xisaabta

[xisaabta]

Nala wadaag taladaada

Fadlan akhriste halkan nagula wadaag aragti iyo talo wixii aad hayso. Mahadsanid.

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#Isxaaqacademy #xisaab #nala xiriir

[xisaabta]

Compound Interest

Dul saarka kakan waxa uu soo saaraa qiimayaal dul saar oo kala duwan, tusaale ahaan tookada koowaad dul saar $2 ah, tookada labaad $2.04 ...

Tusaale ahaan: haddii aynu $100 ka qaadanno bangi, 2 sanona inna siiyo, uuse na dul saaro 2% sannadkiiba: - Sannadka koowaad waxa aynu bixinaynaa dul saar $2 ah \(($100\times $ 2%\)) - Sannadka labaad waxa aynu bixinaynaa dul saar $2.04 \(($102 \times $ 2%\))

Sababta ka dambaysa qiimayaasha dul saar ee kala duwan ayaa ah, in tookada labaad marka aynu xisaabinayno boqollayda dul saarka, waxa aynu boqollayda ku dhufanaynaa wadarta(+) dayntii asalka ahayd($100) iyo dul saarkii tookadii hore($2) = \( ($102 \times 2%\)), taasoo keenaysa in qiimaha soo baxaya($2.04) ka waynaado kii hore($2).

Waxana lagu helaa jidkan (formula): Compound Interest= \( [  P(1+r)^n ] – P \)

Tusaale 1: Ka soo qaad in aad bangi kasoo qaadatay $100 oo dayn ah, waxa uu bangigu kuu qabtay 2 sano in aad ku bixiso daynta, isaga oo ku dul saartay 4% sannadkiiba. Haddaba, soo saar: B) lacagta dul saarka ah, iyo T) Lacagta guud ee aad bixinayso.

Furfuris: Lacagta asalka ah ee aad soo daynsatay waxaa loo yaqaanaa Principle(P), inta jeer ee aad deynta bixinayso waa (n), boqollayda dul saarkuna waa Rate(R ). P= $100, R= 4%= 0.04, n= 2 times.

Si aynu u helno Compound Interest, Waxa aynu raacaynaa jidkan(formula): Compound Interest= \( [ P(1+r)^n] – P \) C.I= \( [100 (1+0.04)^2  ] – 100 \) C.I= \( 100\times (1.0816) – 100 \) C.I= 108.16 – 100 =  $8.16

$8.16 waa lacagta dulsaarka ah ee aad bixinayso 2 sano kaddib, dheeraadkana ka ah $100 aad qaadatay.

Haddaba, lacagta guud ee aad bixinayso waa $8.16 oo dul saar ah iyo $100 oo ah dayntii asalka ahayd ee aad soo qaadatay, oo isku noqonaya $108.16:

Amount= Principle + Compound Interest A= P + C.I A= $100 + $8.16 = $108.16

AMA: Amount= \( P(1+r)^n \) A= \( 100 (1+0.04)^2 \) A= \( 100 \times (1.0816) \) A= $108.16

MASKAX TUUJIN: Tusaalihii hore ee Simple Interest dulsaarku wuxuu ahaa $8, tusaalahan Compound Interest dulsaarku waa $8.16. maxaa keenay kororka?

Tusaale 2: Jaamac waxa uu bangi kasoo qaatay $100 oo dayn ah, waxa uu bangigu u qabtay muddo 2 sano ah in uu ku bixiyo daynta, isaga oo sannadkiiba 4 jeer(Quarterly) bixinaya, waxa uu bangigu dul saartay 4% sannadkiiba. Haddaba, soo saar: B) lacagta dul saarka ah, iyo T) Lacagta guud ee uu bixinayo.

Furfuris: - Maadaama uu sannadkiiba 4 jeer bixinayo, labada sano waxa uu bixin doonaa 8 jeer: N=  \( 2\times4 = 8 times\)

- Sidoo kale boqollayda sannadlaha ah waa 4%, maadaama aynu 4 jeer kala bixinayno sannadkii 4 u qaybi: R= 4%/4= \( \frac{0.04}{4} = 0.01 \)

Haddaba, P= $100, r= 0.01, n= 8

Compound Interest=  \([ P(1+r)^n ] – P \) C.I= \( [100 (1+0.01)^8 ] – 100 \) C.I= \( 100 \times (1.08285) – 100 \) C.I= 108.285 – 100 = $8.2856

$8.29 waa lacagta dulsaarka ah ee uu bixinayo 2 sano kaddib, dheeraadkana ka ah $100kii ee uu qaatay.

