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4.6: Kuzidisha na Gawanya Hesabu Mchanganyiko na FRACTIONS Complex (Sehemu ya 2)

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    Kurahisisha Maneno na Bar Fraction

    Ishara mbaya huenda wapi sehemu? Kawaida, ishara hasi imewekwa mbele ya sehemu, lakini wakati mwingine utaona sehemu na nambari hasi au denominator. Kumbuka kwamba sehemu ndogo zinawakilisha mgawanyiko. Sehemu\(− \dfrac{1}{3}\) inaweza kuwa matokeo ya kugawa\(\dfrac{−1}{3}\), hasi kwa chanya, au ya kugawa\(\dfrac{1}{−3}\), chanya kwa hasi. Wakati nambari na denominator zina ishara tofauti, quotient ni hasi.

    \[\dfrac{-1}{3} = - \dfrac{1}{3} \quad \dfrac{negative}{positive} = negative \quad \dfrac{1}{-3} = - \dfrac{1}{3} \quad \dfrac{positive}{negative} = negative \tag{4.3.46} \nonumber \]

    Ikiwa namba na denominator ni hasi, basi sehemu yenyewe ni chanya kwa sababu tunagawanya hasi kwa hasi.

    \[\dfrac{-1}{-3} = \dfrac{1}{3} \qquad \dfrac{negative}{negative} = positive \tag{4.3.47} \nonumber \]

    Uwekaji wa Ishara Hasi katika Fraction

    Kwa idadi yoyote chanya\(a\) na\(b\),

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

    Mfano\(\PageIndex{11}\): equivalent fractions

    Ni ipi kati ya sehemu zifuatazo ni sawa na\(\dfrac{7}{−8}\)?

    \[\dfrac{-7}{-8}, \dfrac{-7}{8}, \dfrac{7}{8}, - \dfrac{7}{8} \nonumber \]

    Suluhisho

    Quotient ya chanya na hasi ni hasi, hivyo\(\dfrac{7}{−8}\) ni hasi. Ya vipande vilivyoorodheshwa,\(\dfrac{−7}{8}\) na pia\(− \dfrac{7}{8}\) ni hasi.

    Zoezi\(\PageIndex{21}\)

    Ni ipi kati ya sehemu zifuatazo ni sawa na\(\dfrac{-3}{5}\)?

    \[\dfrac{-3}{-5}, \dfrac{3}{5}, - \dfrac{3}{5}, \dfrac{3}{-5} \nonumber \]

    Jibu

    \(-\dfrac{3}{5}, \dfrac{3}{-5}\)

    Zoezi\(\PageIndex{22}\)

    Ni ipi kati ya sehemu zifuatazo ni sawa na\(- \dfrac{2}{7}\)?

    \[\dfrac{-2}{-7}, \dfrac{-2}{7}, \dfrac{2}{7}, \dfrac{2}{-7} \nonumber \]

    Jibu

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

    Sehemu za sehemu hufanya kama alama za makundi. Maneno hapo juu na chini ya bar ya sehemu yanapaswa kutibiwa kama yalikuwa katika mabano. Kwa mfano,\(\dfrac{4 + 8}{5 − 3}\) ina maana\((4 + 8) ÷ (5 − 3)\). Utaratibu wa shughuli unatuambia kurahisisha namba na denominator kwanza-kama kulikuwa na mabenzi-kabla ya kugawanya.

    Tutaongeza baa za sehemu kwenye seti yetu ya alama za makundi kutoka Tumia Lugha ya Algebra ili uwe na seti kamili zaidi hapa.

    Kundi Alama

    Mabano, mabano, braces, ishara ya thamani kamili, na bar ya sehemu huonyeshwa.

    JINSI YA: KURAHISISHA KUJIELEZA NA BAR YA SEHEMU

    Hatua ya 1. Kurahisisha nambari.

    Hatua ya 2. Kurahisisha denominator.

    Hatua ya 3. Kurahisisha sehemu.

    Mfano\(\PageIndex{12}\):

    Kurahisisha:\(\dfrac{4 + 8}{5 − 3}\).

    Suluhisho

    Kurahisisha maneno katika nambari. \(\dfrac{12}{5 - 3}\)
    Kurahisisha usemi katika denominator. \(\dfrac{12}{2}\)
    Kurahisisha sehemu. \(6\)
    Zoezi\(\PageIndex{23}\)

    Kurahisisha:\(\dfrac{4 + 6}{11 − 2}\).

