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4.5: Kuzidisha na Gawanya Hesabu Mchanganyiko na FRACTIONS Complex (Sehemu ya 1)

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    Malengo ya kujifunza
    • Kuzidisha na kugawanya idadi mchanganyiko
    • Tafsiri misemo kwa maneno na sehemu ndogo
    • Kurahisisha sehemu ndogo
    • Kurahisisha maneno yaliyoandikwa na bar ya sehemu
    kuwa tayari!

    Kabla ya kuanza, fanya jaribio hili la utayari.

    1. Gawanya na kupunguza, ikiwa inawezekana:\((4 + 5) ÷ (10 − 7)\). Ikiwa umekosa tatizo hili, kagua Mfano 3.2.8.
    2. Kuzidisha na kuandika jibu kwa fomu rahisi:\(\dfrac{1}{8} \cdot \dfrac{2}{3}\). Ikiwa umekosa tatizo hili, tathmini Mfano 4.2.7.
    3. \(2 \dfrac{3}{5}\)Badilisha kuwa sehemu isiyofaa. Kama amekosa tatizo hili, mapitio Mfano 4.1.11.

    Ongeza na Gawanya Hesabu Mchanganyiko

    Katika sehemu iliyopita, umejifunza jinsi ya kuzidisha na kugawanya sehemu ndogo. Mifano yote huko ilitumia sehemu ndogo au zisizofaa. Ni nini kinachotokea unapoulizwa kuzidisha au kugawanya namba zilizochanganywa? Kumbuka kwamba tunaweza kubadilisha nambari iliyochanganywa kwa sehemu isiyofaa. Na umejifunza jinsi ya kufanya hivyo katika Visualize Fractions.

    Mfano\(\PageIndex{1}\): multiply

    Kuzidisha:\(3 \dfrac{1}{3} \cdot \dfrac{5}{8}\)

    Suluhisho

    \(3 \dfrac{1}{3}\)Badilisha kwa sehemu isiyofaa. \(\dfrac{10}{3} \cdot \dfrac{5}{8}\)
    Kuzidisha. \(\dfrac{10 \cdot 5}{3 \cdot 8}\)
    Angalia mambo ya kawaida. \(\dfrac{\cancel{2} \cdot 5 \cdot 5}{3 \cdot \cancel{2} \cdot 4}\)
    Ondoa mambo ya kawaida. \(\dfrac{5 \cdot 5}{3 \cdot 4}\)
    Kurahisisha. \(\dfrac{25}{12}\)

    Kumbuka kwamba sisi kushoto jibu kama sehemu yasiyofaa\(\dfrac{25}{12}\),, na hakuwa na kubadilisha kwa idadi mchanganyiko. Katika algebra, ni vyema kuandika majibu kama sehemu zisizofaa badala ya namba zilizochanganywa. Hii inepuka machafuko yoyote iwezekanavyo kati\(2 \dfrac{1}{12}\) na\(2 \cdot \dfrac{1}{12}\).

    Zoezi\(\PageIndex{1}\)

    Kuzidisha, na kuandika jibu lako kwa fomu rahisi:\(5 \dfrac{2}{3} \cdot \dfrac{6}{17}\).

    Jibu

    \(2\)

    Zoezi\(\PageIndex{2}\)

    Kuzidisha, na kuandika jibu lako kwa fomu rahisi:\(\dfrac{3}{7} \cdot 5 \dfrac{1}{4}\).

    Jibu

    \(\dfrac{9}{4}\)

    JINSI YA: KUZIDISHA AU KUGAWANYA IDADI YA MCHANGANYIKO

    Hatua ya 1. Badilisha nambari zilizochanganywa kwa sehemu zisizofaa.

    Hatua ya 2. Fuata sheria za kuzidisha sehemu au mgawanyiko.

    Hatua ya 3. Kurahisisha kama inawezekana.

    Mfano\(\PageIndex{2}\):

    Kuzidisha, na kuandika jibu lako kwa fomu rahisi:\(2 \dfrac{4}{5} \left(− 1 \dfrac{7}{8}\right)\).

