প্রথমে চলক সংখ্যা সমূহকে ভগ্নাংশে রূপান্তরিত করতে হবে এবং তারপর যোগ করতে হবে।

ক) $0.4\dot{5} + 0.1\dot{3}\dot{4}$

প্রথম সংখ্যা $0.4\dot{5}$ কে ভগ্নাংশে রূপান্তর করি:

ধরি, $x = 0.4\dot{5} = 0.45555\ldots$

তাহলে, $10x = 4.5555\ldots$ এবং $100x = 45.5555\ldots$

এখন, $100x - 10x = 45.5555\ldots - 4.5555\ldots$ বা, $90x = 41 \implies x = \frac{41}{90}$

আবার ধরি, $y = 0.1\dot{3}\dot{4} = 0.1343434\ldots$

তাহলে, $100y = 13.43434\ldots$ এবং $10000y = 1343.43434\ldots$

এখন, $10000y - 100y = 1343.43434\ldots - 13.43434\ldots$

$$9900y = 1330 \implies y = \frac{1330}{9900} = \frac{133}{990}$$

এখন, যোগ করি:

$$\frac{41}{90} + \frac{133}{990} = \frac{41 \times 11}{990} + \frac{133}{990}\\ = \frac{451}{990} + \frac{133}{990} = \frac{584}{990} = \frac{292}{495}$$

খ) $2.0\dot{5} + 8.0\dot{4} + 7.018$

ধরি, $x = 2.0\dot{5} = 2.05555\ldots$ তাহলে, $10x = 20.5555\ldots$ এবং $100x = 205.5555\ldots$

এখন,

$$100x - 10x = 205.5555\ldots - 20.5555\ldots \\ বা, 90x = 185 \implies x = \frac{185}{90} = \frac{37}{18}$$

এখন ধরি, $y = 8.0\dot{4} = 8.04444\ldots$ তাহলে, $10y = 80.4444\ldots$ এবং $100y = 804.4444\ldots$

এখন,

$$100y - 10y = 804.4444\ldots - 80.4444\ldots \\ বা, 90y = 724 \implies y = \frac{724}{90} = \frac{362}{45}$$

এখন, $7.018$ কে ভগ্নাংশে রূপান্তর করি:

$$7.018 = \frac{7018}{1000}$$

এখন, যোগ করি:

$$\frac{37}{18} + \frac{362}{45} + \frac{7018}{1000}$$

এখানে, $LCM(18, 45, 1000) = 9000$

তাহলে,

$$\frac{37 \times 500}{9000} + \frac{362 \times 200}{9000} + \frac{7018 \times 9}{9000}$$ $$= \frac{18500}{9000} + \frac{72400}{9000} + \frac{63162}{9000}$$ $$= \frac{154062}{9000} = \frac{77031}{4500} = \frac{25677}{1500}$$

গ) $0.00\dot{6} + 0.\dot{9}\dot{2} + 0.\dot{1}3\dot{4}$

ধরি, $x = 0.00\dot{6} = 0.006666\ldots$

তাহলে, $1000x = 6.666\ldots$ এবং $10000x = 66.666\ldots$

এখন,

$$10000x - 1000x = 66.666\ldots - 6.666\ldots$$ $$9000x = 60 \implies x = \frac{60}{9000} = \frac{1}{150}$$

এখন ধরি, $y = 0.\dot{9}\dot{2} = 0.929292\ldots$ তাহলে, $100y = 92.9292\ldots$ এবং $10000y = 9292.9292\ldots$

এখন,

$$10000y - 100y = 9292.9292\ldots - 92.9292\ldots$$ $$9900y = 9200 \implies y = \frac{9200}{9900} = \frac{920}{990} = \frac{46}{49.5} = \frac{92}{99}$$

আবার ধরি, $z = 0.\dot{1}3\dot{4} = 0.1343434\ldots$ তাহলে, $100z = 13.43434\ldots$ এবং $10000z = 1343.43434\ldots$

এখন,

$$10000z - 100z = 1343.43434\ldots - 13.43434\ldots$$ $$9900z = 1330 \implies z = \frac{1330}{9900} = \frac{133}{990}$$

এখন, যোগ করি:

$$\frac{1}{150} + \frac{92}{99} + \frac{133}{990}$$

এখানে, $LCM(150, 99, 990) = 4950$

তাহলে,

$$\frac{1 \times 33}{4950} + \frac{92 \times 50}{4950} + \frac{133 \times 5}{4950}$$ $$= \frac{33}{4950} + \frac{4600}{4950} + \frac{665}{4950}$$ $$= \frac{5298}{4950} = \frac{883}{825}$$

*LCM = ল. সা. গু.