    Jibu

    \(\dfrac{10}{9}\)

    Zoezi\(\PageIndex{24}\)

    Kurahisisha:\(\dfrac{3 + 5}{18− 2}\).

    Jibu

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

    Mfano\(\PageIndex{13}\):

    Kurahisisha:\(\dfrac{4 − 2(3)}{2^{2} + 2}\).

    Suluhisho

    Tumia utaratibu wa shughuli. Kuzidisha katika nambari na kutumia exponent katika denominator. \(\dfrac{4 - 6}{4 + 2}\)
    Kurahisisha nambari na denominator. \(\dfrac{-2}{6}\)
    Kurahisisha sehemu. \(- \dfrac{1}{3}\)
    Zoezi\(\PageIndex{25}\)

    Kurahisisha:\(\dfrac{6 − 3(5)}{3^{2} + 3}\).

    Jibu

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

    Zoezi\(\PageIndex{26}\)

    Kurahisisha:\(\dfrac{4 − 4(6)}{3^{3} + 3}\).

    Jibu

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

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

    Kurahisisha:\(\dfrac{(8 − 4)^{2}}{8^{2} − 4^{2}}\).

    Suluhisho

    Tumia utaratibu wa shughuli (mabano kwanza, kisha wafuasi). \(\dfrac{(4)^{2}}{64 - 16}\)
    Kurahisisha namba na denominator. \(\dfrac{16}{48}\)
    Kurahisisha sehemu. \(\dfrac{1}{3}\)
    Zoezi\(\PageIndex{27}\)

    Kurahisisha:\(\dfrac{(11 − 7)^{2}}{11^{2} − 7^{2}}\).

    Jibu

    \(\dfrac{2}{9}\)

    Zoezi\(\PageIndex{28}\)

    Kurahisisha:\(\dfrac{(6 + 2)^{2}}{6^{2} − 2^{2}}\).

    Jibu

    \(\dfrac{8}{5}\)

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

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

    Suluhisho

    Kuzidisha. \(\dfrac{-12 + (-12)}{-6 - 2}\)
    Kurahisisha. \(\dfrac{-24}{-8} \)
    Gawanya. \(3 \)
    Zoezi\(\PageIndex{29}\)

    Kurahisisha:\(\dfrac{8(−2) + 4(−3)}{−5(2) + 3}\).

    Jibu

    \(4\)

    Zoezi\(\PageIndex{30}\)

    Kurahisisha:\(\dfrac{7(−1) + 9(−3)}{−5(3) + 2}\).

    Jibu

    \(2\)

    Dhana muhimu

    • Panua au ugawanye namba zilizochanganywa.
      1. Badilisha nambari zilizochanganywa kwa sehemu zisizofaa.
      2. Fuata sheria za kuzidisha sehemu au mgawanyiko.
      3. Kurahisisha kama inawezekana.
    • Kurahisisha sehemu tata.
      1. Andika upya sehemu tata kama tatizo la mgawanyiko.
      2. Fuata sheria za kugawa sehemu ndogo.
      3. Kurahisisha kama inawezekana.
    • Uwekaji wa ishara hasi katika sehemu.
      • Kwa idadi yoyote nzuri\(a\) na\(b\),\(\dfrac{-a}{b} = \dfrac{a}{-b} = -\dfrac{a}{b}\).
    • Kurahisisha kujieleza na bar ya sehemu.
      1. Kurahisisha nambari.
      2. Kurahisisha denominator.
      3. Kurahisisha sehemu.

    faharasa

    sehemu ngumu

    Sehemu ngumu ni sehemu ambayo nambari au denominator ina sehemu.

    Mazoezi hufanya kamili

    Ongeza na Gawanya Hesabu Mchanganyiko

    Katika mazoezi yafuatayo, kuzidisha na kuandika jibu kwa fomu rahisi.

    1. \(4 \dfrac{3}{8} \cdot \dfrac{7}{10}\)
    2. \(2 \dfrac{4}{9} \cdot \dfrac{6}{7}\)
    3. \(\dfrac{15}{22} \cdot 3 \dfrac{3}{5}\)
    4. \(\dfrac{25}{36} \cdot 6 \dfrac{3}{10}\)
    5. \(4 \dfrac{2}{3} (−1 \dfrac{1}{8})\)
    6. \(2 \dfrac{2}{5} (−2 \dfrac{2}{9})\)
    7. \(−4 \dfrac{4}{9} \cdot 5 \dfrac{13}{16}\)
    8. \(−1 \dfrac{7}{20} \cdot 2 \dfrac{11}{12}\)

    Katika mazoezi yafuatayo, ugawanye, na uandike jibu lako kwa fomu rahisi.

