অনুশীলনী-৬(ক)

১. অপ্রকৃত ভগ্নাংশে প্রকাশ করো:

(১) 2\frac{2}{3} (২)  3\frac{1}{9} (৩)  5\frac{5}{{11}}

(৪) 6\frac{3}{{10}} (৫) 20\frac{1}{2}

সমাধান:

(১) 2\frac{2}{3}

= \frac{{3 \times 2 + 2}}{3}

= \frac{{6 + 2}}{3}

= \frac{8}{3}

উত্তর: \frac{8}{3}

(২)  3\frac{1}{9}

= \frac{{9 \times 3 + 1}}{9}

= \frac{{27 + 1}}{9}

= \frac{{28}}{9}

উত্তর: \frac{{28}}{9}

(৩)  5\frac{5}{{11}}

= \frac{{11 \times 5 + 5}}{{11}}

= \frac{{55 + 5}}{{11}}

=\frac{{60}}{{11}}

উত্তর:  \frac{{60}}{{11}}                                                                                                                             

(৪) 6\frac{3}{{10}}

= \frac{{10 \times 6 + 3}}{{10}}

= \frac{{60 + 3}}{{10}}

= \frac{{63}}{{10}}

উত্তর: \frac{{63}}{{10}}

(৫) 20\frac{1}{2}

= \frac{{2 \times 20 + 1}}{2}

= \frac{{40 + 1}}{2}

= \frac{{41}}{2}

উত্তর: \frac{{41}}{2}

২. মিশ্র ভগ্নাংশ বা পূর্ণ সংখ্যায় প্রকাশ করো:

(১) \frac{7}{3} (২) \frac{{21}}{5} (৩) \frac{{36}}{9}

(৪) \frac{{78}}{{11}} (৫) \frac{{220}}{{10}}

সমাধান:

(১) \frac{7}{3}

এখন, 3)7(2
        6
——————–
1

\therefore \frac{7}{3} = 2\frac{1}{3}; যা মিশ্র ভগ্নাংশ

উত্তর: 2\frac{1}{3}


৩.  হিসাব করো:

(১) \frac{4}{6} + \frac{3}{6}             (২) 1\frac{2}{3} + 2\frac{2}{3}

(৩) \frac{5}{6} + \frac{3}{6}             (৪) \frac{3}{2} + \frac{1}{4}

(৫) \frac{1}{3} + \frac{4}{5}             (৬) 1\frac{1}{3} + \frac{1}{6}

(৭) \frac{4}{{15}} + 1\frac{1}{{12}}     (৮) 1\frac{7}{{15}} + \frac{3}{5}

(৯) \frac{8}{7} - \frac{5}{7}             (১০) 1\frac{2}{5} - \frac{4}{5}

(১১) 3 - \frac{3}{4}   (১২) \frac{7}{6} - \frac{1}{4}

(১৩) 3\frac{2}{3} - \frac{4}{5}           (১৪) 2\frac{1}{3} - \frac{8}{{15}}

(১৫) 3\frac{7}{{12}} - 1\frac{5}{6}

সমাধান:

(১) \frac{4}{6} + \frac{3}{6}

= \frac{{4 + 3}}{6}

= \frac{7}{6}

= 1\frac{1}{6}

উত্তর: 1\frac{1}{6}    

(২) 1\frac{2}{3} + 2\frac{2}{3}      

= \frac{5}{3} + \frac{8}{3}

= \frac{{5 + 8}}{3}

= \frac{{13}}{3}

=4\frac{1}{3}

উত্তর: 4\frac{1}{3}

(৩) \frac{5}{6} + \frac{3}{6}

= \frac{{5 + 3}}{6}

= \frac{8}{6}

= 1\frac{1}{3}

উত্তর: 1\frac{1}{3}   

(৪) \frac{3}{2} + \frac{1}{4}

এখানে, ভগ্নাংশগুলোর হর 2,4

এদের ল.সা.গু =4

এখন, 4 \div 2 = 2

\therefore \frac{3}{2} = \frac{{3 \times 2}}{{2 \times 2}} = \frac{6}{4}

4 \div 4 = 1

\therefore \frac{1}{4} = \frac{{1 \times 1}}{{4 \times 1}} = \frac{1}{4}

\therefore \frac{3}{2} + \frac{1}{4}

= \frac{6}{4} + \frac{1}{4}

= \frac{{6 + 1}}{4}

= \frac{7}{4}

= 1\frac{3}{4}

উত্তর: 1\frac{3}{4}

(৫) \frac{1}{3} + \frac{4}{5}

এখানে, ভগ্নাংশগুলোর হর 3,5

এদের ল.সা.গু = 3 \times 5 = 15

এখন, 15 \div 3 = 5

\therefore \frac{1}{3} = \frac{{1 \times 5}}{{3 \times 5}} = \frac{5}{{15}}

15 \div 5 = 3

\therefore \frac{4}{5} = \frac{{4 \times 3}}{{5 \times 3}} = \frac{{12}}{{15}}

\therefore \frac{1}{3} + \frac{4}{5}

= \frac{5}{{15}} + \frac{{12}}{{15}}

= \frac{{5 + 12}}{{15}}

= \frac{{17}}{{15}}

= 1\frac{2}{{15}}

উত্তর: 1\frac{2}{{15}}          

(৬) 1\frac{1}{3} + \frac{1}{6}

= \frac{{3 \times 1 + 1}}{3} + \frac{1}{6}

= \frac{4}{3} + \frac{1}{6}

এখানে, ভগ্নাংশগুলোর হর 3,6

এদের ল.সা.গু = 6

এখন, 6 \div 3 = 2

\therefore \frac{4}{3} = \frac{{4 \times 2}}{{3 \times 2}} = \frac{8}{6}

6 \div 6 = 1

60 \div 15 = 4

\therefore \frac{4}{{15}} = \frac{{4 \times 4}}{{15 \times 4}} = \frac{{16}}{{60}}

60 \div 12 = 5

\therefore \frac{{13}}{{12}} = \frac{{13 \times 5}}{{12 \times 5}} = \frac{{65}}{{60}}

\therefore \frac{4}{{15}} + 1\frac{1}{{12}}

= \frac{{16}}{{60}} + \frac{{65}}{{60}}

= \frac{{15 + 65}}{{60}}

= \frac{{81}}{{60}}

= \frac{{27}}{{20}}

= 1\frac{7}{{20}}

উত্তর: 1\frac{7}{{20}}

(৭) \frac{4}{{15}} + 1\frac{1}{{12}}

=\frac{4}{{15}} + \frac{{12 \times 1 + 1}}{{12}}

= \frac{4}{{15}} + \frac{{13}}{{12}}

এখানে, ভগ্নাংশগুলোর হর 12,15

\therefore ল.সা.গু = 60

এখন, 60 \div 15 = 4

\therefore \frac{4}{{15}} = \frac{{4 \times 4}}{{15 \times 4}} = \frac{{16}}{{60}}

60 \div 12 = 5

\therefore \frac{{13}}{{12}} = \frac{{13 \times 5}}{{12 \times 5}} = \frac{{65}}{{60}}

\therefore \frac{4}{{15}} + 1\frac{1}{{12}}

= \frac{{16}}{{60}} + \frac{{65}}{{60}}

= \frac{{16 + 65}}{{60}}

= \frac{{81}}{{60}}

= \frac{{27}}{{20}}

= 1\frac{7}{{20}}

উত্তর: 1\frac{7}{{20}}  

(৮) 1\frac{7}{{15}} + \frac{3}{5}

= \frac{{15 \times 1 + 7}}{{15}} + \frac{3}{5}

= \frac{{22}}{{15}} + \frac{3}{5}

এখানে, ভগ্নাংশগুলোর হর 5,15

\therefore ল.সা.গু = 15

এখন, 15 \div 15 = 1

\therefore \frac{{22}}{{15}} = \frac{{22 \times 1}}{{15 \times 1}} = \frac{{22}}{{15}}