    Suluhisho

    Badilisha nambari zilizochanganywa kwa sehemu zisizofaa. \(\dfrac{14}{5} \left(-1 \dfrac{7}{8}\right)\)
    Kuzidisha. \(- \dfrac{14 \cdot 15}{5 \cdot 8}\)
    Angalia mambo ya kawaida. \(- \dfrac{\cancel{2} \cdot 7 \cdot \cancel{5} \cdot 3}{\cancel{5} \cdot \cancel{2} \cdot 4}\)
    Ondoa mambo ya kawaida. \(- \dfrac{7 \cdot 3}{4}\)
    Kurahisisha. \(- \dfrac{21}{4}\)
    Zoezi\(\PageIndex{3}\)

    Panua, na uandike jibu lako kwa fomu rahisi. \(5 \dfrac{5}{7} \left(− 2 \dfrac{5}{8}\right)\).

    Jibu

    \(-15\)

    Zoezi\(\PageIndex{4}\)

    Panua, na uandike jibu lako kwa fomu rahisi. \(-3 \dfrac{2}{5} \cdot 4 \dfrac{1}{6}\).

    Jibu

    \(-\dfrac{85}{6}\)

    Mfano\(\PageIndex{3}\): divide

    Gawanya, na uandike jibu lako kwa fomu rahisi:\(3 \dfrac{4}{7} ÷ 5\).

    Suluhisho

    Badilisha nambari zilizochanganywa kwa sehemu zisizofaa. \(\dfrac{25}{7} \div \dfrac{5}{1}\)
    Panua sehemu ya kwanza kwa usawa wa pili. \(\dfrac{25}{7} \cdot \dfrac{1}{5}\)
    Kuzidisha. \(\dfrac{25 \cdot 1}{7 \cdot 5}\)
    Angalia mambo ya kawaida. \(\dfrac{\cancel{5} \cdot 5 \cdot 1}{7 \cdot \cancel{5}}\)
    Ondoa mambo ya kawaida. \(\dfrac{5 \cdot 1}{7}\)
    Kurahisisha. \(\dfrac{5}{7}\)
    Zoezi\(\PageIndex{5}\)

    Gawanya, na uandike jibu lako kwa fomu rahisi:\(4 \dfrac{3}{8} ÷ 7\).

    Jibu

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

    Zoezi\(\PageIndex{6}\)

    Gawanya, na uandike jibu lako kwa fomu rahisi:\(2 \dfrac{5}{8} ÷ 3\).

    Jibu

    \(\dfrac{7}{8}\)

    Mfano\(\PageIndex{4}\): divide

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

    Suluhisho

    Badilisha nambari zilizochanganywa kwa sehemu zisizofaa. \(\dfrac{5}{2} \div \dfrac{5}{4}\)
    Panua sehemu ya kwanza kwa usawa wa pili. \(\dfrac{5}{2} \cdot \dfrac{4}{5}\)
    Kuzidisha. \(\dfrac{5 \cdot 4}{2 \cdot 5}\)
    Angalia mambo ya kawaida. \(\dfrac{\cancel{5} \cdot \cancel{2} \cdot 2}{\cancel{2} \cdot 1 \cdot \cancel{5}}\)
    Ondoa mambo ya kawaida. \(\dfrac{2}{1}\)
    Kurahisisha. \(2\)
    Zoezi\(\PageIndex{7}\)

    Gawanya, na uandike jibu lako kwa fomu rahisi:\(2 \dfrac{2}{3} \div 1 \dfrac{1}{3}\).

    Jibu

    \(2\)

    Zoezi\(\PageIndex{8}\)

    Gawanya, na uandike jibu lako kwa fomu rahisi:\(3 \dfrac{3}{4} \div 1 \dfrac{1}{2}\).

    Jibu

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

    Tafsiri Maneno kwa Maneno na Fractions

    Maneno quotient na uwiano mara nyingi hutumiwa kuelezea sehemu ndogo. Katika Ondoa Hesabu nzima, sisi defined quotient kama matokeo ya mgawanyiko. Quotient ya\(a\) na\(b\) ni matokeo ya kupata kutoka kugawa\(a\) na\(b\), au\(\dfrac{a}{b}\). Hebu tufanye kutafsiri maneno fulani katika maneno ya algebraic kwa kutumia maneno haya.