    1. \(5 \dfrac{1}{3}\)÷ 4
    2. \(13 \dfrac{1}{2}\)÷ 9
    3. -12 ÷\(3 \dfrac{3}{11}\)
    4. -7 ÷\(5 \dfrac{1}{4}\)
    5. \(6 \dfrac{3}{8} \div 2 \dfrac{1}{8}\)
    6. \(2 \dfrac{1}{5} \div 1 \dfrac{1}{10}\)
    7. \(−9 \dfrac{3}{5} \div (−1 \dfrac{3}{5})\)
    8. \(−18 \dfrac{3}{4} \div (−3 \dfrac{3}{4})\)

    Tafsiri Maneno kwa Maneno na Fractions

    Katika mazoezi yafuatayo, tafsiri kila maneno ya Kiingereza katika kujieleza kwa algebraic.

    1. quotient ya 5u na 11
    2. quotient ya 7v na 13
    3. quotient ya p na q
    4. quotient ya a na b
    5. quotient ya r na jumla ya s na 10
    6. quotient ya A na tofauti ya 3 na B

    Kurahisisha sehemu tata

    Katika mazoezi yafuatayo, kurahisisha sehemu tata.

    1. \(\dfrac{\dfrac{2}{3}}{\dfrac{8}{9}}\)
    2. \(\dfrac{\dfrac{4}{5}}{\dfrac{8}{15}}\)
    3. \(\dfrac{− \dfrac{8}{21}}{\dfrac{12}{35}}\)
    4. \(\dfrac{− \dfrac{9}{16}}{\dfrac{33}{40}}\)
    5. \(\dfrac{− \dfrac{4}{5}}{2}\)
    6. \(\dfrac{− \dfrac{9}{10}}{3}\)
    7. \(\dfrac{\dfrac{2}{5}}{8}\)
    8. \(\dfrac{\dfrac{5}{3}}{10}\)
    9. \(\dfrac{\dfrac{m}{3}}{\dfrac{n}{2}}\)
    10. \(\dfrac{\dfrac{r}{5}}{\dfrac{s}{3}}\)
    11. \(\dfrac{− \dfrac{x}{6}}{− \dfrac{8}{9}}\)
    12. \(\dfrac{− \dfrac{3}{8}}{− \dfrac{y}{12}}\)
    13. \(\dfrac{2 \dfrac{4}{5}}{\dfrac{1}{10}}\)
    14. \(\dfrac{4 \dfrac{2}{3}}{\dfrac{1}{6}}\)
    15. \(\dfrac{\dfrac{7}{9}}{−2 \dfrac{4}{5}}\)
    16. \(\dfrac{\dfrac{3}{8}}{−6 \dfrac{3}{4}}\)

    Kurahisisha Maneno na Bar Fraction

    Katika mazoezi yafuatayo, tambua sehemu ndogo sawa.

    1. Ni ipi kati ya sehemu zifuatazo ni sawa na\(\dfrac{5}{−11}\)? $$\ dfrac {-5} {-11},\ dfrac {-5} {11},\ dfrac {5} {5} {1} {11} {11} $
    2. Ni ipi kati ya sehemu zifuatazo ni sawa na\(\dfrac{−4}{9}\)? $$\ dfrac {-4} {9-9},\ dfrac {-4} {9},\ dfrac {4} {4} {9}, -\ dfrac {4} {9} $
    3. Ni ipi kati ya sehemu zifuatazo ni sawa na\(− \dfrac{11}{3}\)? $$\ dfrac {-11} {3},\ dfrac {1} {3},\ dfrac {-11} {—3},\ dfrac {1} {-3} $$
    4. Ni ipi kati ya sehemu zifuatazo ni sawa na\(− \dfrac{13}{6}\)? $$\ dfrac {13} {6},\ dfrac {13} {-6},\ dfrac {-13} {-6},\ dfrac {-13} {6} $

    Katika mazoezi yafuatayo, kurahisisha.