15 \div 5 = 3

\therefore \frac{3}{5} = \frac{{3 \times 3}}{{5 \times 3}} = \frac{9}{{15}}

\therefore 1\frac{7}{{15}} + \frac{3}{5}

= \frac{{22}}{{15}} + \frac{9}{{15}}

= \frac{{22 + 9}}{{15}}

= \frac{{31}}{{15}}

= 2\frac{1}{{15}}

উত্তর: 2\frac{1}{{15}}

(৯) \frac{8}{7} - \frac{5}{7}

= \frac{{8 - 5}}{7}

= \frac{3}{7}

(১০) 1\frac{2}{5} - \frac{4}{5}

= \frac{{5 \times 1 + 2}}{5} - \frac{4}{5}

= \frac{7}{5} - \frac{4}{5}

= \frac{{7 - 4}}{5}

=\frac{3}{5}

উত্তর: \frac{3}{5}

(১১) 3 - \frac{3}{4}   

=\frac{3}{1} - \frac{3}{4}

এখানে, ভগ্নাংশগুলোর হর 1,4

\therefore ল.সা.গু = 4

এখন, 4 \div 1 = 4

\therefore \frac{3}{1} = \frac{{3 \times 4}}{{1 \times 4}} = \frac{12}{4}

4 \div 4 = 1

\therefore \frac{3}{4} = \frac{{3 \times 1}}{{4 \times 1}} = \frac{3}{4}

\therefore 3 - \frac{3}{4}

= \frac{{12}}{4} - \frac{3}{4}

= \frac{{12 - 3}}{4}

= \frac{9}{4}

= 2\frac{1}{4}

উত্তর: 2\frac{1}{4}                 

(১২) \frac{7}{6} - \frac{1}{4}

এখানে, ভগ্নাংশগুলোর হর 4,6

\therefore ল.সা.গু = 12

এখন, 12 \div 6 = 2

\therefore \frac{7}{6} = \frac{{7 \times 2}}{{6 \times 2}} = \frac{{14}}{{12}}

12 \div 4 = 3

\therefore \frac{1}{4} = \frac{{1 \times 3}}{{4 \times 3}} = \frac{3}{{12}}

\therefore \frac{7}{6} - \frac{1}{4}

= \frac{{14}}{{12}} - \frac{3}{{12}}

= \frac{{14 - 3}}{{12}}

= \frac{{11}}{{12}}

উত্তর: \frac{{11}}{{12}}

(১৩) 3\frac{2}{3} - \frac{4}{5}

= \frac{{3 \times 2 + 2}}{3} - \frac{4}{5}

= \frac{8}{3} - \frac{4}{5}

এখানে, ভগ্নাংশগুলোর হর 3,5

\therefore ল.সা.গু = 15

এখন, 15 \div 3 = 5

\therefore \frac{8}{3} = \frac{{8 \times 5}}{{3 \times 5}} = \frac{{40}}{{15}}

15 \div 5 = 3

\therefore \frac{4}{5} = \frac{{4 \times 3}}{{5 \times 3}} = \frac{{12}}{{15}}

\therefore 2\frac{2}{3} - \frac{4}{5}

= \frac{8}{3} - \frac{4}{5}

= \frac{{40}}{{15}} - \frac{{12}}{{15}}

= \frac{{40 - 12}}{{15}}

= \frac{{28}}{{15}}

= 1\frac{{13}}{{15}}

উত্তর: 1\frac{{13}}{{15}}

১৪) 2\frac{1}{3} - \frac{8}{{15}}

= \frac{{3 \times 2 + 1}}{3} - \frac{8}{{15}}

= \frac{7}{3} - \frac{8}{{15}}

এখানে, ভগ্নাংশগুলোর হর 3,15

\therefore ল.সা.গু = 15

এখন, 15 \div 3 = 5

\therefore \frac{7}{3} = \frac{{7 \times 5}}{{3 \times 5}} = \frac{{35}}{{15}}

15 \div 15 = 1

\therefore \frac{8}{{15}} = \frac{{8 \times 1}}{{15 \times 1}} = \frac{8}{{15}}