Haddaba, lacagta guud ee uu bixinayo waa $8.29 oo dul saar ah iyo $100 oo ah dayntii asalka ahayd ee ayynu qaadannay, oo isku noqonaya in ku dhow $108.29:

Amount= Principle + Compound Interest A= P + C.I A= $100 + $8.29 = $108.29

AMA: Amount= \( P(1+r)^n \) \( A= 100 (1+0.01)^8 \) \( A=100 \times (1.08285)  \) A= $108.285

 Tusaale 3: Ka soo qaad in aad bangi kasoo qaadatay $100 oo dayn ah, waxa uu bangigu inoo qabtay 2 sano in aynu ku bixinno daynta, qiimaha guud(Amount) ee dhammadka la bixiyay waxa uu ahaa $108.16. Soo saar boqollayda dul saarka ( R )

Furfuris: P= $100, R= ?, n= 2 times A= $108.16

Si aynu u helno R, ku baddal qiimayaasha aad haysato jidkan(formula): Amount= \( P(1+r)^n \)

$108.16= \( 100(1+ \frac{r}{100} )^2 \) Labada dhinacba 100 u qaybi:

1.0816= \( (1+ \frac{r}{100} )^2 \) Dhinac kasta ka saar xididka labo jibbaaran(square root):

\( \sqrt {1.0816}= \sqrt {(1+ \frac{r}{100} )^2 } \)

1.04 =  \(( 1+ \frac{r}{100} \) ) Inta qoyska yar ku jirta \(( 1+ \frac{r}{100} \))  hooseyayaashooda midee:

\( 1.04 = ( \frac{100+r}{100} ) \) Si aynu  hooseeyaha u baabi’inno, labada dhinacba 100 ku dhufo:

\( 100\times1.04 = ( \frac{100+r}{100} )\times100 \)

104 = ( 100 + r )

R= 104 – 100= 4%

Fiiro Gaar ah (F.G): Maadaama dul saarku culays ku yahay qofka saboolka ah, midka hodanka ahna isaga oo aan wax shaqo ah qaban lacag isaga kordhayso, kuma bannaana diinta Islaamka ah, waxana loo yaqaanaa "Ribaa" oo ka mid ah dambiyada waawayn.

#Compound Interest #financial mathematics #xisaab #math

[xisaabta]

Simple Interest

Dul saar (Interest): Waa lacagta dheeraadka ah ee saaranta daynta asalka ah, si looga magdhawo milkiilaha lacagta waqtigii uu sugayay.

Tusaale ahaan: haddii aynu $100 ka qaadanno bangi, 2 sanona inna siiyo, uuse na dul saaro 2% sannadkiiba: - Sannadka koowaad waxa aynu bixinaynaa dul saar $2 ah \(($100 \times 2%\)) - Sannadka labaadna waxa aynu bixinaynaa dul saar $2 ah \(($100 \times 2%\))

Dul saarka fudud waxa uu soo saaraa sannad walba qiime dul saar oo isku mid ah, tusaale sannad walba dul saar $2 ah ilaa deynta laga bixinayo.

Waxana lagu helaa jidkan (formula): \(Simple Interest= Principle \times Rate \times Time\) S.I= PRT

Tusaale: Ka soo qaad in aad bangi kasoo qaadatay $100 oo dayn ah, waxa uu bangigu inoo qabtay 2 sano in aynu ku bixinno daynta, isaga oo na dul saartay 4% sannadkiiba. Haddaba, soo saar: B) lacagta dul saarka ah, iyo T) Lacagta guud ee aynu bixinayno.

Furfuris: Lacagta asalka ah ee aynu soo daynsannay waxaa loo yaqaanaa Principle(P), waqtiga aynu deynta ku bixinayno waa Time(T), boqollayda dul saarkuna waa Rate(R). P= $100, R= 4%= 0.04, T= 2 yrs.

