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4.4: Kuzidisha na Gawanya sehemu ndogo (Sehemu ya 2)

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    Kupata Reprecipals

    Sehemu ndogo\(\dfrac{2}{3}\) na\(\dfrac{3}{2}\) zinahusiana kwa kila mmoja kwa njia maalum. Hivyo ni\(− \dfrac{10}{7}\) na\(− \dfrac{7}{10}\). Je! Unaona jinsi gani? Mbali na hilo kuangalia kama matoleo kichwa-chini ya mtu mwingine, kama tulikuwa na kuzidisha jozi hizi za FRACTIONS, bidhaa itakuwa 1.

    \[\dfrac{2}{3} \cdot \dfrac{3}{2} = 1 \quad and \quad - \dfrac{10}{7} \left(- \dfrac{7}{10}\right) = 1 \tag{4.2.53} \nonumber \]

    Jozi hizo za namba zinaitwa kurudi.

    Ufafanuzi: Recipural

    Upeo wa sehemu\(\dfrac{a}{b}\) ni\(\dfrac{b}{a}\), wapi\(a ≠ 0\) na\(b ≠ 0\).

    Nambari na usawa wake una bidhaa ya\(1\).

    \[\dfrac{a}{b} \cdot \dfrac{b}{a} = 1 \tag{4.2.54}\]

    Ili kupata usawa wa sehemu, tunabadilisha sehemu. Hii inamaanisha kwamba tunaweka namba katika denominator na denominator katika nambari.

    Ili kupata matokeo mazuri wakati wa kuzidisha namba mbili, namba lazima ziwe na ishara sawa. Hivyo kurudia lazima iwe na ishara sawa.

    “a” juu ya “b” imeongezeka kwa “b” juu ya “a” sawa na moja chanya.

    Ili kupata usawa, endelea ishara sawa na ugeuke sehemu. Nambari ya sifuri haina usawa. Kwa nini? Nambari na kurudi kwake kuongezeka kwa\(1\). Je, kuna idadi yoyote\(r\) ili\(0 • r = 1\)? Hapana. Kwa hivyo,\(0\) nambari haina usawa.

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

    Pata usawa wa kila nambari. Kisha angalia kwamba bidhaa ya kila nambari na usawa wake ni\(1\).

    1. \(\dfrac{4}{9}\)
    2. \(− \dfrac{1}{6}\)
    3. \(− \dfrac{14}{5}\)
    4. \(7\)

    Suluhisho

    Ili kupata vizuizi, tunaweka ishara na kugeuza sehemu ndogo.

    Kupata usawa wa\(\dfrac{4}{9}\). Usawa wa\(\dfrac{4}{9}\) ni\(\dfrac{9}{4}\).

    Angalia:

    Panua idadi na usawa wake. \(\dfrac{4}{9} \cdot \dfrac{9}{4}\)
    Kuzidisha nambari na denominators. \(\dfrac{36}{36} \)
    Kurahisisha. \(1 \; \checkmark \)
    Kupata usawa wa\(- \dfrac{1}{6}\). Usawa wa\(- \dfrac{1}{6}\) ni\(\dfrac{6}{1}\).
    Kurahisisha. \(-6 \)
    Angalia. \(- \dfrac{1}{6} \cdot (-6) = 1 \; \checkmark \)
    Kupata usawa wa\(- \dfrac{14}{5}\). \(- \dfrac{5}{14} \)
    Angalia. \(- \dfrac{14}{5} \cdot \left(- \dfrac{5}{14}\right) = \dfrac{70}{70} = 1 \; \checkmark \)
    Pata usawa wa 7.  
    Andika 7 kama sehemu. \(\dfrac{7}{1}\)
    Andika usawa wa\(\dfrac{7}{1}\). \(\dfrac{1}{7} \)
    Angalia. \(7 \cdot \left(\dfrac{1}{7}\right) = 1 \; \checkmark \)
    Zoezi\(\PageIndex{21}\)

    Pata usawa:

    1. \(\dfrac{5}{7}\)
    2. \(− \dfrac{1}{8}\)
    3. \(− \dfrac{11}{4}\)
    4. \(14\)
    Jibu

    \(\dfrac{7}{5}\)

    Jibu b

    \(-8\)

    Jibu c

    \(-\dfrac{4}{11}\)

    Jibu d

    \(\dfrac{1}{14}\)

    Zoezi\(\PageIndex{22}\)

    Pata usawa:

    1. \(\dfrac{3}{7}\)
    2. \(− \dfrac{1}{12}\)
    3. \(− \dfrac{14}{9}\)
    4. \(21\)
    Jibu

    \(\dfrac{7}{3}\)

    Jibu b

    \(-12\)

    Jibu c

    \(-\dfrac{9}{14}\)

    Jibu d

    \(\dfrac{1}{21}\)

    Katika sura iliyotangulia, tulifanya kazi na kupinga na maadili kamili. Jedwali\(\PageIndex{1}\) linalinganisha kupinga, maadili kamili, na kurudi.