\therefore 2\frac{1}{3} - \frac{8}{{15}}

= \frac{7}{3} - \frac{8}{{15}}

= \frac{{35}}{{15}} - \frac{8}{{15}}

= \frac{{35 - 8}}{{15}}

= \frac{{27}}{{15}}

= \frac{9}{5}

= 1\frac{4}{5}

উত্তর: 1\frac{4}{5}

(১৫) 3\frac{7}{{12}} - 1\frac{5}{6}

= \frac{{12 \times 3 + 7}}{{12}} - \frac{{6 \times 1 + 5}}{6}

= \frac{{43}}{{12}} - \frac{{11}}{6}

 এখানে, ভগ্নাংশগুলোর হর 6,12

\therefore ল.সা.গু = 12

এখন, 12 \div 12 = 1

\therefore \frac{{43}}{{12}} = \frac{{43 \times 1}}{{12 \times 1}} = \frac{{43}}{{12}}

12 \div 6 = 2

\therefore \frac{{11}}{6} = \frac{{11 \times 2}}{{6 \times 2}} = \frac{{22}}{{12}}

\therefore 3\frac{7}{{12}} - 1\frac{5}{6}

= \frac{{43}}{{12}} - \frac{{11}}{6}

= \frac{{43}}{{12}} - \frac{{22}}{{12}}

= \frac{{43 - 22}}{{12}}

= \frac{{21}}{{12}}

= \frac{7}{4}

= 1\frac{3}{4}

উত্তর: 1\frac{3}{4}

৪. হিসাব করো:

(১) \frac{1}{7} + \frac{3}{7} + \frac{5}{7}           (২) \frac{1}{{18}} + \frac{2}{9} + \frac{5}{6}

(৩) 2\frac{2}{3} + 1\frac{1}{4} + 1\frac{5}{6}      (৪) \frac{{20}}{{11}} - \frac{7}{{11}} - \frac{8}{{11}}

(৫) \frac{5}{2} - \frac{1}{3} - \frac{5}{6}          (৬) 5\frac{1}{{15}} + 1\frac{3}{5} + 2\frac{2}{3}

(৭) \frac{7}{{13}} - \frac{6}{{13}} + \frac{5}{{13}}  (৮) \frac{3}{{14}} + \frac{7}{8} - \frac{{11}}{{12}}

(৯) 1\frac{1}{3} + 3\frac{1}{4} - 2\frac{5}{6} - \frac{3}{4}

সমাধান:

(১) \frac{1}{7} + \frac{3}{7} + \frac{5}{7}  

= \frac{{1 + 3 + 5}}{7}

= \frac{9}{7}

= 1\frac{2}{7}

উত্তর: 1\frac{2}{7}        

(২) \frac{1}{{18}} + \frac{2}{9} + \frac{5}{6}

এখানে, ভগ্নাংশগুলোর হর 6,9,18

\therefore ল.সা.গু = 54

এখন, 54 \div 18 = 3

\therefore \frac{1}{{18}} = \frac{{1 \times 3}}{{18 \times 3}} = \frac{3}{{54}}

54 \div 9 = 6

\therefore \frac{2}{9} = \frac{{2 \times 6}}{{9 \times 6}} = \frac{{12}}{{54}}

54 \div 6 = 9

\therefore \frac{5}{6} = \frac{{5 \times 9}}{{6 \times 9}} = \frac{{45}}{{54}}

\therefore \frac{1}{{18}} + \frac{2}{9} + \frac{5}{6}

= \frac{3}{{54}} + \frac{{12}}{{54}} + \frac{{45}}{{54}}

= \frac{{3 + 12 + 45}}{{54}}

= \frac{{60}}{{54}}

= \frac{{10}}{9}

= 1\frac{1}{9}

উত্তর: 1\frac{1}{9}

(৩) 2\frac{2}{3} + 1\frac{1}{4} + 1\frac{5}{6}

 = \frac{{3 \times 2 + 2}}{3} + \frac{{4 \times 1 + 1}}{4} + \frac{{6 \times 1 + 5}}{6}