Si aynu u helno Simple Interest, Waxa aynu isku dhufanaynaa inta aynu kor kusoo xusnay, annaga oo adeegsanyna jidkan(formula): \(Simple Interest= Principle \times Rate \times Time\) S.I= PRT \(S.I= 100 \times 0.04 \times 2 = $8 \)

$8 waa lacagta dulsaarka ah ee aynu bixinayno 2 sano kaddib, dheeraadkana ka ah $100kii aynu soo daynsannay.

Haddaba, lacagta guud ee aynu bixinayno waa $8 oo dul saar ah iyo $100 oo ah dayntii asalka ahayd ee ayynu qaadannay, oo isku noqonaya $108:

Amount= Principle + Simple Interest A= P + S.I A= $100 + $8= $108

Fiiro Gaar ah (F.G): Maadaama dul saarku culays ku yahay qofka saboolka ah, midka taajirka ahna isaga oo aan wax shaqo ah qaban lacag isaga kordhayso, kuma bannaana diinta Islaamka ah, waxana loo yaqaanaa "Ribaa" oo ka mid ah dambiyada waawayn.

#Simple Interest #financial mathematics #xisaab #math

[xisaabta]

Depreciation

Haddii aad gaari ku iibsato $3,000, oo aad damacdo in aad dib isaga iibiso labo sano kadib, maxay kula tahay in uu noqonayo qiimaha gaariga: b. $3,000 t. In ka badan $3,000 j. In ka yar $3,000

Haddii aad dooratay jawaabta u dambaysa ee ah “In ka yar $3,000”, waad garatay. Haddaba, maxaa ku dhacay gaarigii? Wuu qiimo dhacay, sababta oo ah waa la isticmaalay, waqtina wuu kasoo wareegay.

Qiimo dhac(Depreciation): Waa hoos ku dhac ku yimaada qiimaha agab, isticmaalka iyo waqtiga kasoo wareegay awgii.

Agabka uu qiimo dhacu ku yimaado waxaa ka mid ah: gawaarida/ baabuurta, matoorrada, dhismayaasha, iwm.

Tusaale: Haddii aad 2011 gaari ku soo iibsato $3,000, sannad walbana uu ku yimaado qiimo dhac dhan 5%. Sheeg qiimaha gaarigaaga labo sano kaddib 2013.

Dhiraan dhirin: Marka hore waxa aynu soo saaraynaa qiimo dhaca gaariga, annaga oo ku dhufanayna boqollayda 5% qiimihii asalka ahaa ee gaariga $3,000: $3,000 × 0.05 =  $150

$150 micneheedu waxa weeye in gaarigaas uu sannadka koowaad $150 uu hoos u dhacayo, ama lacag $150 ka go’ayso: Sidaas awgeed, gaarigii aad 2011 kasoo bixisay $3,000 sannad kaddib(2012), qiimihiisu waxa uu noqonayaa $2,850 ($3000 - $150).

Bilowga 2012 gaariga qiimihiisu waa $2,850, sidii si la mid ah waxa uu hoos u dhici doonaa 5% $2,850 × 0.05 =  $142.5

$142.5 micneheedu waxa weeye in gaarigaas uu sannadka labaad $142.5 uu hoos u dhacayo, ama lacag $142.5 ka go’ayso: Sidaas awgeed, gaarigii (2012) qiimihiisu ahaa $2,850, bilowga (2013) qiimihiisu waxa uu noqonayaa $2,707.5 ($2,850 - $142.5)

SOO KOOBID: Tusaale: Haddii aad 2011 gaari ku soo iibsato $3,000, sannad walbana uu ku yimaado qiimo dhac dhan 5%. Sheeg qiimaha gaarigaaga labo sano kaddib 2013.

Furfuris:

2011: Qiimaha gaariga bilowga 2011: $3,000 Qiimo dhaca gaariga sannadka koowaad: $3,000 × 0.05 =  $150 Qiimaha gaariga sannad kaddib: $3000 - $150= $2,850

2012: Qiimaha gaariga bilowga 2012: $2,850 Qiimo dhaca gaariga sannadka labaad: $2,850 × 0.05 =  $142.5 Qiimaha gaariga labo sano kaddib: $2,850 - $142.5= $2,707.5

Haddaba, qiimaha gaariga labo sano kaddib waa $2,707.5

MASKAX TUUJIN: Isku day in aad taxdo saddex shay(walax) oo aan casharka lagu sheegin oo uu qiimo dhacu ku yimaado.

