10. Fractions & Decimals (Terminating / Recurring)

Fractions aur Decimals ka complete concept: types of fractions, decimal conversion, terminating vs recurring decimals, p/q rules, recurring to fraction tricks, SSC CGL examples & practice with answers.

---

1) Fraction kya hota hai?

Fraction ka form hota hai p/q, jahan p = numerator, q = denominator (q ≠ 0). Fraction basically “part of a whole” show karta hai.

2) Types of Fractions (SSC basics)

• Proper fraction: p < q (example 3/7)
• Improper fraction: p ≥ q (example 9/4)
• Mixed number: 2 1/3 = 7/3
• Equivalent fractions: 1/2 = 2/4 = 3/6
• Like fractions: same denominator (3/11, 7/11)
• Unlike fractions: different denominators (2/3, 5/7)

3) Fraction → Decimal conversion

Simple: p ÷ q karo.

• 1/4 = 0.25
• 3/8 = 0.375
• 1/3 = 0.333... (recurring)

4) Terminating vs Recurring decimal

• Terminating: decimal end ho jata hai (0.25, 2.375)
• Recurring/Repetitive: digits repeat hote hain (0.333..., 1.2727...)
• Non-terminating + Non-recurring: irrational (√2, π) — (yaha reminder)

5) The most important rule (p/q test)

Fraction p/q ko lowest form me karo. Agar denominator q ka prime factorization sirf 2 aur 5 me ho: q = 2^a × 5^b, tab decimal terminating hoga. Otherwise decimal recurring hoga.

Examples

• 7/40 (lowest) → 40 = 2³×5 ⇒ terminating
• 3/12 = 1/4 → 4 = 2² ⇒ terminating
• 7/6 → 6 = 2×3 (3 present) ⇒ recurring
• 5/14 → 14 = 2×7 (7 present) ⇒ recurring

6) Decimal → Fraction (very important)

A) Terminating decimal to fraction

Digits ko numerator banao, point ke digits count ke according denominator 10, 100, 1000... phir simplify.

• 0.75 = 75/100 = 3/4
• 2.125 = 2125/1000 = 17/8

B) Pure recurring decimal to fraction (repeat starts immediately)

Trick: If x = 0.\u0305a (1 digit repeat) ⇒ x = a/9 If x = 0.\u0305ab (2 digits repeat) ⇒ x = ab/99 If x = 0.\u0305abc (3 digits repeat) ⇒ x = abc/999

• 0.\u03053 = 3/9 = 1/3
• 0.\u030527 = 27/99 = 3/11
• 0.\u0305125 = 125/999

C) Mixed recurring decimal (repeat after some non-repeating digits)

Standard method (always works): Let x = number. Multiply by 10^(non-repeat digits) and 10^(total digits till one repeat block), subtract.

Example: x = 0.1\u03056 (i.e., 0.1666...)

• x = 0.1666...
• 10x = 1.666...
• 100x = 16.666...
• (100x − 10x) = 15
• 90x = 15 ⇒ x = 15/90 = 1/6

7) Common traps

• Before applying q=2^a×5^b rule, fraction ko lowest form me karo.
• 2.5 aur 2.50 same hote hain (trailing zeros allowed).
• Recurring decimal always rational hota hai.

---

8) Practice (SSC CGL) + Answers

1. Decide: 7/20 terminating hai ya recurring?
2. Convert to decimal: 3/8
3. Convert to fraction: 0.045
4. Convert to fraction: 0.\u030536
5. Convert to fraction: 1.2\u03053 (1.2333...)
6. Which is bigger: 5/6 or 0.82?

Show Answers

1. 20=2²×5 ⇒ terminating
2. 3/8 = 0.375
3. 0.045 = 45/1000 = 9/200
4. 0.\u030536 = 36/99 = 4/11
5. x=1.2333...; 10x=12.333...; 100x=123.333...; 100x-10x=111 ⇒ 90x=111 ⇒ x=111/90=37/30
6. 5/6 = 0.8333... > 0.82