    1. \(\dfrac{4 + 11}{8}\)
    2. \(\dfrac{9 + 3}{7}\)
    3. \(\dfrac{22 + 3}{10}\)
    4. \(\dfrac{19 − 4}{6}\)
    5. \(\dfrac{48}{24 − 15}\)
    6. \(\dfrac{46}{4 + 4}\)
    7. \(\dfrac{−6 + 6}{8 + 4}\)
    8. \(\dfrac{−6 + 3}{17 − 8}\)
    9. \(\dfrac{22 − 14}{19 − 13}\)
    10. \(\dfrac{15 + 9}{18 + 12}\)
    11. \(\dfrac{5 \cdot 8}{−10}\)
    12. \(\dfrac{3 \cdot 4}{−24}\)
    13. \(\dfrac{4 \cdot 3}{6 \cdot 6}\)
    14. \(\dfrac{6 \cdot 6}{9 \cdot 2}\)
    15. \(\dfrac{4^{2} − 1}{25}\)
    16. \(\dfrac{7^{2} + 1}{60}\)
    17. \(\dfrac{8 \cdot 3 + 2 \cdot 9}{14 + 3}\)
    18. \(\dfrac{9 \cdot 6 − 4 \cdot 7}{22 + 3}\)
    19. \(\dfrac{15 \cdot 5 − 5^{2}}{2 \cdot 10}\)
    20. \(\dfrac{12 \cdot 9 − 3^{2}}{3 \cdot 18}\)
    21. \(\dfrac{5 \cdot 6 − 3 \cdot 4}{4 \cdot 5 − 2 \cdot 3}\)
    22. \(\dfrac{8 \cdot 9 − 7 \cdot 6}{5 \cdot 6 − 9 \cdot 2}\)
    23. \(\dfrac{5^{2} − 3^{2}}{3 − 5}\)
    24. \(\dfrac{6^{2} − 4^{2}}{4 − 6}\)
    25. \(\dfrac{2 + 4(3)}{−3 − 2^{2}}\)
    26. \(\dfrac{7 + 3(5)}{−2 − 3^{2}}\)
    27. \(\dfrac{7 \cdot 4 − 2(8 − 5)}{9 \cdot 3 − 3 \cdot 5}\)
    28. \(\dfrac{9 \cdot 7 − 3(12 − 8)}{8 \cdot 7 − 6 \cdot 6}\)
    29. \(\dfrac{9(8 − 2)−3(15 − 7)}{6(7 − 1)−3(17 − 9)}\)
    30. \(\dfrac{8(9 − 2)−4(14 − 9)}{7(8 − 3)−3(16 − 9)}\)

    kila siku Math

    1. Kuoka Kichocheo cha vidakuzi vya chip chokoleti huita\(2 \dfrac{1}{4}\) vikombe vya unga. Graciela anataka mara mbili mapishi.
      1. Ni kiasi gani cha unga ambacho Graciela kitahitaji? Onyesha hesabu yako. Andika matokeo yako kama sehemu isiyofaa na kama namba iliyochanganywa.
      2. Kupima vikombe kawaida kuja katika seti na vikombe kwa\(\dfrac{1}{8}, \dfrac{1}{4}, \dfrac{1}{3}, \dfrac{1}{2}\), na 1 kikombe. Chora mchoro kuonyesha njia mbili tofauti ambazo Graciela angeweza kupima unga unahitajika mara mbili ya mapishi.
    2. Baking kibanda katika kata haki anauza fudge na pauni. Tuzo yao ya kushinda “Chocolate Overdose” fudge ina\(2 \dfrac{2}{3}\) vikombe vya chips chocolate kwa pauni.
      1. Ni vikombe ngapi vya chips vya chokoleti viko katika pound ya nusu ya fudge?
      2. Wamiliki wa kibanda hufanya fudge katika makundi 10 ya pound. Je, ni chips ngapi za chokoleti wanahitaji kufanya kundi la 10 la pound? Andika matokeo yako kama sehemu zisizofaa na kama namba zilizochanganywa.

    Mazoezi ya kuandika

    1. Eleza jinsi ya kupata usawa wa nambari iliyochanganywa.
    2. Eleza jinsi ya kuzidisha namba zilizochanganywa.
    3. Randy anadhani kwamba\(3 \dfrac{1}{2} \cdot 5 \dfrac{1}{4}\) ni\(15 \dfrac{1}{8}\). Eleza nini kibaya na kufikiri Randy ya.
    4. Eleza kwa nini\(− \dfrac{1}{2}, \dfrac{−1}{2}\), na\(\dfrac{1}{−2}\) ni sawa.

    Self Check

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

    (b) Orodha hii inakuambia nini kuhusu ustadi wako wa sehemu hii? Ni hatua gani utachukua ili kuboresha?

    Wachangiaji na Majina