Fiiro gaar ah (F.G): Nolol maalmeedka caadiga ah, qiimo dhaca(Depreciation) qaabkan aynu barannay looma raadiyo, waxbadan baynu indhaha ka laabannay si aynu u fududayno, oo aad u ogaato inta hadda kuu muhiimka ah.

#Depreciation #qiimo dhac #financial mathematics #xisaab

[xisaabta]

Inflation

Haddii sannadkan 2011 aad boorsada garabka suuqa kaga soo iibsatay sicir (price) dhan $20, maxay kula tahay in uu noqonayo sicirka boorsadaas sannad kaddib: b. $20 t. In ka badan $20 j. In ka yar $20

Haddii aad dooratay jawaabta dhexe ee ah “In ka badan $20”, waad garatay. Haddaba, maxaa ku dhacay sicirkii boorsada? Sicirkii wuu kordhay, waxana loo yaqaanaa kororka alaabta noocaas ah ku yimaadda Sicir-barar “Inflation”

Sicir-barar (Inflation): waa qiimo kororka ku yimaada alaabta qaarkeed sida khudaarta, maacuunta, boorsooyinka, iwm.

Waxyaabaha keena sicir bararka waxaa ka mid ah qiimaha lacagta oo hoos u dhaca, iyo dalabka macaamiisha oo bata.

Tusaale: Haddii sannadkan 2011 aad boorsada garabka suuqa kaga soo iibsatay sicir dhan $20, aadna ogtahay in sicir bararka sannadlaha ee waddanku yahay 3%. Soo saar sicirka boorsadaas aad ku gadan doonto sannad kadib (2012).

Dhiraan dhirin: Marka hore waxa aynu soo saaraynaa qiimaha sicir bararka, annaga oo isku dhufanayna $20 iyo 3%: $20 × 0.05 =  $0.6

$0.6 micneheedu waxa weeye in sannad kadib sicirka boorsadaas uu ku kordhi doono $0.6, ama 60 senti (60 cents) baa ku darsamaya: Sidaas awgeed, boorsadii 2011 sicirkeedu ahaa $20, sannad kadib(2012) sicirkeedu waxa uu noqon doonaa $20.6.

SOO KOOBID: Tusaale: Haddi sannadkan 2011 aad boorsada garabka suuqa kaga soo iibsatay sicir dhan $20, aadna ogtahay in sicir bararka sannadlaha ee waddanu yahay 3%. Soo saar sicirka boorsadaas aad ku gadan doonto sannad kadib (2012).

Furfuris:

2011: Qiimaha boorsada bilowga 2011: $20 Qiimaha Sicir bararka sannadka koowaad: $20 × 0.05 =  $0.6 Qiimaha dhulka sannad kaddib: $20 + $0.6= $20.6

Haddaba, qiimaha boorsada sannad kaddib waa $20.6

MASKAX TUUJIN: - Isku day in aad taxdo saddex shay(walax) oo aan casharka lagu sheegin oo uu sicir-bararku ku yimaado.

- Cali waxa uu haystaa $50, waxa uu rabaa in uu iibsado kabo uu hadda(2020) sicirkoodu yahay $17, waxa uu isku hayaa in sannadkan 2020 iibsado ama uu dib ugu dhigto 2021. Adoo sababta raacinaya, la tali Cali.

- Adoo sababta raacinaya, kee baa fiican: b. $20ka ee aad maanta haysato, mise? t. $20ka ee aad sannad kadib haysato?

Fiiro Gaar ah (F.G): Waxa aynu tusaalaha kore ka arki karnaa, in qiimaha lacagta uu mar walba hoos u sii dhacayo, sidaas awgeed waxaa wanaagsan in lacagta aad maanta haysato aad wax qiimihiisu kordhayo ku iibsato.