    Jedwali\(\PageIndex{1}\)
    Kinyume Thamani kamili kurudisha nyuma
    ina ishara kinyume kamwe hasi ina ishara sawa, sehemu inverts
    Mfano\(\PageIndex{12}\): fractions

    Jaza chati kwa kila sehemu katika safu ya kushoto:

    Idadi Kinyume Thamani kamili kurudisha nyuma
    \(- \dfrac{3}{8}\)      
    \(\dfrac{1}{2}\)      
    \(\dfrac{9}{5}\)      
    \(-5\)      

    Suluhisho

    Ili kupata kinyume, mabadiliko ya ishara. Ili kupata thamani kamili, kuondoka namba nzuri sawa, lakini kuchukua kinyume cha idadi hasi. Ili kupata usawa, weka ishara sawa na ugeuke sehemu.

    Idadi Kinyume Thamani kamili kurudisha nyuma
    \(- \dfrac{3}{8}\) \(\dfrac{3}{8}\) \(\dfrac{3}{8}\) \(- \dfrac{8}{3}\)
    \(\dfrac{1}{2}\) \(- \dfrac{1}{2}\) \(\dfrac{1}{2}\) \(2\)
    \(\dfrac{9}{5}\) \(- \dfrac{9}{5}\) \(\dfrac{9}{5}\) \(\dfrac{5}{9}\)
    \(-5\) \(5\) \(5\) \(- \dfrac{1}{5}\)
    Zoezi\(\PageIndex{23}\)

    Jaza chati kwa kila namba iliyotolewa:

    Idadi Kinyume Thamani kamili kurudisha nyuma
    \(- \dfrac{5}{8}\)      
    \(\dfrac{1}{4}\)      
    \(\dfrac{8}{3}\)      
    \(-8\)      
    Jibu
    Idadi Kinyume Thamani kamili kurudisha nyuma
    \(-\dfrac{5}{8}\) \(\dfrac{5}{8}\) \(\dfrac{5}{8}\) \(-\dfrac{8}{5}\)
    \(\dfrac{1}{4}\) \(-\dfrac{1}{4}\) \(\dfrac{1}{4}\) \(4\)
    \(\dfrac{8}{3}\) \(-\dfrac{8}{3}\) \(\dfrac{8}{3}\) \(\dfrac{3}{8}\)
    \(-8\) \(8\) \(8\) \(-\dfrac{1}{8}\)
    Zoezi\(\PageIndex{24}\)

    Jaza chati kwa kila namba iliyotolewa:

    Idadi Kinyume Thamani kamili kurudisha nyuma
    \(- \dfrac{4}{7}\)      
    \(\dfrac{1}{8}\)      
    \(\dfrac{9}{4}\)      
    \(-1\)      
    Jibu
    Idadi Kinyume Thamani kamili kurudisha nyuma
    \(-\dfrac{4}{7}\) \(\dfrac{4}{7}\) \(\dfrac{4}{7}\) \(- \dfrac{7}{4}\)
    \(\dfrac{1}{8}\) \(-\dfrac{1}{8}\) \(\dfrac{1}{8}\) \(8\)
    \(\dfrac{9}{4}\) \(-\dfrac{9}{4}\) \(\dfrac{9}{4}\) \(\dfrac{4}{9}\)
    \(-1\) \(1\) \(1\) \(-\dfrac{1}{1}\)

    Gawanya FRACTIONS

    Kwa nini\(12 ÷ 3 = 4\)? Sisi hapo awali tulielezea hili kwa counters. Ni makundi ngapi ya\(3\) counters yanaweza kufanywa kutoka kwa kikundi cha\(12\) counters?

    Ovals nne nyekundu zinaonyeshwa. Ndani ya kila mviringo ni duru tatu za kijivu.

