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4.9: Kuongeza na Ondoa sehemu na Denominators tofauti (Sehemu ya 2)

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    Tambua na Tumia Uendeshaji wa Fraction

    Kwa sasa katika sura hii, umefanya mazoezi ya kuzidisha, kugawanya, kuongeza, na kuondoa sehemu ndogo. Jedwali lifuatayo linafupisha shughuli hizi nne za sehemu. Kumbuka: Unahitaji denominator ya kawaida ili kuongeza au kuondoa sehemu ndogo, lakini si kuzidisha au kugawanya sehemu ndogo.

    Muhtasari wa Uendeshaji wa Sehemu

    Kuzidisha sehemu: Kuzidisha nambari na kuzidisha denominators.

    \[\dfrac{a}{b} \cdot \dfrac{c}{d} = \dfrac{ac}{bd}\]

    Mgawanyiko wa sehemu: Panua sehemu ya kwanza kwa usawa wa pili.

    \[\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \cdot \dfrac{d}{c}\]

    sehemu Aidha: Kuongeza nambari na mahali jumla juu ya denominator kawaida. Ikiwa sehemu ndogo zina denominators tofauti, kwanza ubadilishe kwa fomu sawa na LCD.

    \[\dfrac{a}{c} + \dfrac{b}{c} = \dfrac{a + b}{c}\]

    Kuondoa sehemu: Ondoa nambari na uweke tofauti juu ya denominator ya kawaida. Ikiwa sehemu ndogo zina denominators tofauti, kwanza ubadilishe kwa fomu sawa na LCD.

    \[\dfrac{a}{c} - \dfrac{a}{c} = \dfrac{a - b}{c}\]

    Mfano\(\PageIndex{11}\): simplify

    Kurahisisha:

    1. \(− \dfrac{1}{4} + \dfrac{1}{6}\)
    2. \(− \dfrac{1}{4} \div \dfrac{1}{6}\)

    Suluhisho

    Kwanza tunajiuliza, “Kazi ni nini?”

    1. Uendeshaji ni kuongeza. Je, sehemu ndogo zina denominator ya kawaida? Hapana.
    Kupata LCD.
    Andika upya kila sehemu kama sehemu sawa na LCD. \(- \dfrac{1 \cdot \textcolor{red}{3}}{4 \cdot \textcolor{red}{3}} + \dfrac{1 \cdot \textcolor{red}{2}}{6 \cdot \textcolor{red}{2}} \)
    Kurahisisha nambari na denominators. \(- \dfrac{3}{12} + \dfrac{2}{12} \)
    Ongeza nambari na uweke jumla juu ya denominator ya kawaida. \(- \dfrac{1}{12} \)
    Angalia ili uone kama jibu linaweza kuwa rahisi. Haiwezi.  
    1. Uendeshaji ni mgawanyiko. Hatuna haja ya denominator ya kawaida.
    Ili kugawanya sehemu ndogo, kuzidisha sehemu ya kwanza kwa usawa wa pili. \(- \dfrac{1}{4} \cdot \dfrac{6}{1}\)
    Kuzidisha. \(- \dfrac{6}{4}\)
    Kurahisisha. \(- \dfrac{3}{2} \)
    Zoezi\(\PageIndex{21}\)

    Kurahisisha:

    1. \(− \dfrac{3}{4} - \dfrac{1}{6}\)
    2. \(− \dfrac{3}{4} \cdot \dfrac{1}{6}\)
    Jibu

    \(-\dfrac{11}{12}\)

    Jibu b

    \(-\dfrac{1}{8}\)

    Zoezi\(\PageIndex{22}\)

    Kurahisisha:

    1. \(\dfrac{5}{6} \div \left(- \dfrac{1}{4}\right)\)
    2. \(\dfrac{5}{6} - \left(- \dfrac{1}{4}\right)\)
    Jibu

    \(-\dfrac{10}{3}\)

    Jibu b

    \(\dfrac{13}{12}\)

    Mfano\(\PageIndex{12}\): simplify

    Kurahisisha:

    1. \(\dfrac{5x}{6} - \dfrac{3}{10}\)
    2. \(\dfrac{5x}{6} \cdot \dfrac{3}{10}\)

    Suluhisho

    1. Uendeshaji ni kuondoa. Sehemu ndogo hazina denominator ya kawaida.
    Andika upya kila sehemu kama sehemu sawa na LCD, 30. \(\dfrac{5x \cdot \textcolor{red}{5}}{6 \cdot \textcolor{red}{5}} - \dfrac{3 \cdot \textcolor{red}{3}}{10 \cdot \textcolor{red}{3}} \)
      \(\dfrac{25x}{30} - \dfrac{9}{30} \)
    Ondoa nambari na uweke tofauti juu ya denominator ya kawaida. \(\dfrac{25x - 9}{30} \)
    1. Uendeshaji ni kuzidisha; hakuna haja ya denominator ya kawaida.
    Ili kuzidisha sehemu ndogo, kuzidisha nambari na kuzidisha denominators. \(\dfrac{5x \cdot 3}{ 6 \cdot 10} \)
    Andika upya, kuonyesha mambo ya kawaida. \(\dfrac{\cancel{5} \cdot x \cdot \cancel{3}}{2 \cdot \cancel{3} \cdot 2 \cdot \cancel{5}} \)
    Ondoa mambo ya kawaida ili kurahisisha. \(\dfrac{x}{4} \)
    Zoezi\(\PageIndex{23}\)

    Kurahisisha:

    1. \(\dfrac{3a}{4} - \dfrac{8}{9}\)
    2. \(\dfrac{3a}{4} \cdot \dfrac{8}{9}\)
    Jibu

    \(\dfrac{27a-32}{36}\)

    Jibu b

    \(\dfrac{2a}{3}\)

    Zoezi\(\PageIndex{24}\)

    Kurahisisha:

    1. \(\dfrac{4k}{5} + \dfrac{5}{6}\)
    2. \(\dfrac{4k}{5} \div \dfrac{5}{6}\)
    Jibu

    \(\dfrac{24k+25}{30}\)

    Jibu b

    \(\dfrac{24k}{25}\)

    Tumia Utaratibu wa Uendeshaji ili kurahisisha sehemu ndogo

    Katika Kuzidisha na Gawanya Hesabu Mchanganyiko na Fractions Complex, tuliona kwamba sehemu tata ni sehemu ambayo namba au denominator ina sehemu. Sisi kilichorahisisha sehemu tata kwa kuandika tena kama matatizo ya mgawanyiko. Kwa mfano,

    \[\dfrac{\dfrac{3}{4}}{\dfrac{5}{8}} = \dfrac{3}{4} \div \dfrac{5}{8} \nonumber \]

    Sasa tutaangalia sehemu ndogo ambazo namba au denominator inaweza kuwa rahisi. Ili kufuata utaratibu wa shughuli, sisi kurahisisha nambari na denominator tofauti kwanza. Kisha tunagawanya nambari na denominator.

    JINSI YA: KURAHISISHA SEHEMU NDOGO

    Hatua ya 1. Kurahisisha nambari.

    Hatua ya 2. Kurahisisha denominator.

    Hatua ya 3. Gawanya nambari kwa denominator.

    Hatua ya 4. Kurahisisha kama inawezekana.

    Mfano\(\PageIndex{13}\): simplify

    Kurahisisha:\(\dfrac{\left(\dfrac{1}{2}\right)^{2}}{4 + 3^{2}}\).

    Suluhisho

    Kurahisisha nambari. \(\dfrac{\dfrac{1}{4}}{4 + 3^{2}}\)
    Kurahisisha neno na exponent katika denominator. \(\dfrac{\dfrac{1}{4}}{4 + 9} \)
    Ongeza maneno katika denominator. \(\dfrac{\dfrac{1}{4}}{13} \)
    Gawanya nambari kwa denominator. \(\dfrac{1}{4} \div 13 \)
    Andika upya kama kuzidisha kwa usawa. \(\dfrac{1}{4} \cdot \dfrac{1}{13} \)
    Kuzidisha. \(\dfrac{1}{52}\)
    Zoezi\(\PageIndex{25}\)

    Kurahisisha:\(\dfrac{\left(\dfrac{1}{3}\right)^{2}}{2^{3} + 2}\).