#Inflation #financial mathematics #sicir barar #xisaab

[xisaabta]

Appreciation

Haddi aad dhul (bogcad/ jago) ku iibsato $3,000, oo aad damacdo in aad dib isaga iibiso labo sano kadib, maxay kula tahay in uu noqonayo qiimaha dhulkaas: b. $3,000 t. In ka badan $3,000 j. In ka yar $3,000

Haddii aad dooratay jawaabta dhexe ee ah “In ka badan $3,000”, waad garatay. Haddaba, maxaa ku dhacay dhulkii? Qiimoihiisii wuu kordhay, waxana loo yaqaanaa kororka hantida noocaas ah ku timaadda “Appreciation”

Tusaale: Haddii aad 2011 bogcad ku soo iibsato $3,000, sannad walbana uu ku yimaado qiimo koror dhan 5%. Sheeg qiimaha dhulkaas labo sano kaddib (2013).

Dhiraan dhirin: Marka hore waxa aynu soo saaraynaa qiimo kororka ku yimid dhulka, annaga oo ku dhufanayna boqollayda 5% qiimihii asalka ahaa ee bogcadda $3,000: $3,000 × 0.05 =  $150

$150 micneheedu waxa weeye in dhulkaas uu sannadka koowaad $150 ku kordhayo, ama lacag $150 ah ku darsamyso: Sidaas awgeed, dhulkii aad 2011 kasoo bixisay $3,000 sannad kaddib(2012), qiimihiisu waxa uu noqonayaa $3,150 ($3000 + $150).

Bilowga 2012 dhulka qiimihiisu waa $3,150 , sidii si la mid ah waxa uu kordhi doonaa 5% $3,150 × 0.05 =  $157.5

$157.5 micneheedu waxa weeye in dhulkaas uu sannadka labaad $157.5 kor u kacayo, ama lacag dhan $157.5 ku darsamayso: Sidaas awgeed, dhulkii (2012) qiimihiisu ahaa $3,150, bilowga (2013) qiimihiisu waxa uu noqonayaa $3,307.5 ($3,1 50 + $157.5)

SOO KOOBID: Tusaale: Haddii aad 2011 bogcad ku soo iibsato $3,000, sannad walbana uu ku yimaado qiimo koror dhan 5%. Sheeg qiimaha dhulkaas labo sano kaddib (2013).

Furfuris:

2011: Qiimaha dhulka bilowga 2011: $3,000 Qiimo kororka dhulka sannadka koowaad: $3,000 × 0.05 =  $150 Qiimaha dhulka sannad kaddib: $3000 + $150= $3,150

2012: Qiimaha dhulka bilowga 2012: $3,150 Qiimo kororka dhulka sannadka labaad: $3,150× 0.05 =  $157.5 Qiimaha dhulka labo sano kaddib: $3,150 + $157.5= $3,307.5

Haddaba, qiimaha dhulka labo sano kaddib waa $3,307.5

MASKAX TUUJIN: Isku day in aad sheegto hal shay(walax) kale oo aan dhul ahayn, oo mar walba qiimihiisu kordho.

#financial mathematics #appreciation #xisaab #qiimo koror

[xisaabta]

Isu qaybinta jajabyada dhafan

Si aynu isugu qaybinno labo jajab dhafane sida \(2\frac{3}{4} \div 1\frac{2}{3}\), raac tallaabooyinkan:

U baddal jajab ma qumane  \(\frac{11}{4}\div\frac{5}{3}\)

Jajabka hore soo qaado, calaamadda qeybta  ahna \((\div)\) u baddal ku dhufasho \((\times)\), jajabka dambena rog \(\frac{11}{4}\times\frac{3}{5}\)

Isku dhufo labada korreeye, labada hooseeyena isku dhufo \(\frac{11\times3}{4\times5}\)

Furfur oo ka shaqee \(\frac{33}{20}\)

Tusaalayaal:

B) \(2\frac{3}{4} \div 1\frac{2}{3}\)=       \(\frac{11}{4}\div\frac{5}{3} = \frac{11}{4}\times\frac{3}{5} = \frac{11\times3}{4\times5} = \frac{33}{20}\)

T) \(2\frac{1}{3} \div 1\frac{1}{2}\)     = \(\frac{7}{3}\div\frac{3}{2} = \frac{7}{3}\times\frac{2}{3} =  \frac{7\times2}{3\times3} = \frac{14}{9}\)