    Kielelezo\(\PageIndex{2}\)

    Kuna\(4\) makundi ya\(3\) counters. Kwa maneno mengine, kuna\(3\) s nne katika\(12\). Kwa hiyo,\(12 ÷ 3 = 4\).

    Nini kuhusu kugawa sehemu ndogo? Tuseme tunataka kupata quotient:\(\dfrac{1}{2} \div \dfrac{1}{6}\). Tunahitaji kufikiri\(\dfrac{1}{6}\) wangapi s kuna katika\(\dfrac{1}{2}\). Tunaweza kutumia tiles sehemu kwa mfano mgawanyiko huu. Tunaanza kwa kuunganisha tiles nusu na sehemu ya sita kama inavyoonekana kwenye Kielelezo\(\PageIndex{3}\). Taarifa, kuna\(\dfrac{1}{6}\) tiles tatu katika\(\dfrac{1}{2}\), hivyo\(\dfrac{1}{2} \div \dfrac{1}{6} = 3\).

    Mstatili unaonyeshwa, ulioandikwa kama nusu moja. Chini ni mstatili unaofanana umegawanyika katika vipande vitatu sawa, kila kinachoitwa kama moja ya sita.

    Kielelezo\(\PageIndex{3}\)

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

    Mfano:\(\dfrac{1}{4} \div \dfrac{1}{8}\).

    Suluhisho

    Tunataka kuamua jinsi wengi\(\dfrac{1}{8}\) s ni katika\(\dfrac{1}{4}\). Anza na\(\dfrac{1}{4}\) tile moja. Weka\(\dfrac{1}{8}\) tiles chini ya\(\dfrac{1}{4}\) tile.

    Zoezi\(\PageIndex{25}\)

    Mfano:\(\dfrac{1}{3} \div \dfrac{1}{6}\).

    Jibu

    Ex 4.2.25.png

    Zoezi\(\PageIndex{26}\)

    Mfano:\(\dfrac{1}{2} \div \dfrac{1}{4}\).

    Jibu

    Ex 4.2.26.png

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

    Mfano:\(2 ÷ \dfrac{1}{4}\).

    Suluhisho

    Tunajaribu kuamua\(\dfrac{1}{4}\) wangapi s kuna katika\(2\). Tunaweza mfano huu kama inavyoonekana.

    Mstatili mbili zinaonyeshwa, kila kinachoitwa kama 1. Chini ni mstatili wawili unaofanana, kila mgawanyiko katika vipande vinne. Kila moja ya vipande nane ni kinachoitwa kama moja ya nne.

    Kwa sababu kuna nane\(\dfrac{1}{4}\) s katika\(2\),\(2 ÷ \dfrac{1}{4} = 8\).

    Zoezi\(\PageIndex{27}\)

    Mfano:\(2 ÷ \dfrac{1}{3}\)

    Jibu

    Ex 4.2.27.png

    Zoezi\(\PageIndex{28}\)

    Mfano:\(3 ÷ \dfrac{1}{2}\)

    Jibu

    Ex 4.2.28.png

    Hebu tumia pesa kwa mfano\(2 ÷ \dfrac{1}{4}\) kwa njia nyingine. Sisi mara nyingi kusoma\(\dfrac{1}{4}\) kama 'robo', na tunajua kwamba robo ni moja ya nne ya dola kama inavyoonekana katika Kielelezo\(\PageIndex{4}\). Kwa hiyo tunaweza kufikiria\(2 ÷ \dfrac{1}{4}\) kama, “Ni robo ngapi zipo katika dola mbili?” Dola moja ni\(4\) robo, hivyo\(2\) dola itakuwa\(8\) robo. Hivyo tena,\(2 ÷ \dfrac{1}{4} = 8\).

    Picha ya robo ya Marekani imeonyeshwa.

    Takwimu: Sarafu ya\(\PageIndex{4}\) Marekani inayoitwa robo ni ya thamani ya moja ya nne ya dola.

    Kutumia tiles sehemu, sisi ilionyesha kuwa\(\dfrac{1}{2} \div \dfrac{1}{6} = 3\). Kumbuka kwamba\(\dfrac{1}{2} \cdot \dfrac{6}{1} = 3\) pia. Ni jinsi gani\(\dfrac{1}{6}\) na\(\dfrac{6}{1}\) kuhusiana? Wao ni wafuasi. Hii inatuongoza kwenye utaratibu wa mgawanyiko wa sehemu.

    ufafanuzi: Sehemu Division

    Kama\(a, b, c,\) na\(d\) ni idadi ambapo\(b ≠ 0\),\(c ≠ 0\), na\(d ≠ 0\), basi

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

    Ili kugawanya sehemu ndogo, kuzidisha sehemu ya kwanza kwa usawa wa pili.

