in the form p/q, where p is the Numerator (upper part) and q is the Denominator (lower part), and q ≠ 0.

Types of Fractions:

• Proper Fraction: Numerator < Denominator (e.g., 3/5, 7/11). Its value is always less than 1.
• Improper Fraction: Numerator ≥ Denominator (e.g., 8/5, 13/4). Its value is equal to or greater than 1.
• Mixed Fraction: A combination of a whole number and a proper fraction (e.g., 2 3/5 = 2 + 3/5 = 13/5).
• Like Fractions: Fractions having the same denominator (e.g., 2/7, 5/7, 6/7).
• Unlike Fractions: Fractions having different denominators (e.g., 2/3, 4/5, 5/9).
• Equivalent Fractions: Fractions that represent the same value (e.g., 1/2 = 2/4 = 3/6 = 4/8).
• Continued Fraction (Ladder Fraction): A fraction that contains a fraction in its denominator, cascading downwards.

2. Important Formulas, Rules, and Facts

• Addition / Subtraction:
• To add or subtract unlike fractions, find the LCM of the denominators.
• Formula: (a/b) ± (c/d) = [(a × d) ± (b × c)] ÷ (b × d)

• Multiplication:
• (a/b) × (c/d) = (a × c) ÷ (b × d)

• Division:
• To divide, multiply by the reciprocal of the divisor.
• (a/b) ÷ (c/d) = (a/b) × (d/c) = (a × d) ÷ (b × c)

• HCF of Fractions:
• HCF of Fractions = HCF of Numerators ÷ LCM of Denominators

• LCM of Fractions:
• LCM of Fractions = LCM of Numerators ÷ HCF of Denominators

(Always reduce each individual fraction to its lowest terms before calculating HCF or LCM).

• Converting Recurring Decimals to Fractions:
• Pure Recurring Decimal: 0.aaaa... = a/9; 0.abab... = ab/99
• Mixed Recurring Decimal: 0.abbb... = (ab − a) ÷ 90; 0.abcbc... = (abc − a) ÷ 990

• Rules for Comparison:
• If denominators are equal: The fraction with the larger numerator is larger (e.g., 5/7 > 3/7).
• If numerators are equal: The fraction with the smaller denominator is larger (e.g., 5/7 > 5/9).

3. Solved Examples (RRB NTPC Exam Style)

Example 1: Arranging Fractions in Order

Question: Arrange the following fractions in ascending order: 3/5, 7/9, 11/13, 1/2.

Solution:

• Method (Difference Method):
• Check the difference between denominator and numerator (D − N) for each proper fraction:
• For 3/5: 5 − 3 = 2
• For 7/9: 9 − 7 = 2
• For 11/13: 13 − 11 = 2
• When (D − N) is constant, the fraction with the larger numerator is larger.
• Therefore: 3/5 < 7/9 < 11/13
• Now compare 1/2 with 3/5:
• 1/2 = 0.50
• 3/5 = 0.60
• So, 1/2 < 3/5
• Final Order: 1/2 < 3/5 < 7/9 < 11/13.

Example 2: HCF and LCM of Fractions

Question: Find the HCF and LCM of 2/3, 8/9, 16/81, and 10/27.

Solution:

• Step 1: Check lowest terms
• All fractions (2/3, 8/9, 16/81, 10/27) are already in their simplest form.
• Step 2: Find HCF
• Numerators: 2, 8, 16, 10 → HCF(2, 8, 16, 10) = 2
• Denominators: 3, 9, 81, 27 → LCM(3, 9, 81, 27) = 81
• HCF of fractions = HCF(Numerators) ÷ LCM(Denominators) = 2/81
• Step 3: Find LCM
• LCM(Numerators) = LCM(2, 8, 16, 10) = 80
• HCF(Denominators) = HCF(3, 9, 81, 27) = 3
• LCM of fractions = LCM(Numerators) ÷ HCF(Denominators) = 80/3

Example 3: Continued Fraction Simplification

Question: Simplify: 1 + 1 ÷ [1 + 1 ÷ (1 + 1/3)]

Solution:

• Work from the bottom up:
• Step 1: 1 + 1/3 = 4/3
• Step 2: 1 ÷ (4/3) = 3/4
• Step 3: 1 + 3/4 = 7/4
• Step 4: 1 ÷ (7/4) = 4/7
• Step 5: 1 + 4/7 = 11/7
• Answer: 11/7 (or 1 4/7)

Example 4: Word Problem (Remaining Value)

Question: A student spends 1/3 of his pocket money on books, 1/4 of the remaining on snacks, and is left with Rs. 450. What was his total pocket money?

Solution:

• Let total pocket money = 1
• Spent on books = 1/3
• Remaining money = 1 − 1/3 = 2/3
• Spent on snacks = 1/4 of (2/3) = (1/4) × (2/3) = 2/12 = 1/6
• Final remaining money = (2/3) − (1/6) = (4 − 1) ÷ 6 = 3/6 = 1/2
• According to the question:
• 1/2 of total = Rs. 450
• Total pocket money = 450 × 2 = Rs. 900
• Answer: Rs. 900

4. Shortcuts and Tricks

1. Cross-Multiplication for Fast Comparison:

• To compare two fractions a/b and c/d:
• Calculate (a × d) and (b × c).
• If a × d > b × c, then a/b > c/d.
• Example: Compare 4/7 and 5/9 → 4 × 9 = 36; 7 × 5 = 35. Since 36 > 35, 4/7 > 5/9.

1. Same Difference Rule (Proper Fractions):

• For proper fractions (N < D), if (D − N) is the same for all:
• The fraction with the largest numerator is the greatest.
• The fraction with the smallest numerator is the smallest.

1. Unit Fraction Operations:

• Adding two unit fractions: (1/a) + (1/b) = (a + b) ÷ (a × b)
• Subtracting two unit fractions: (1/a) − (1/b) = (b − a) ÷ (a × b)
• Example: (1/4) + (1/5) = (4 + 5) ÷ (4 × 5) = 9/20.

1. Handling 'Remaining' Word Problems with Values:

• Assume total money/work as the product of the denominators to avoid fraction calculations.
• In Example 4, denominators are 3 and 4. Assume total = 12 units.
• Books = 1/3 of 12 = 4 units. Remaining = 8 units.
• Snacks = 1/4 of 8 = 2 units. Remaining = 6 units.
• 6 units = Rs. 450 → 1 unit = 75 → 12 units = 12 × 75 = Rs. 900.

5. Common Mistakes to Avoid

• Direct Addition of Numerators and Denominators:
• Incorrect: (1/2) + (1/3) = (1 + 1) ÷ (2 + 3) = 2/5.
• Correct: (1/2) + (1/3) = (3 + 2) ÷ 6 = 5/6.

• Skipping Reduction Before Finding HCF/LCM:
• Finding HCF of 4/6 and 8/12 directly without simplifying them to 2/3 and 2/3 will yield incorrect answers. Always convert to simplest form first.

• Inverting the Wrong Term in Division:
• (a/b) ÷ (c/d) means inverting only the second fraction (divisor), resulting in (a/b) × (d/c). Do not invert the first fraction.

• Misapplying the 'Same Difference' Rule to Improper Fractions:
• For improper fractions (N > D) where (N − D) is constant, the fraction with the largest numerator is the smallest (e.g., 7/5 = 1.4, while 5/3 = 1.66; so 7/5 < 5/3). The rule reverses for improper fractions.