    Mfano\(\PageIndex{5}\): translate

    Tafsiri maneno katika usemi wa algebraic: “quotient ya\(3x\) na\(8\).”

    Suluhisho

    Keyword ni quotient; inatuambia kwamba operesheni ni mgawanyiko. Angalia maneno ya na kupata idadi ya kugawanya.

    Quotient ya\(3x\) na\(8\).

    Hii inatuambia kwamba tunahitaji kugawanya\(3x\) na\(8\). \(\dfrac{3x}{8}\)

    Zoezi\(\PageIndex{9}\)

    Tafsiri maneno katika usemi wa algebraic: quotient ya\(9s\) na\(14\).

    Jibu

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

    Zoezi\(\PageIndex{10}\)

    Tafsiri maneno katika kujieleza algebraic: quotient ya\(5y\) na\(6\).

    Jibu

    \(\dfrac{5y}{6}\)

    Mfano\(\PageIndex{6}\):

    Tafsiri maneno katika kujieleza kwa algebraic: quotient ya tofauti ya\(m\) na\(n\), na\(p\).

    Suluhisho

    Sisi ni kuangalia kwa quotient ya tofauti ya\(m\) na\(n\), na\(p\). Hii ina maana tunataka kugawanya tofauti ya\(m\) na\(n\) kwa\(p\).

    \[\dfrac{m − n}{p} \nonumber \]

    Zoezi\(\PageIndex{11}\)

    Tafsiri maneno katika kujieleza kwa algebraic: quotient ya tofauti ya\(a\) na\(b\), na\(cd\).

    Jibu

    \(\dfrac{a-b}{cd}\)

    Zoezi\(\PageIndex{12}\)

    Tafsiri maneno katika kujieleza kwa algebraic: quotient ya jumla ya\(p\) na\(q\), na\(r\).

    Jibu

    \(\dfrac{p+q}{r}\)

    Kurahisisha sehemu tata

    Kazi yetu na sehemu ndogo hadi sasa imejumuisha sehemu ndogo, sehemu zisizofaa, na namba zilizochanganywa. Aina nyingine ya sehemu inaitwa sehemu tata, ambayo ni sehemu ambayo namba au denominator ina sehemu. Baadhi ya mifano ya sehemu ndogo ni:

    \[\dfrac{\dfrac{6}{7}}{3} \quad \dfrac{\dfrac{3}{4}}{\dfrac{5}{8}} \quad \dfrac{\dfrac{x}{2}}{\dfrac{5}{6}} \nonumber \]

    Ili kurahisisha sehemu ngumu, kumbuka kwamba bar ya sehemu ina maana ya mgawanyiko. Hivyo sehemu tata\(\dfrac{\dfrac{3}{4}}{\dfrac{5}{8}}\) inaweza kuandikwa kama\(\dfrac{3}{4} \div \dfrac{5}{8}\).

    Mfano\(\PageIndex{7}\): simplify

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

    Suluhisho

    Andika upya kama mgawanyiko. \(\dfrac{3}{4} \div \dfrac{5}{8}\)
    Panua sehemu ya kwanza kwa usawa wa pili. \(\dfrac{3}{4} \cdot \dfrac{8}{5}\)
    Kuzidisha. \(\dfrac{3 \cdot 8}{4 \cdot 5}\)
    Angalia mambo ya kawaida. \(\dfrac{3 \cdot \cancel{4} \cdot 2}{\cancel{4} \cdot 5}\)
    Ondoa mambo ya kawaida na kurahisisha. \(\dfrac{6}{5}\)
    Zoezi\(\PageIndex{13}\)

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

    Jibu

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

    Zoezi\(\PageIndex{14}\)

    Kurahisisha:\(\dfrac{\dfrac{3}{7}}{\dfrac{6}{11}}\).