    Tunahitaji kusema\(b ≠ 0\),\(c ≠ 0\) na kuwa na\(d ≠ 0\) uhakika hatuwezi kugawanya kwa sifuri.

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

    Gawanya, na uandike jibu kwa fomu rahisi:\ (\ dfrac {2} {5}\ div\ kushoto (-\ dfrac {3} {7}\ kulia).

    Suluhisho

    Panua sehemu ya kwanza kwa usawa wa pili. \(\dfrac{2}{5} \left(- \dfrac{7}{3}\right) \)
    Kuzidisha. Bidhaa hiyo ni hasi. \(- \dfrac{14}{15}\)
    Zoezi\(\PageIndex{29}\)

    Gawanya, na uandike jibu kwa fomu rahisi:\(\dfrac{3}{7} \div \left(− \dfrac{2}{3}\right)\).

    Jibu

    \(-\dfrac{9}{14}\)

    Zoezi\(\PageIndex{30}\)

    Gawanya, na uandike jibu kwa fomu rahisi:\(\dfrac{2}{3} \div \left(− \dfrac{7}{5}\right)\).

    Jibu

    \(-\dfrac{10}{21}\)

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

    Gawanya, na uandike jibu kwa fomu rahisi:\(\dfrac{2}{3} \div \dfrac{n}{5}\).

    Suluhisho

    Panua sehemu ya kwanza kwa usawa wa pili. \(\dfrac{2}{3} \div \dfrac{5}{n} \)
    Kuzidisha. \(\dfrac{10}{3n}\)
    Zoezi\(\PageIndex{31}\)

    Gawanya, na uandike jibu kwa fomu rahisi:\(\dfrac{3}{5} \div \dfrac{p}{7}\).

    Jibu

    \(\dfrac{21}{5p}\)

    Zoezi\(\PageIndex{32}\)

    Gawanya, na uandike jibu kwa fomu rahisi:\(\dfrac{5}{8} \div \dfrac{q}{3}\).

    Jibu

    \(\dfrac{15}{8q}\)

    Mfano\(\PageIndex{17}\): divide

    Gawanya, na uandike jibu kwa fomu rahisi:\(− \dfrac{3}{4} \div \left(− \dfrac{7}{8}\right)\).

    Suluhisho

    Panua sehemu ya kwanza kwa usawa wa pili. \(- \dfrac{3}{4} \cdot \left(- \dfrac{8}{7}\right) \)
    Kuzidisha. Kumbuka kuamua ishara kwanza. \(\dfrac{3 \cdot 8}{4 \cdot 7}\)
    Andika upya ili kuonyesha mambo ya kawaida. \(\dfrac{3 \cdot \cancel{4} \cdot 2}{\cancel{4} \cdot 7} \)
    Ondoa mambo ya kawaida na kurahisisha. \(\dfrac{6}{7} \)
    Zoezi\(\PageIndex{33}\)

    Gawanya, na uandike jibu kwa fomu rahisi:\(− \dfrac{2}{3} \div \left(− \dfrac{5}{6}\right)\).

    Jibu

    \(\dfrac{4}{5}\)

    Zoezi\(\PageIndex{34}\)

    Gawanya, na uandike jibu kwa fomu rahisi:\(− \dfrac{5}{6} \div \left(− \dfrac{2}{3}\right)\).

    Jibu

    \(\dfrac{5}{4}\)

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

    Gawanya, na uandike jibu kwa fomu rahisi:\(\dfrac{7}{18} \div \dfrac{14}{27}\).

    Suluhisho

    Panua sehemu ya kwanza kwa usawa wa pili. \(\dfrac{7}{18} \cdot \dfrac{27}{14} \)
    Kuzidisha. \(\dfrac{7 \cdot 27}{18 \cdot 14} \)
    Andika upya kuonyesha mambo ya kawaida. \(\dfrac{\cancel{\textcolor{red}{7}} \cdot \cancel{\textcolor{red}{9}} \cdot 3}{\cancel{\textcolor{red}{9}} \cdot \cancel{\textcolor{red}{7}} \cdot 2}\)
    Ondoa mambo ya kawaida. \(\dfrac{3}{2 \cdot 2} \)
    Kurahisisha. \(\dfrac{3}{4} \)
    Zoezi\(\PageIndex{35}\)

    Gawanya, na uandike jibu kwa fomu rahisi:\(\dfrac{7}{27} \div \dfrac{35}{36}\).