    Jibu

    \(\dfrac{1}{90}\)

    Zoezi\(\PageIndex{26}\)

    Kurahisisha:\(\dfrac{1 + 4^{2}}{\left(\dfrac{1}{4}\right)^{2}}\).

    Jibu

    \(272\)

    Mfano\(\PageIndex{14}\): simplify

    Kurahisisha:\(\dfrac{\dfrac{1}{2} + \dfrac{2}{3}}{\dfrac{3}{4} - \dfrac{1}{6}}\).

    Suluhisho

    Andika upya nambari na LCD ya 6 na denominator na LCD ya 12. \(\dfrac{\dfrac{3}{6} + \dfrac{4}{6}}{\dfrac{9}{12} - \dfrac{2}{12}} \)
    Ongeza kwenye nambari. Ondoa katika denominator. \(\dfrac{\dfrac{7}{6}}{\dfrac{7}{12}} \)
    Gawanya nambari kwa denominator. \(\dfrac{7}{6} \div \dfrac{7}{12}\)
    Andika upya kama kuzidisha kwa usawa. \(\dfrac{7}{6} \cdot \dfrac{12}{7} \)
    Andika upya, kuonyesha mambo ya kawaida. \(\dfrac{\cancel{7} \cdot \cancel{6} \cdot 2}{\cancel{6} \cancel{7} \cdot 1} \)
    Kurahisisha. \(2 \)
    Zoezi\(\PageIndex{27}\)

    Kurahisisha:\(\dfrac{\dfrac{1}{3} + \dfrac{1}{2}}{\dfrac{3}{4} - \dfrac{1}{3}}\).

    Jibu

    \(2\)

    Zoezi\(\PageIndex{28}\)

    Kurahisisha:\(\dfrac{\dfrac{2}{3} - \dfrac{1}{2}}{\dfrac{1}{4} + \dfrac{1}{3}}\).

    Jibu

    \(\dfrac{2}{7}\)

    Tathmini Maneno ya kutofautiana na FRACTIONS

    Tumepima maneno kabla, lakini sasa tunaweza pia kutathmini maneno na sehemu ndogo. Kumbuka, kutathmini maneno, sisi badala ya thamani ya kutofautiana katika kujieleza na kisha kurahisisha.

    Mfano\(\PageIndex{15}\): evaluate

    Tathmini\(x + \dfrac{1}{3}\) wakati

    1. \(x = - \dfrac{1}{3}\)
    2. \(x = - \dfrac{3}{4}\)

    Suluhisho

    1. Kutathmini\(x + \dfrac{1}{3}\) wakati\(x = − \dfrac{1}{3}\), badala\(− \dfrac{1}{3}\) ya\(x\) katika kujieleza.
    Mbadala\(\textcolor{red}{- \dfrac{1}{3}}\) kwa ajili ya x. \(\textcolor{red}{- \dfrac{1}{3}} + \dfrac{1}{3} \)
    Kurahisisha. \(0 \)
    1. Kutathmini\(x + \dfrac{1}{3}\) wakati\(x = − \dfrac{3}{4}\), sisi badala\(− \dfrac{3}{4}\) ya\(x\) katika kujieleza.
    Mbadala\(\textcolor{red}{- \dfrac{3}{4}}\) kwa ajili ya x. \(\textcolor{red}{- \dfrac{1}{3}} + \dfrac{1}{3}\)
    Andika upya kama sehemu ndogo sawa na LCD, 12. \(- \dfrac{3 \cdot 3}{4 \cdot 3} + \dfrac{1 \cdot 4}{3 \cdot 4} \)
    Kurahisisha nambari na denominators. \(- \dfrac{9}{12} + \dfrac{4}{12} \)
    Ongeza. \(- \dfrac{5}{12} \)
    Zoezi\(\PageIndex{29}\)