    Jibu

    \(\dfrac{11}{14}\)

    JINSI YA: KURAHISISHA SEHEMU NGUMU

    Hatua ya 1. Andika upya sehemu tata kama tatizo la mgawanyiko.

    Hatua ya 2. Fuata sheria za kugawa sehemu ndogo.

    Hatua ya 3. Kurahisisha kama inawezekana.

    Mfano\(\PageIndex{8}\): simplify

    Kurahisisha:\(\dfrac{− \dfrac{6}{7}}{3}\).

    Suluhisho

    Andika upya kama mgawanyiko. \(- \dfrac{6}{7} \div 3\)
    Panua sehemu ya kwanza kwa usawa wa pili. \(- \dfrac{6}{7} \cdot \dfrac{1}{3}\)
    Panua; bidhaa itakuwa hasi. \(- \dfrac{6 \cdot 1}{7 \cdot 3}\)
    Angalia mambo ya kawaida. \(- \dfrac{\cancel{3} \cdot 2 \cdot 1}{7 \cdot \cancel{3}}\)
    Ondoa mambo ya kawaida na kurahisisha. \(- \dfrac{2}{7}\)
    Zoezi\(\PageIndex{15}\)

    Kurahisisha:\(\dfrac{− \dfrac{8}{7}}{4}\).

    Jibu

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

    Zoezi\(\PageIndex{16}\)

    Kurahisisha:\(− \dfrac{3}{\dfrac{9}{10}}\).

    Jibu

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

    Mfano\(\PageIndex{9}\): simplify

    Kurahisisha:\(\dfrac{\dfrac{x}{2}}{\dfrac{xy}{6}}\).

    Suluhisho

    Andika upya kama mgawanyiko. \(\dfrac{x}{2} \div \dfrac{xy}{6}\)
    Panua sehemu ya kwanza kwa usawa wa pili. \(\dfrac{x}{2} \cdot \dfrac{6}{xy}\)
    Kuzidisha. \(\dfrac{x \cdot 6}{2 \cdot xy}\)
    Angalia mambo ya kawaida. \(\dfrac{\cancel{x} \cdot 3 \cdot \cancel{2}}{\cancel{2} \cdot \cancel{x} \cdot y}\)
    Ondoa mambo ya kawaida na kurahisisha. \(\dfrac{3}{y}\)
    Zoezi\(\PageIndex{17}\)

    Kurahisisha:\(\dfrac{\dfrac{a}{8}}{\dfrac{ab}{6}}\).

    Jibu

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

    Zoezi\(\PageIndex{18}\)

    Kurahisisha:\(\dfrac{\dfrac{p}{2}}{\dfrac{pq}{8}}\).

    Jibu

    \(\dfrac{4}{q}\)

    Mfano\(\PageIndex{10}\): simplify

    Kurahisisha:\(\dfrac{2 \dfrac{3}{4}}{\dfrac{1}{8}}\).

    Suluhisho

    Andika upya kama mgawanyiko. \(2 \dfrac{3}{4} \div \dfrac{1}{8}\)
    Badilisha nambari iliyochanganywa kwa sehemu isiyofaa. \(\dfrac{11}{4} \div \dfrac{1}{8}\)
    Panua sehemu ya kwanza kwa usawa wa pili. \(\dfrac{11}{4} \cdot \dfrac{8}{1}\)
    Kuzidisha. \(\dfrac{11 \cdot 8}{4 \cdot 1}\)
    Angalia mambo ya kawaida. \(\dfrac{11 \cdot \cancel{4} \cdot 2}{\cancel{4} \cdot 1}\)
    Ondoa mambo ya kawaida na kurahisisha. \(22\)
    Zoezi\(\PageIndex{19}\)

    Kurahisisha:\(\dfrac{\dfrac{5}{7}}{1 \dfrac{2}{5}}\).

    Jibu

    \(\dfrac{25}{49}\)

    Zoezi\(\PageIndex{20}\)

    Kurahisisha:\(\dfrac{\dfrac{8}{5}}{3 \dfrac{1}{5}}\).

    Jibu

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