    Jibu

    \(\dfrac{4}{15}\)

    Zoezi\(\PageIndex{36}\)

    Gawanya, na uandike jibu kwa fomu rahisi:\(\dfrac{5}{14} \div \dfrac{15}{28}\).

    Jibu

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

    Dhana muhimu

    • Sawa FRACTIONS Mali
    • Ikiwa\(a, b, c\) ni namba wapi\(b\neq 0, c\neq 0\), basi\(\dfrac{a}{b} = \dfrac{a\cdot c}{b\cdot c}\) na\(\dfrac{a\cdot c}{b\cdot c} = \dfrac{a}{b}\)
    • Kurahisisha sehemu.
      1. Andika upya nambari na denominator ili kuonyesha mambo ya kawaida. Ikiwa inahitajika, fikiria nambari na denominator katika idadi kubwa.
      2. Kurahisisha, kwa kutumia sehemu sawa mali, kwa kuondoa mambo ya kawaida.
      3. Panua mambo yoyote iliyobaki.
    • Kuzidisha sehemu
      • Kama\(a, b, c,\) na
    • kurudisha nyuma
      • Nambari na usawa wake una bidhaa ya 1. \(\frac{a}{b} \cdot \frac{b}{a} = 1\)
      • Kinyume Thamani kamili kurudisha nyuma
        ina ishara kinyume kamwe hasi ina ishara sawa, sehemu inverts
    • Sehemu Idara
      • Kama\(a, b, c,\) na\(d\) ni idadi ambapo\(b\neq 0\),\(c\neq 0\), na\(d\neq 0\), basi\(\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b}\cdot \dfrac{d}{c}\)
      • Ili kugawanya sehemu ndogo, kuzidisha sehemu ya kwanza kwa usawa wa pili.

    faharasa

    kurudisha nyuma

    Upeo wa sehemu\(\dfrac{a}{b}\) ni\(\dfrac{b}{a}\) wapi\(a\neq 0\) na\(b\neq 0\).

    sehemu kilichorahisishwa

    Sehemu inachukuliwa kuwa rahisi ikiwa hakuna mambo ya kawaida katika nambari na denominator.

    Mazoezi hufanya kamili

    Kurahisisha Fractions

    Katika mazoezi yafuatayo, kurahisisha kila sehemu. Usibadili sehemu yoyote isiyofaa kwa namba zilizochanganywa.

    1. \(\dfrac{7}{21}\)
    2. \(\dfrac{8}{24}\)
    3. \(\dfrac{15}{20}\)
    4. \(\dfrac{12}{18}\)
    5. \(- \dfrac{40}{88}\)
    6. \(- \dfrac{63}{99}\)
    7. \(- \dfrac{108}{63}\)
    8. \(- \dfrac{104}{48}\)
    9. \(\dfrac{120}{252}\)
    10. \(\dfrac{182}{294}\)
    11. \(- \dfrac{168}{192}\)
    12. \(- \dfrac{140}{224}\)
    13. \(\dfrac{11x}{11y}\)
    14. \(\dfrac{15a}{15b}\)
    15. \(− \dfrac{3x}{12y}\)
    16. \(− \dfrac{4x}{32y}\)
    17. \(\dfrac{14x^{2}}{21y}\)
    18. \(\dfrac{24a}{32b^{2}}\)

    Kuzidisha vipande

    Katika mazoezi yafuatayo, tumia mchoro wa mfano.