    Tathmini\(x + \dfrac{3}{4}\) wakati:

    1. \(x = - \dfrac{7}{4}\)
    2. \(x = - \dfrac{5}{4}\)
    Jibu

    \(-1\)

    Jibu b

    \(-\dfrac{1}{2}\)

    Zoezi\(\PageIndex{30}\)

    Tathmini\(y + \dfrac{1}{2}\) wakati:

    1. \(y = \dfrac{2}{3}\)
    2. \(y = - \dfrac{3}{4}\)
    Jibu

    \(\dfrac{7}{6}\)

    Jibu b

    \(-\dfrac{1}{4}\)

    Mfano\(\PageIndex{16}\): evaluate

    Tathmini\(y − \dfrac{5}{6}\) wakati\(y = - \dfrac{2}{3}\).

    Suluhisho

    Sisi badala\(− \dfrac{2}{3}\) ya\(y\) katika kujieleza.

    Mbadala\(\textcolor{red}{- \dfrac{2}{3}}\) kwa y. \(\textcolor{red}{- \dfrac{2}{3}} - \dfrac{5}{6}\)
    Andika upya kama sehemu ndogo sawa na LCD, 6. \(- \dfrac{4}{6} - \dfrac{5}{6} \)
    Ondoa. \(- \dfrac{9}{6} \)
    Kurahisisha. \(- \dfrac{3}{2} \)
    Zoezi\(\PageIndex{31}\)

    Tathmini\(y − \dfrac{1}{2}\) wakati\(y = - \dfrac{1}{4}\).

    Jibu

    \(-\dfrac{3}{4}\)

    Zoezi\(\PageIndex{32}\)

    Tathmini\(x − \dfrac{3}{8}\) wakati\(x = - \dfrac{5}{2}\).

    Jibu

    \(-\dfrac{23}{8}\)

    Mfano\(\PageIndex{17}\):

    Tathmini\(2x^2y\) wakati\(x = \dfrac{1}{4}\) na\(y = − \dfrac{2}{3}\).

    Suluhisho

    Badilisha maadili katika maneno. Katika\(2x^2y\), exponent inatumika tu kwa\(x\).

    Mbadala\(\textcolor{red}{\dfrac{1}{4}}\) ya x na\(\textcolor{blue}{- \dfrac{2}{3}}\) kwa y. \(2 \left(\textcolor{red}{\dfrac{1}{4}}\right)^{2} \left(\textcolor{blue}{- \dfrac{2}{3}}\right) \)
    Kurahisisha watetezi kwanza. \(2 \left(\dfrac{1}{16}\right) \left(- \dfrac{2}{3}\right)\)
    Kuzidisha. Bidhaa itakuwa hasi. \(- \dfrac{2}{1} \cdot \dfrac{1}{16} \cdot \dfrac{2}{3} \)
    Kurahisisha. \(- \dfrac{4}{48} \)
    Ondoa mambo ya kawaida. \(- \dfrac{1 \cdot \cancel{4}}{\cancel{4} \cdot 12} \)
    Kurahisisha. \(- \dfrac{1}{12} \)
    Zoezi\(\PageIndex{33}\)

    Tathmini:\(3ab^2\) wakati\(a = − \dfrac{2}{3}\) na\(b = − \dfrac{1}{2}\).

    Jibu

    \(-\dfrac{1}{2}\)

    Zoezi\(\PageIndex{34}\)

    Tathmini:\(4c^3d\) wakati\(c = − \dfrac{1}{2}\) na\(d = − \dfrac{4}{3}\).

    Jibu

    \(\dfrac{2}{3}\)

    Mfano\(\PageIndex{18}\): evaluate

    Tathmini:\(\dfrac{p + q}{r}\) wakati\(p = −4\),\(q = −2\), na\(r = 8\).

    Suluhisho

    Sisi badala ya maadili katika kujieleza na kurahisisha.