= \frac{8}{3} + \frac{5}{4} + \frac{{11}}{6}

এখানে, ভগ্নাংশগুলোর হর 3,4,6

\therefore ল.সা.গু = 12

এখন, 12 \div 3 = 4

\therefore \frac{8}{3} = \frac{{8 \times 4}}{{3 \times 4}} = \frac{{32}}{{12}}

12 \div 4 = 3

\therefore \frac{5}{4} = \frac{{5 \times 3}}{{4 \times 3}} = \frac{{15}}{{12}}

12 \div 6 = 2

\therefore \frac{{11}}{6} = \frac{{11 \times 2}}{{6 \times 2}} = \frac{{22}}{{12}}

\therefore 2\frac{2}{3} + 1\frac{1}{4} + 1\frac{5}{6}

= \frac{8}{3} + \frac{5}{4} + \frac{{11}}{6}

= \frac{{32}}{{12}} + \frac{{15}}{{12}} + \frac{{22}}{{12}}

= \frac{{32 + 15 + 22}}{{12}}

= \frac{{69}}{{12}}

= \frac{{23}}{4}

= 5\frac{3}{4}

উত্তর: 5\frac{3}{4}

(৪) \frac{{20}}{{11}} - \frac{7}{{11}} - \frac{8}{{11}}

= \frac{{20 - 7 - 8}}{{11}}

= \frac{{20 - 15}}{{11}}

= \frac{5}{{11}}

উত্তর: \frac{5}{{11}}

(৫) \frac{5}{2} - \frac{1}{3} - \frac{5}{6}

এখানে, ভগ্নাংশগুলোর হর 2,3,6

\therefore ল.সা.গু = 6

এখন, 6 \div 2 = 3

\therefore \frac{5}{2} = \frac{{5 \times 3}}{{2 \times 3}} = \frac{{15}}{6}

6 \div 3 = 2

\therefore \frac{1}{3} = \frac{{1 \times 2}}{{3 \times 2}} = \frac{2}{6}

6 \div 6 = 1

\therefore \frac{5}{6} = \frac{{5 \times 1}}{{6 \times 1}} = \frac{5}{6}

\therefore \frac{5}{2} - \frac{1}{3} - \frac{5}{6}

= \frac{8}{3} + \frac{5}{4} + \frac{{11}}{6}

= \frac{{15}}{6} + \frac{2}{6} + \frac{5}{6}

= \frac{{15 + 2 + 5}}{6}

= \frac{{15 - 7}}{6}

= \frac{8}{6} = \frac{4}{3}

= 1\frac{1}{3}

উত্তর: 1\frac{1}{3}       

(৬) 5\frac{1}{{15}} + 1\frac{3}{5} + 2\frac{2}{3}

= \frac{{76}}{{15}} - \frac{8}{5} - \frac{8}{3}

= \frac{{76}}{{15}} - \frac{{8 \times 3}}{{5 \times 3}} - \frac{{8 \times 5}}{{3 \times 5}}

= \frac{{76}}{{15}} - \frac{{24}}{{15}} - \frac{{40}}{{15}}

= \frac{{76 - 24 - 40}}{{15}}

= \frac{{76 - 64}}{{15}}

= \frac{{12}}{{15}}

= \frac{4}{5}

উত্তর: \frac{4}{5}

(৭) \frac{7}{{13}} - \frac{6}{{13}} + \frac{5}{{13}}

= \frac{{7 - 6 + 5}}{{13}}

= \frac{{1 + 5}}{{13}}

= \frac{6}{{13}}

= \frac{4}{5}

উত্তর: \frac{4}{5}

(৮) \frac{3}{{14}} + \frac{7}{8} - \frac{{11}}{{12}}

= \frac{{3 \times 6}}{{4 \times 6}} + \frac{{7 \times 3}}{{8 \times 3}} - \frac{{11 \times 2}}{{12 \times 2}}

