১৫সাধারণ সমস্যা

যোগ কর: ক) 0.45˙+0.13˙4˙0.4\dot{5} + 0.1\dot{3}\dot{4} খ) 2.05˙+8.04˙+7.0182.0\dot{5} + 8.0\dot{4} + 7.018 গ) 0.006˙+0.9˙2˙+0.1˙34˙0.00\dot{6} + 0.\dot{9}\dot{2} + 0.\dot{1}3\dot{4}

সমাধান

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

ক) 0.45˙+0.13˙4˙0.4\dot{5} + 0.1\dot{3}\dot{4}

প্রথম সংখ্যা 0.45˙0.4\dot{5} কে ভগ্নাংশে রূপান্তর করি:

ধরি, x=0.45˙=0.45555x = 0.4\dot{5} = 0.45555\ldots

তাহলে, 10x=4.555510x = 4.5555\ldots এবং 100x=45.5555100x = 45.5555\ldots

এখন, 100x10x=45.55554.5555100x - 10x = 45.5555\ldots - 4.5555\ldots বা, 90x=41    x=419090x = 41 \implies x = \frac{41}{90}

আবার ধরি, y=0.13˙4˙=0.1343434y = 0.1\dot{3}\dot{4} = 0.1343434\ldots

তাহলে, 100y=13.43434100y = 13.43434\ldots এবং 10000y=1343.4343410000y = 1343.43434\ldots

এখন, 10000y100y=1343.4343413.4343410000y - 100y = 1343.43434\ldots - 13.43434\ldots

9900y=1330    y=13309900=1339909900y = 1330 \implies y = \frac{1330}{9900} = \frac{133}{990}

এখন, যোগ করি:

4190+133990=41×11990+133990=451990+133990=584990=292495\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) 2.05˙+8.04˙+7.0182.0\dot{5} + 8.0\dot{4} + 7.018

ধরি, x=2.05˙=2.05555x = 2.0\dot{5} = 2.05555\ldots তাহলে, 10x=20.555510x = 20.5555\ldots এবং 100x=205.5555100x = 205.5555\ldots

এখন,

100x10x=205.555520.5555বা,90x=185    x=18590=3718100x - 10x = 205.5555\ldots - 20.5555\ldots \\ বা, 90x = 185 \implies x = \frac{185}{90} = \frac{37}{18}

এখন ধরি, y=8.04˙=8.04444y = 8.0\dot{4} = 8.04444\ldots তাহলে, 10y=80.444410y = 80.4444\ldots এবং 100y=804.4444100y = 804.4444\ldots

এখন,

100y10y=804.444480.4444বা,90y=724    y=72490=36245100y - 10y = 804.4444\ldots - 80.4444\ldots \\ বা, 90y = 724 \implies y = \frac{724}{90} = \frac{362}{45}

এখন, 7.0187.018 কে ভগ্নাংশে রূপান্তর করি:

7.018=701810007.018 = \frac{7018}{1000}

এখন, যোগ করি:

3718+36245+70181000\frac{37}{18} + \frac{362}{45} + \frac{7018}{1000}

এখানে, LCM(18,45,1000)=9000LCM(18, 45, 1000) = 9000

তাহলে,

37×5009000+362×2009000+7018×99000\frac{37 \times 500}{9000} + \frac{362 \times 200}{9000} + \frac{7018 \times 9}{9000} =185009000+724009000+631629000= \frac{18500}{9000} + \frac{72400}{9000} + \frac{63162}{9000} =1540629000=770314500=256771500= \frac{154062}{9000} = \frac{77031}{4500} = \frac{25677}{1500}

গ) 0.006˙+0.9˙2˙+0.1˙34˙0.00\dot{6} + 0.\dot{9}\dot{2} + 0.\dot{1}3\dot{4}

ধরি, x=0.006˙=0.006666x = 0.00\dot{6} = 0.006666\ldots

তাহলে, 1000x=6.6661000x = 6.666\ldots এবং 10000x=66.66610000x = 66.666\ldots

এখন,

10000x1000x=66.6666.66610000x - 1000x = 66.666\ldots - 6.666\ldots 9000x=60    x=609000=11509000x = 60 \implies x = \frac{60}{9000} = \frac{1}{150}

এখন ধরি, y=0.9˙2˙=0.929292y = 0.\dot{9}\dot{2} = 0.929292\ldots তাহলে, 100y=92.9292100y = 92.9292\ldots এবং 10000y=9292.929210000y = 9292.9292\ldots

এখন,

10000y100y=9292.929292.929210000y - 100y = 9292.9292\ldots - 92.9292\ldots 9900y=9200    y=92009900=920990=4649.5=92999900y = 9200 \implies y = \frac{9200}{9900} = \frac{920}{990} = \frac{46}{49.5} = \frac{92}{99}

আবার ধরি, z=0.1˙34˙=0.1343434z = 0.\dot{1}3\dot{4} = 0.1343434\ldots তাহলে, 100z=13.43434100z = 13.43434\ldots এবং 10000z=1343.4343410000z = 1343.43434\ldots

এখন,

10000z100z=1343.4343413.4343410000z - 100z = 1343.43434\ldots - 13.43434\ldots 9900z=1330    z=13309900=1339909900z = 1330 \implies z = \frac{1330}{9900} = \frac{133}{990}

এখন, যোগ করি:

1150+9299+133990\frac{1}{150} + \frac{92}{99} + \frac{133}{990}

এখানে, LCM(150,99,990)=4950LCM(150, 99, 990) = 4950

তাহলে,

1×334950+92×504950+133×54950\frac{1 \times 33}{4950} + \frac{92 \times 50}{4950} + \frac{133 \times 5}{4950} =334950+46004950+6654950= \frac{33}{4950} + \frac{4600}{4950} + \frac{665}{4950} =52984950=883825= \frac{5298}{4950} = \frac{883}{825}

*LCM = ল. সা. গু.