    Mbadala\(\textcolor{red}{-4}\) kwa p,\(\textcolor{blue}{-2}\) kwa q na\(\textcolor{magenta}{8}\) kwa r. \(\dfrac{\textcolor{red}{-4} + \textcolor{blue}{(-2)}}{\textcolor{magenta}{8}} \)
    Ongeza kwenye nambari ya kwanza. \(- \dfrac{6}{8}\)
    Kurahisisha. \(- \dfrac{3}{4}\)
    Zoezi\(\PageIndex{35}\)

    Tathmini:\(\dfrac{a + b}{c}\) wakati\(a = −8\),\(b = −7\), na\(c = 6\).

    Jibu

    \(-\dfrac{5}{2}\)

    Zoezi\(\PageIndex{36}\)

    Tathmini:\(\dfrac{x + y}{z}\) wakati\(x = 9\),\(y = −18\), na\(z =- 6\).

    Jibu

    \(\dfrac{3}{2}\)

    Mazoezi hufanya kamili

    Kupata angalau kawaida Denominator (LCD)

    Katika mazoezi yafuatayo, pata denominator ya kawaida (LCD) kwa kila seti ya sehemu ndogo.

    1. \(\dfrac{2}{3}\)na\(\dfrac{3}{4}\)
    2. \(\dfrac{3}{4}\)na\(\dfrac{2}{5}\)
    3. \(\dfrac{7}{12}\)na\(\dfrac{5}{8}\)
    4. \(\dfrac{9}{16}\)na\(\dfrac{7}{12}\)
    5. \(\dfrac{13}{30}\)na\(\dfrac{25}{42}\)
    6. \(\dfrac{23}{30}\)na\(\dfrac{5}{48}\)
    7. \(\dfrac{21}{35}\)na\(\dfrac{39}{56}\)
    8. \(\dfrac{18}{35}\)na\(\dfrac{33}{49}\)
    9. \(\dfrac{2}{3}, \dfrac{1}{6}\)na\(\dfrac{3}{4}\)
    10. \(\dfrac{2}{3}, \dfrac{1}{4}\)na\(\dfrac{3}{5}\)

    Badilisha FRACTIONS kwa FRACTIONS sawa na LCD

    Katika mazoezi yafuatayo, kubadilisha kwa sehemu sawa kwa kutumia LCD.

    1. \(\dfrac{1}{3}\)na\(\dfrac{1}{4}\), LCD = 12
    2. \(\dfrac{1}{4}\)na\(\dfrac{1}{5}\), LCD = 20
    3. \(\dfrac{5}{12}\)na\(\dfrac{7}{8}\), LCD = 24
    4. \(\dfrac{7}{12}\)na\(\dfrac{5}{8}\), LCD = 24
    5. \(\dfrac{13}{16}\)na\(- \dfrac{11}{12}\), LCD = 48
    6. \(\dfrac{11}{16}\)na\(- \dfrac{5}{12}\), LCD = 48
    7. \(\dfrac{1}{3}, \dfrac{5}{6}\), na\(\dfrac{3}{4}\), LCD = 12
    8. \(\dfrac{1}{3}, \dfrac{3}{4}\), na\(\dfrac{3}{5}\), LCD = 60

    Kuongeza na Ondoa Fractions na Denominators tofauti

    Katika mazoezi yafuatayo, ongeza au uondoe. Andika matokeo kwa fomu rahisi.

