knowledge-base/class_10/ch1/0161_numbers.md

metadata

source: ncert_exercises
topic: numbers
class_level: class_10
chapter: ch1
exercise: ex1.3
difficulty: beginner

Class 10 | Ch1: Real Numbers | Exercise 1.3

Decimal Expansions: p/q terminates iff q = 2ᵐ×5ⁿ (after simplification). If q has other prime factors → non-terminating repeating.

Q1. Without performing division, state whether decimal expansions terminate or not: (i) 13/3125 (ii) 17/8 (iii) 64/455 (iv) 15/1600 (v) 29/343 (vi) 23/(2³×5²) (vii) 129/(2²×5⁷×7⁵) (viii) 6/15 (ix) 35/50 (x) 77/210 Answer: (i) 3125=5⁵. Terminates. ✅ (ii) 8=2³. Terminates. ✅ (iii) 455=5×7×13. Has 7,13 → Non-terminating. ✅ (iv) 1600=2⁶×5². Terminates. ✅ (v) 343=7³. Non-terminating. ✅ (vi) q=2³×5². Terminates. ✅ (vii) Has 7⁵ → Non-terminating. ✅ (viii) 6/15=2/5. q=5. Terminates. ✅ (ix) 35/50=7/10=7/(2×5). Terminates. ✅ (x) 77/210=11/30=11/(2×3×5). Has 3 → Non-terminating. ✅

Q2. Write decimal expansions of the terminating ones above: (i) 13/3125=13/5⁵=13×2⁵/10⁵=416/100000=0.00416. ✅ (ii) 17/8=17×125/1000=2125/1000=2.125. ✅ (iv) 15/1600=15/(2⁶×5²)=15×5⁴/2⁶×5⁶=15×625/10⁶=9375/10⁶=0.009375. ✅ (vi) 23/(2³×5²)=23×5/(2³×5³)=115/1000=0.115. ✅ (viii) 2/5=4/10=0.4. ✅ (ix) 7/10=0.7. ✅