Decimals & Fractions — Concept

1) Decimals: Definition & Place Value

दशमलव संख्या \(a.bcd\ldots\) लाई whole भाग \(a\) र fractional भागबाट बनाइन्छ। उदाहरण: \( 3.45 = 3 + \frac{45}{100} \).

स्थान-मूल्य: दायाँतर्फका अङ्कहरू tenths \((10^{-1})\), hundredths \((10^{-2})\), thousandths \((10^{-3})\) …

\[ 45.327 = 45 + \frac{3}{10} + \frac{2}{100} + \frac{7}{1000}. \]

2) Decimals \(\leftrightarrow\) Fractions

Terminating (finite): \(0.125 = \frac{125}{1000} = \frac{1}{8}\).
Repeating: यदि \(x = 0.\overline{3}\) भने \(10x = 3.\overline{3}\Rightarrow 9x=3\Rightarrow x=\frac{1}{3}\).

\[ \text{उदाहरण: } 0.7\overline{2}\ \Rightarrow\ x=0.7222\ldots,\ 100x=72.222\ldots,\ 10x=7.222\ldots \Rightarrow 90x=65 \Rightarrow x=\frac{13}{18}. \]

3) Operations on Decimals

Add/Subtract: दशमलव बिन्दु align गरेर जोड/घटाउ।
Multiply: पहिला integer जस्तै गुणा; अन्त्यमा दुवै संख्याका दशमलव अङ्कहरूको कुल बराबर स्थान सारेर बिन्दु राख्ने।
Divide: विभाजकलाई पूर्ण बनाउने गरी दुवैलाई समान स्थान shift गर्ने।

\[ 2.4\times 1.25 = \frac{240\times 125}{10^4} = 3.00. \]

4) Rounding & Ranges

नजिकको hundredth मा round गर्दा सीमा \([k.005,\ k.015)\) प्रकारका हुन्छन्। उदाहरण: \(10.45\) पाउन वास्तविक मान \([10.445,\ 10.455)\).

नियम: “अर्को अंक \(\ge 5\)” भए बढाउने, नभए उही राख्ने।

5) Fractions: Definition & Forms

भिन्न \(\frac{p}{q}\) जहाँ \(q\ne 0\). Proper \((p

\[ A\tfrac{r}{q} = \frac{Aq+r}{q}, \qquad \text{Simplest form: } \gcd(p,q)=1. \]

6) Comparing Fractions

Cross-multiply: \(\frac{a}{b}\) vs \(\frac{c}{d}\) (\(b,d>0\)) ⇒ \(ad\) र \(bc\) तुलना।
वा common denominator बनाएर तुलना।

\[ \frac{5}{12} \text{ vs } \frac{3}{7}:\ 5\cdot 7=35,\ 3\cdot 12=36 \Rightarrow \frac{5}{12} < \frac{3}{7}. \]

7) Operations on Fractions

\[ \frac{a}{b}\pm \frac{c}{d} = \frac{ad \pm bc}{bd},\quad \frac{a}{b}\cdot \frac{c}{d} = \frac{ac}{bd},\quad \frac{a}{b}\div \frac{c}{d} = \frac{a}{b}\cdot \frac{d}{c}. \]

\[ \frac{7}{12}+\frac{5}{18} = \frac{21+10}{36} = \frac{31}{36}\ (\text{simplest}). \]

8) Decimals terminate कहिले?

\(\tfrac{p}{q}\) simplest form मा हुँदा decimal terminate हुन्छ त्यतिखेर मात्र जब \(q\) का prime factors \(2\) र \(5\) मात्र हुन्छन्; अन्यथा repeating हुन्छ।

\[ \frac{1}{20}=0.05\ (\because 20=2^2\cdot 5), \qquad \frac{1}{6}=0.1\overline{6}\ (\because 6=2\cdot 3). \]

9) Worked Examples

(a) Decimal \(\to\) Fraction (finite): \(0.375=\frac{375}{1000}=\frac{3}{8}\).

(b) Decimal \(\to\) Fraction (repeating): \(x=0.\overline{12}\Rightarrow 100x=12.\overline{12}\Rightarrow 99x=12\Rightarrow x=\frac{12}{99}=\frac{4}{33}\).

(c) Fraction \(\to\) Decimal: \(\frac{7}{16}\Rightarrow 0.4375\) (terminate; \(16=2^4\)).

(d) Compare mixed forms: \(1.4\) vs \(\frac{7}{5}\). Since \(\frac{7}{5}=1.4\Rightarrow\) equal.

10) Repeating Block & Position

Repeating block को लम्बाइ \(k\) भए, दायाँतर्फको \(n\)-औँ अंक: \(r=n\bmod k\) (शून्य शेष ⇒ ब्लकको अन्तिम अंक)।

\[ 0.\overline{123} \text{ मा } k=3,\quad 50\bmod 3 = 2 \Rightarrow 50\text{-औँ अंक }=2. \]

11) Common Pitfalls & Tips

Decimal multiplication पछि दशमलव अङ्कको कुल संख्या राख्न बिर्सिनु।
Rounding boundary नबुझी गलत निर्णय (जस्तै \([10.445,10.455)\)).
Repeating block को लम्बाइ छुट्टयाउनुअघि position निकाल्दा गल्ती।
Fractions add गर्दा simplify/LCM नसोची ठूलो संख्यामा फसिनु—पहिले cross-cancel सोच्नु।

Quick Reference

\[ \tfrac{1}{2}=0.5,\quad \tfrac{1}{4}=0.25,\quad \tfrac{3}{4}=0.75 \] \[ \tfrac{1}{5}=0.2,\ \tfrac{2}{5}=0.4,\ \tfrac{3}{5}=0.6 \] \[ \tfrac{1}{8}=0.125,\ \tfrac{3}{8}=0.375,\ \tfrac{5}{8}=0.625,\ \tfrac{7}{8}=0.875 \] \[ \tfrac{1}{3}=0.\overline{3},\quad \tfrac{2}{3}=0.\overline{6} \]