    1. \(\dfrac{1}{3} + \dfrac{1}{5}\)
    2. \(\dfrac{1}{4} + \dfrac{1}{5}\)
    3. \(\dfrac{1}{2} + \dfrac{1}{7}\)
    4. \(\dfrac{1}{3} + \dfrac{1}{8}\)
    5. \(\dfrac{1}{3} - \left(- \dfrac{1}{9}\right)\)
    6. \(\dfrac{1}{4} - \left(- \dfrac{1}{8}\right)\)
    7. \(\dfrac{1}{5} - \left(- \dfrac{1}{10}\right)\)
    8. \(\dfrac{1}{2} - \left(- \dfrac{1}{6}\right)\)
    9. \(\dfrac{2}{3} + \dfrac{3}{4}\)
    10. \(\dfrac{3}{4} + \dfrac{2}{5}\)
    11. \(\dfrac{7}{12} + \dfrac{5}{8}\)
    12. \(\dfrac{5}{12} + \dfrac{3}{8}\)
    13. \(\dfrac{7}{12} - \dfrac{9}{16}\)
    14. \(\dfrac{7}{16} - \dfrac{5}{12}\)
    15. \(\dfrac{11}{12} - \dfrac{3}{8}\)
    16. \(\dfrac{5}{8} - \dfrac{7}{12}\)
    17. \(\dfrac{2}{3} - \dfrac{3}{8}\)
    18. \(\dfrac{5}{6} - \dfrac{3}{4}\)
    19. \(− \dfrac{11}{30} + \dfrac{27}{40}\)
    20. \(− \dfrac{9}{20} + \dfrac{17}{30}\)
    21. \(− \dfrac{13}{30} + \dfrac{25}{42}\)
    22. \(− \dfrac{23}{30} + \dfrac{5}{48}\)
    23. \(− \dfrac{39}{56} - \dfrac{22}{35}\)
    24. \(− \dfrac{33}{49} - \dfrac{18}{35}\)
    25. \(- \dfrac{2}{3} - \left(- \dfrac{3}{4}\right)\)
    26. \(- \dfrac{3}{4} - \left(- \dfrac{4}{5}\right)\)
    27. \(- \dfrac{9}{16} - \left(- \dfrac{4}{5}\right)\)
    28. \(- \dfrac{7}{20} - \left(- \dfrac{5}{8}\right)\)
    29. 1 +\(\dfrac{7}{8}\)
    30. 1 +\(\dfrac{5}{6}\)
    31. 1-\(\dfrac{5}{9}\)
    32. 1-\(\dfrac{3}{10}\)
    33. \(\dfrac{x}{3} + \dfrac{1}{4}\)
    34. \(\dfrac{y}{2} + \dfrac{2}{3}\)
    35. \(\dfrac{y}{4} - \dfrac{3}{5}\)
    36. \(\dfrac{x}{5} - \dfrac{1}{4}\)

    Tambua na Tumia Uendeshaji wa Fraction

    Katika mazoezi yafuatayo, fanya shughuli zilizoonyeshwa. Andika majibu yako kwa fomu rahisi.

    1. (a)\(\dfrac{3}{4} + \dfrac{1}{6}\) (b)\(\dfrac{3}{4} \div \dfrac{1}{6}\)
    2. (a)\(\dfrac{2}{3} + \dfrac{1}{6}\) (b)\(\dfrac{2}{3} \div \dfrac{1}{6}\)
    3. (a)\(- \dfrac{2}{5} - \dfrac{1}{8}\) (b)\(- \dfrac{2}{5} \cdot \dfrac{1}{8}\)
    4. (a)\(- \dfrac{4}{5} - \dfrac{1}{8}\) (b)\(- \dfrac{4}{5} \cdot \dfrac{1}{8}\)
    5. (a)\(\dfrac{5n}{6} \div \dfrac{8}{15}\) (b)\(\dfrac{5n}{6} - \dfrac{8}{15}\)
    6. (a)\(\dfrac{3a}{8} \div \dfrac{7}{12}\) (b)\(\dfrac{3a}{8} - \dfrac{7}{12}\)
    7. (a)\(\dfrac{9}{10} \cdot \left(− \dfrac{11d}{12}\right)\) (b)\(\dfrac{9}{10} + \left(− \dfrac{11d}{12}\right)\)
    8. (a)\(\dfrac{4}{15} \cdot \left(− \dfrac{5}{q}\right)\) (b)\(\dfrac{4}{15} + \left(− \dfrac{5}{q}\right)\)
    9. \(- \dfrac{3}{8} \div \left(- \dfrac{3}{10}\right)\)
    10. \(- \dfrac{5}{12} \div \left(- \dfrac{5}{9}\right)\)
    11. \(- \dfrac{3}{8} + \dfrac{5}{12}\)
    12. \(- \dfrac{1}{8} + \dfrac{7}{12}\)
    13. \(\dfrac{5}{6} − \dfrac{1}{9}\)
    14. \(\dfrac{5}{9} − \dfrac{1}{6}\)
    15. \(\dfrac{3}{8} \cdot \left(− \dfrac{10}{21}\right)\)
    16. \(\dfrac{7}{12} \cdot \left(− \dfrac{8}{35}\right)\)
    17. \(− \dfrac{7}{15} - \dfrac{y}{4}\)
    18. \(− \dfrac{3}{8} - \dfrac{x}{11}\)
    19. \(\dfrac{11}{12a} \cdot \dfrac{9a}{16}\)
    20. \(\dfrac{10y}{13} \cdot \dfrac{8}{15y}\)