    1. \(\dfrac{1}{2} \cdot \dfrac{2}{3}\)
    2. \(\dfrac{1}{2} \cdot \dfrac{5}{8}\)
    3. \(\dfrac{1}{3} \cdot \dfrac{5}{6}\)
    4. \(\dfrac{1}{3} \cdot \dfrac{2}{5}\)

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

    1. \(\dfrac{2}{5} \cdot \dfrac{1}{3}\)
    2. \(\dfrac{1}{2} \cdot \dfrac{3}{8}\)
    3. \(\dfrac{3}{4} \cdot \dfrac{9}{10}\)
    4. \(\dfrac{4}{5} \cdot \dfrac{2}{7}\)
    5. \(− \dfrac{2}{3} \left(− \dfrac{3}{8}\right)\)
    6. \(− \dfrac{3}{4} \left(− \dfrac{4}{9}\right)\)
    7. \(- \dfrac{5}{9} \cdot \dfrac{3}{10}\)
    8. \(- \dfrac{3}{8} \cdot \dfrac{4}{15}\)
    9. \(− \dfrac{7}{12} \left(− \dfrac{8}{21}\right)\)
    10. \(\dfrac{5}{12} \left(− \dfrac{8}{15}\right)\)
    11. \(\left(− \dfrac{14}{15}\right) \left(\dfrac{9}{20}\right)\)
    12. \(\left(− \dfrac{9}{10}\right) \left(\dfrac{25}{33}\right)\)
    13. \(\left(− \dfrac{63}{84}\right) \left(- \dfrac{44}{90}\right)\)
    14. \(\left(− \dfrac{33}{60}\right) \left(- \dfrac{40}{88}\right)\)
    15. \(4 \cdot \dfrac{5}{11}\)
    16. \(5 \cdot \dfrac{8}{3}\)
    17. \(\dfrac{3}{7} \cdot 21n\)
    18. \(\dfrac{5}{6} \cdot 30m\)
    19. \(−28p \left(− \dfrac{1}{4}\right)\)
    20. \(−51q \left(− \dfrac{1}{3}\right)\)
    21. \(−8 \left(\dfrac{17}{4}\right)\)
    22. \(\dfrac{14}{5} (−15)\)
    23. \(−1 \left(− \dfrac{3}{8}\right)\)
    24. \((−1) \left(- \dfrac{6}{7}\right)\)
    25. \(\left(\dfrac{2}{3}\right)^{3}\)
    26. \(\left(\dfrac{4}{5}\right)^{2}\)
    27. \(\left(\dfrac{6}{5}\right)^{4}\)
    28. \(\left(\dfrac{4}{7}\right)^{4}\)

    Pata Recipurals Katika mazoezi yafuatayo, pata usawa.

    1. \(\dfrac{3}{4}\)
    2. \(\dfrac{2}{3}\)
    3. \(− \dfrac{5}{17}\)
    4. \(− \dfrac{6}{19}\)
    5. \(\dfrac{11}{8}\)
    6. -13
    7. 19-19
    8. -1
    9. 1
    10. Jaza chati.
      Kinyume Thamani kamili kurudisha nyuma
      \(- \dfrac{7}{11}\)      
      \(\dfrac{4}{5}\)      
      \(\dfrac{10}{7}\)      
      \(-8\)      
    11. Jaza chati.
      Kinyume Thamani kamili kurudisha nyuma
      \(- \dfrac{3}{13}\)      
      \(\dfrac{9}{14}\)      
      \(\dfrac{15}{7}\)      
      \(-9\)      

    Gawanya FRACTIONS

    Katika mazoezi yafuatayo, mfano kila mgawanyiko wa sehemu.

    1. \(\dfrac{1}{2} \div \dfrac{1}{4}\)
    2. \(\dfrac{1}{2} \div \dfrac{1}{8}\)
    3. \(2 \div \dfrac{1}{5}\)
    4. \(3 \div \dfrac{1}{4}\)

