---
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. ✅