    Tumia Utaratibu wa Uendeshaji ili kurahisisha sehemu ndogo

    Katika mazoezi yafuatayo, kurahisisha.

    1. \(\dfrac{\left(\dfrac{1}{5} \right)^{2}}{2 + 3^{2}}\)
    2. \(\dfrac{\left(\dfrac{1}{3} \right)^{2}}{5 + 2^{2}}\)
    3. \(\dfrac{2^{3} + 4^{2}}{\left(\dfrac{2}{3}\right)^{2}}\)
    4. \(\dfrac{3^{3} - 3^{2}}{\left(\dfrac{3}{4}\right)^{2}}\)
    5. \(\dfrac{\left(\dfrac{3}{5} \right)^{2}}{\left(\dfrac{3}{7}\right)^{2}}\)
    6. \(\dfrac{\left(\dfrac{3}{4} \right)^{2}}{\left(\dfrac{5}{8}\right)^{2}}\)
    7. \(\dfrac{2}{\dfrac{1}{3} + \dfrac{1}{5}}\)
    8. \(\dfrac{5}{\dfrac{1}{4} + \dfrac{1}{3}}\)
    9. \(\dfrac{\dfrac{2}{3} + \dfrac{1}{2}}{\dfrac{3}{4} - \dfrac{2}{3}}\)
    10. \(\dfrac{\dfrac{3}{4} + \dfrac{1}{2}}{\dfrac{5}{6} - \dfrac{2}{3}}\)
    11. \(\dfrac{\dfrac{7}{8} - \dfrac{2}{3}}{\dfrac{1}{2} + \dfrac{3}{8}}\)
    12. \(\dfrac{\dfrac{3}{4} - \dfrac{3}{5}}{\dfrac{1}{4} + \dfrac{2}{5}}\)

    Mazoezi ya mchanganyiko

    Katika mazoezi yafuatayo, kurahisisha.

    1. \(\dfrac{1}{2} + \dfrac{2}{3} \cdot \dfrac{5}{12}\)
    2. \(\dfrac{1}{3} + \dfrac{2}{5} \cdot \dfrac{3}{4}\)
    3. 1-\(\dfrac{3}{5} \div \dfrac{1}{10}\)
    4. 1-\(\dfrac{5}{6} \div \dfrac{1}{12}\)
    5. \(\dfrac{2}{3} + \dfrac{1}{6} + \dfrac{3}{4}\)
    6. \(\dfrac{2}{3} + \dfrac{1}{4} + \dfrac{3}{5}\)
    7. \(\dfrac{3}{8} - \dfrac{1}{6} + \dfrac{3}{4}\)
    8. \(\dfrac{2}{5} + \dfrac{5}{8} - \dfrac{3}{4}\)
    9. 12\(\left(\dfrac{9}{20} − \dfrac{4}{15}\right)\)
    10. 8\(\left(\dfrac{15}{16} − \dfrac{5}{6}\right)\)
    11. \(\dfrac{\dfrac{5}{8} + \dfrac{1}{6}}{\dfrac{19}{24}}\)
    12. \(\dfrac{\dfrac{1}{6} + \dfrac{3}{10}}{\dfrac{14}{30}}\)
    13. \(\left(\dfrac{5}{9} + \dfrac{1}{6}\right) \div \left(\dfrac{2}{3} − \dfrac{1}{2}\right)\)
    14. \(\left(\dfrac{3}{4} + \dfrac{1}{6}\right) \div \left(\dfrac{5}{8} − \dfrac{1}{3}\right)\)

    Katika mazoezi yafuatayo, tathmini maneno yaliyotolewa. Eleza majibu yako kwa fomu rahisi, ukitumia sehemu zisizofaa ikiwa ni lazima.