= \frac{{18}}{{24}} + \frac{{21}}{{24}} - \frac{{22}}{{24}}

= \frac{{18 + 21 - 22}}{{24}}

= \frac{{39 - 22}}{{24}}

= \frac{{17}}{{24}}

উত্তর: \frac{{17}}{{24}}

(৯) 1\frac{1}{3} + 3\frac{1}{4} - 2\frac{5}{6} - \frac{3}{4}

= \frac{4}{3} + \frac{{13}}{4} - \frac{{17}}{6} - \frac{3}{4}

= \frac{{4 \times 4}}{{3 \times 4}} + \frac{{13 \times 3}}{{4 \times 3}} - \frac{{17 \times 2}}{{6 \times 2}} - \frac{{3 \times 3}}{{4 \times 3}}

= \frac{{16}}{{12}} + \frac{{39}}{{12}} - \frac{{34}}{{12}} - \frac{9}{{12}}

= \frac{{16 + 39 - 34 - 9}}{{12}}

= \frac{{55 - 43}}{{12}}

= \frac{{12}}{{12}}

= 1

উত্তর: 1

৫. 3\frac{3}{4} মি. ও 2\frac{1}{3} মি. দৈর্ঘ্যের দুইটি ফিতা একত্রে কত মিটার ?

সমাধান:

3\frac{3}{4} + 2\frac{1}{3}

= \frac{{4 \times 3 + 3}}{4} + \frac{{3 \times 2 + 1}}{3}

= \frac{{12 + 3}}{4} + \frac{{6 + 1}}{3}

= \frac{{15}}{4} + \frac{7}{3}

এখানে, ভগ্নাংশগুলোর হর 3,4

\therefore ল.সা.গু = 12

এখন, 12 \div 4 = 3

\therefore \frac{{15}}{4} = \frac{{15 \times 3}}{{4 \times 3}} = \frac{{45}}{{12}}

12 \div 3 = 4

\therefore \frac{7}{3} = \frac{{7 \times 4}}{{3 \times 4}} = \frac{{28}}{{12}}

\therefore 3\frac{3}{4} + 2\frac{1}{3}

= \frac{{15}}{4} + \frac{7}{3}

= \frac{{45}}{{12}} + \frac{{28}}{{12}}

= \frac{{45 + 28}}{{12}}

= \frac{{73}}{{12}}

= 6\frac{1}{{12}}

\therefore দুইটি ফিতা একত্রে 6\frac{1}{{12}} মিটার।

উত্তর: 6\frac{1}{{12}}

৬. গিতার কাছে 1\frac{5}{6} লিটার ও মানুনের কাছে \frac{{13}}{8} লিটার জুস আছে। কার জুসের পরিমান বেশি এবং কত বেশি ?

সমাধান:

গিতার আছে 1\frac{5}{6} লিটার = \frac{{11}}{6} লিটার জুস

মানুনের আছে \frac{{13}}{8} লিটার জুস

এখন, ভগ্নাংশগুলোর হর 6,8

\therefore ল.সা.গু= 24

ভগ্নাংশ দুটি সমহর বিশিষ্ট ভগ্নাংশে প্রবেশ করি।

এখন, 24 \div 6 = 4

\therefore \frac{{11}}{6} = \frac{{11 \times 4}}{{6 \times 4}} = \frac{{44}}{{24}}

24 \div 8 = 3

\therefore \frac{{13}}{8} = \frac{{13 \times 3}}{{8 \times 3}} = \frac{{39}}{{24}}

এখানে, 44>39   [একই হর বিশিষ্ট দুটি ভগ্নাংশের মধ্যে যে ভগন্ংশটির লব বড় সে ভগ্নাংশটি বড় হয়]   

\Rightarrow \frac{{44}}{{24}} > \frac{{39}}{{24}}

\therefore \frac{{11}}{6} > \frac{{13}}{8}

সুতরাং গিতার জুসের পরিমান বেশি।

এখন, মামুনের অপেক্ষা গিতার জুস বেশি,

\frac{{11}}{6} - \frac{{13}}{8}  

= \frac{{44}}{{24}} - \frac{{39}}{{24}}

= \frac{{44 - 39}}{{24}}

= \frac{5}{{24}}

\therefore গিতার জুসের পরিমাণ মামুনের চেয়ে \frac{5}{{24}} লিটার বেশি।

উত্তর: \frac{5}{{24}} লিটার।                     

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