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

    1. \(\dfrac{1}{2} \div \dfrac{1}{4}\)
    2. \(\dfrac{1}{2} \div \dfrac{1}{8}\)
    3. \(\dfrac{3}{4} \div \dfrac{2}{3}\)
    4. \(\dfrac{4}{5} \div \dfrac{3}{4}\)
    5. \(- \dfrac{4}{5} \div \dfrac{4}{7}\)
    6. \(- \dfrac{3}{4} \div \dfrac{3}{5}\)
    7. \(− \dfrac{7}{9} \div \left(- \dfrac{7}{9}\right)\)
    8. \(− \dfrac{5}{6} \div \left(- \dfrac{5}{6}\right)\)
    9. \(\dfrac{3}{4} \div \dfrac{x}{11}\)
    10. \(\dfrac{2}{5} \div \dfrac{y}{9}\)
    11. \(\dfrac{5}{8} \div \dfrac{a}{10}\)
    12. \(\dfrac{5}{6} \div \dfrac{c}{15}\)
    13. \(\dfrac{5}{18} \div \left(- \dfrac{15}{24}\right)\)
    14. \(\dfrac{7}{18} \div \left(- \dfrac{14}{27}\right)\)
    15. \(\dfrac{7p}{12} \div \dfrac{21p}{8}\)
    16. \(\dfrac{5q}{12} \div \dfrac{15q}{8}\)
    17. \(\dfrac{8u}{15} \div \dfrac{12v}{25}\)
    18. \(\dfrac{12r}{25} \div \dfrac{18s}{35}\)
    19. \(-5 \div \dfrac{1}{2}\)
    20. \(-3 \div \dfrac{1}{4}\)
    21. \(\dfrac{3}{4} \div (-12)\)
    22. \(\dfrac{2}{5} \div (-10)\)
    23. \(−18 \div \left(− \dfrac{9}{2}\right)\)
    24. \(−15 \div \left(− \dfrac{5}{3}\right)\)
    25. \(\dfrac{1}{2} \div \left(- \dfrac{3}{4}\right) \div \dfrac{7}{8}\)
    26. \(\dfrac{11}{2} \div \dfrac{7}{8} \cdot \dfrac{2}{11}\)

    kila siku Math

    1. Baking Kichocheo cha cookies chip chokoleti wito kwa 3 4 kikombe kahawia Imelda anataka mara mbili mapishi.
      1. Ni kiasi gani cha sukari ya kahawia ambayo Imelda itahitaji? Onyesha hesabu yako. Andika matokeo yako kama sehemu isiyofaa na kama namba iliyochanganywa.
      2. Kupima vikombe kawaida kuja katika seti ya\(\dfrac{1}{8}, \dfrac{1}{4}, \dfrac{1}{3}, \dfrac{1}{2}\), na 1 kikombe. Chora mchoro wa kuonyesha njia mbili tofauti ambazo Imelda angeweza kupima sukari ya kahawia inayohitajika mara mbili ya mapishi.
    2. Baking Nina ni kufanya 4 sufuria ya fudge kutumikia baada ya recital muziki. Kwa kila sufuria, anahitaji kikombe cha 2 3 cha maziwa yaliyohifadhiwa.
      1. Ni kiasi gani cha maziwa kilichopunguzwa ambacho Nina atahitaji? Onyesha hesabu yako. Andika matokeo yako kama sehemu isiyofaa na kama namba iliyochanganywa.
      2. Kupima vikombe kawaida kuja katika seti ya\(\dfrac{1}{8}, \dfrac{1}{4}, \dfrac{1}{3}, \dfrac{1}{2}\), na 1 kikombe. Chora mchoro ili kuonyesha njia mbili tofauti ambazo Nina angeweza kupima maziwa yaliyopunguzwa anayohitaji.
    3. Sehemu Don kununuliwa mfuko wingi wa pipi kwamba weighs 5 paundi. Anataka kuuza pipi katika mifuko kidogo kwamba kushikilia\(\dfrac{1}{4}\) pauni. Ni mifuko ngapi ndogo ya pipi anaweza kujaza kutoka kwenye mfuko wa wingi?
    4. Sehemu Kristen ina\(\dfrac{3}{4}\) yadi ya Ribbon. Anataka kukata kwa sehemu sawa ili kufanya nyuzi za nywele kwa dolls 6 za binti yake. Je! Ribbon ya nywele za kila doll itakuwa muda gani?

    Mazoezi ya kuandika

    1. Eleza jinsi unavyopata usawa wa sehemu.
    2. Eleza jinsi unavyopata usawa wa sehemu hasi.
    3. Rafael alitaka kuagiza pizza nusu ya kati kwenye mgahawa. Mhudumu alimwambia kuwa pizza ya kati inaweza kukatwa katika vipande 6 au 8. Je, angependa vipande 3 kati ya 6 au 4 kati ya vipande 8? Rafael alijibu kwamba kwa kuwa hakuwa na njaa sana, angependelea vipande 3 kati ya 6. Eleza nini kibaya na mawazo ya Rafael.
    4. Kutoa mfano kutoka kwa maisha ya kila siku ambayo inaonyesha jinsi\(\dfrac{1}{2} \cdot \dfrac{2}{3}\) ilivyo\(\dfrac{1}{3}\).

    Self Check

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

    (b) Baada ya kuchunguza orodha hii, utafanya nini ili uwe na ujasiri kwa malengo yote?

    Wachangiaji na Majina