    1. x +\(\dfrac{1}{2}\) wakati
      1. x =\(− \dfrac{1}{8}\)
      2. x =\(− \dfrac{1}{2}\)
    2. x +\(\dfrac{2}{3}\) wakati
      1. x =\(− \dfrac{1}{6}\)
      2. x =\(− \dfrac{5}{3}\)
    3. x +\(\left(− \dfrac{5}{6}\right)\) wakati
      1. x =\(\dfrac{1}{3}\)
      2. x =\(− \dfrac{1}{6}\)
    4. x +\(\left(− \dfrac{11}{12}\right)\) wakati
      1. x =\(\dfrac{11}{12}\)
      2. x =\(\dfrac{3}{4}\)
    5. x -\(\dfrac{2}{5}\) wakati
      1. x =\(\dfrac{3}{5}\)
      2. x =\(- \dfrac{3}{5}\)
    6. x -\(\dfrac{1}{3}\) wakati
      1. x =\(\dfrac{2}{3}\)
      2. x =\(- \dfrac{2}{3}\)
    7. \(\dfrac{7}{10}\)- w wakati
      1. w =\(\dfrac{1}{2}\)
      2. w =\(- \dfrac{1}{2}\)
    8. \(\dfrac{5}{12}\)- w wakati
      1. w =\(\dfrac{1}{4}\)
      2. w =\(- \dfrac{1}{4}\)
    9. 4p 2 q wakati p =\(- \dfrac{1}{2}\) na q =\(\dfrac{5}{9}\)
    10. 5m 2 n wakati m =\(- \dfrac{2}{5}\) na n =\(\dfrac{1}{3}\)
    11. 2x 2 y 3 wakati x =\(- \dfrac{2}{3}\) na y =\(- \dfrac{1}{2}\)
    12. 8u 2 v 3 wakati u =\(- \dfrac{3}{4}\) na v =\(- \dfrac{1}{2}\)
    13. \(\dfrac{u + v}{w}\)wakati u = -4, v = -8, w = 2
    14. \(\dfrac{m + n}{p}\)wakati m = -6, n = -2, p = 4
    15. \(\dfrac{a + b}{a - b}\)wakati = -3, b = 8
    16. \(\dfrac{r - s}{r + s}\)wakati r = 10, s = -5

    kila siku Math

    1. Mapambo Laronda ni kufanya inashughulikia kwa mito kutupa juu ya sofa yake. Kwa kila kifuniko cha mto, anahitaji\(\dfrac{3}{16}\) yadi ya kitambaa cha kuchapisha na\(\dfrac{3}{8}\) yadi ya kitambaa imara. Je, ni jumla ya kitambaa cha Laronda kinachohitaji kwa kila kifuniko cha mto?
    2. Vanessa ya kuoka ni kuoka biskuti za chokoleti na cooki Anahitaji\(1 \dfrac{1}{4}\) vikombe vya sukari kwa cookies chip chokoleti, na\(1 \dfrac{1}{8}\) vikombe kwa cookies oatmeal Ni kiasi gani sukari anahitaji kabisa?

    Mazoezi ya kuandika

    1. Eleza kwa nini ni muhimu kuwa na denominator ya kawaida ili kuongeza au kuondoa sehemu ndogo.
    2. Eleza jinsi ya kupata LCD ya vipande viwili.

    Self Check

    (a) Baada ya kukamilisha mazoezi, tumia orodha hii ili kutathmini ujuzi wako wa malengo ya sehemu hii.

    (b) Baada ya kuangalia orodha, unafikiri umeandaliwa vizuri kwa sehemu inayofuata? Kwa nini au kwa nini?

    Wachangiaji na Majina