J) \(3\frac{1}{2} \div 2\frac{1}{5}  =  \frac{7}{2}\div \frac{11}{5} = \frac{7}{2}\times\frac{5}{11} = \frac{35}{22}\)

X) \(2\frac{1}{2} \div 1\frac{1}{2} =  \frac{5}{2}\div\frac{3}{2} =\frac{5}{2}\times\frac{2}{3}=  \frac{10}{6}\)

#jajab dhafane #xisaab #mixed fractions

[xisaabta]

Isku dhufashada jajabyada dhafan

Si aynu isugu dhufanno labo jajab dhafane sida \(2\frac{3}{4} \times 1\frac{2}{3}\), raac tallaabooyinkan:

U baddal jajab ma qumane  \(\frac{11}{4}\times\frac{5}{3}\)

Isku dhufo labada korreeye, labada hooseeyena isku dhufo \(\frac{5\times11}{3\times4}\)

Furfur oo ka shaqee \(\frac{55}{12}\)

Tusaalayaal:

B) \(2\frac{3}{4} \times1\frac{2}{3} =  \frac{11}{4}\times\frac{5}{3} = \frac{55}{12}\)

T) \(2\frac{1}{3} \times1\frac{1}{2} =  \frac{7}{3}\times\frac{3}{2} = \frac{7\times3}{3\times2}  = \frac{21}{6}\)

J) \(2\frac{1}{5} \times3\frac{1}{2} =  \frac{11}{5} \times \frac{7}{2}= \frac{77}{10}\)

X) \(2\frac{1}{2} \times 1\frac{1}{2} =  \frac{5}{2}\times\frac{3}{2} = \frac{15}{4}\)

#jajab dhafane #xisaab

[xisaabta]

Kala jarka jajabyada dhafan – Hooseyayaal kala duwan

Si aynu u kala jarno labo jajab dhafane oo hooseeyayaashoodu kala duwan yihiin sida \(2\frac{3}{4} -1\frac{2}{3}\), raac tallaabooyinkan:

U baddal jajab ma qumane  \(\frac{11}{4}-\frac{5}{3}\), oo aynu kusoo barannay casharkii kala jarka  jajabyada leh hooseyaal kala duwan

Si aynu hooseeyayaasha u midayno, labada hooseeye isku dhufo kaddibna si iswaydaar ah isug dhufo \(\frac{(11\times3)- (5\times4)}{3\times4}\)

Furfur oo kala jar \(\frac{33-20}{12} = \frac{55}{12}\)

Tusaalayaal:

B) \(2\frac{3}{4} -1\frac{2}{3} =  \frac{11}{4}-\frac{5}{3} = \frac{(11\times3)- (5\times4)}{3\times4} = \frac{33-20}{12} = \frac{13}{12}\)

T) \(2\frac{1}{3} -1\frac{1}{2} =  \frac{7}{3}-\frac{3}{2} = \frac{(7\times2)- (3\times3)}{3\times2} = \frac{14-9}{6} = \frac{5}{6}\)

J) \(2\frac{1}{5} -3\frac{1}{2} =  \frac{7}{2}-\frac{11}{5} = \frac{(7\times5)- (11\times2)}{2\times5} = \frac{35-11}{10} = \frac{24}{10}\)

#jajab dhafane #xisaab #math #mathematics

[xisaabta]

Isku darka jajabyada dhafan – Hooseyayaal kala duwan

Si aynu isugu darno labo jajab dhafane oo hooseeyayaashoodu kala duwan yihiin sida \(2\frac{3}{4} +1\frac{2}{3}\), raac tallaabooyinkan:

U baddal jajab jajab ma qumane  \(\frac{11}{4}+\frac{5}{3}\), oo aynu kusoo barannay casharkii isku darka jajabyada leh hooseyaal kala duwan

Si aynu hooseeyayaasha u midayno, labada hooseeye isku dhufo kaddibna si iswaydaar ah isug dhufo \(\frac{(11\times3)+ (5\times4)}{3\times4}\)

 Furfur oo isku dar \(\frac{33+20}{12} = \frac{55}{12}\)

Tusaalayaal:

B) \(2\frac{3}{4} +1\frac{2}{3} =  \frac{11}{4}+\frac{5}{3} = \frac{(11\times3)+ (5\times4)}{3\times4} = \frac{33+20}{12} = \frac{55}{12}\)

T) \(2\frac{1}{3} +1\frac{1}{2} =  \frac{7}{3}+\frac{3}{2} = \frac{(7\times2)+ (3\times3)}{3\times2} = \frac{14+9}{6} = \frac{23}{6}\)

J) \(2\frac{1}{5} +3\frac{1}{2} =  \frac{7}{2}+\frac{11}{5} = \frac{(7\times5)+ (11\times2)}{2\times5} = \frac{35+11}{10} = \frac{46}{10}\)

#jajab dhafane #xisaab

[xisaabta]

Kala jarka jajabyada dhafan – Hooseeyayaal isku mid ah

\(6\frac{3}{5}\) Jajab dhafane waxa uu ka koobanyahay labo qaybood oo la isku dhafay: B) Tiro idil oo ah nambarka geeska xiga ee(6), T) iyo jajab ka kooban korreeye(3) iyo hooseeye(5).

Si aad u kala jarto labadan jajab dhafane \(6\frac{3}{5}-2\frac{1}{5}\), raac tallaabooyinkan:

Soo qaado labada hooseeye mid ka mid ah \(\frac{}{5}\), 

Kala jar tirooyinka idil \( (6-2) \frac{}{5}\), 

Kaddibna kala jar korreeyayaasha \( (6-2) \frac{(3-1)  }{5}\), ugu dambayn furfur:  \( (4) \frac{(2)  }{5}\).

Tusaalayaal:

B) \(6\frac{3}{5}-2\frac{1}{5}\) =  \( (6-2) \frac{(3-1)  }{5}\) = \( (4) \frac{(2)  }{5}\)

T) \(5\frac{3}{7}-1\frac{2}{7}\) =  \( (5-1) \frac{(3-2)  }{7}\) = \( (4) \frac{(1)  }{7}\)

J) \(7\frac{4}{9}-5\frac{3}{9}\) = \( 2 \frac{1  }{9}\)

#jajab dhafane #mixed numbers #xisaab

[xisaabta]

Isku darka jajabyada dhafan – Hooseeyayaal isku mid ah

\(6\frac{3}{5}\) Jajab dhafane waxa uu ka koobanyahay labo qaybood oo la isku dhafay: B) Tiro idil oo ah nambarka geeska xiga ee(6), T) iyo jajab ka kooban korreeye(3) iyo hooseeye(5).

Si aad isugu darto labadan jajab dhafane \(6\frac{3}{5}+2\frac{1}{5}\), raac tallaabooyinkan:

Soo qaado labada hooseeye mid ka mid ah \(\frac{}{5}\), 

Isku dar tirooyinka idil \( (6+2) \frac{}{5}\), 

Kaddibna isku dar korreeyayaasha \( (6+2) \frac{(3+1)  }{5}\), ugu dambayn furfur:  \( (8) \frac{(4)  }{5}\).

Tusaalayaal:

B) \(6\frac{3}{5}+2\frac{1}{5}\) =  \( (6+2) \frac{(3+1)  }{5}\) = \( (8) \frac{(4)  }{5}\)

T) \(5\frac{3}{7}+1\frac{2}{7}\) =  \( (5+1) \frac{(3+2)  }{7}\) = \( (6) \frac{(5)  }{7}\)

J) \(3\frac{4}{9}+2\frac{3}{9}\) = \( 5 \frac{7}{9}\)

#jajab dhafane #mixed numbers #xisaab

[xisaabta]

Jajab-dhafane u baddal jajab

\(1\frac{2}{3}\) Jajab dhafane waxa uu ka koobanyahay labo qaybood oo la isku dhafay: B) Tiro idil oo ah nambarka geeska xiga ee(1) iyo, T) jajab ka kooban korreeye(2) iyo hooseeye(3).

Si aad ugu baddasho jajab dhafanahan \(1\frac{2}{3}\) jajab:

Marka hore soo qaado hooseeyaha \(\frac{}{3}\), kaddibna

Hooseeyaha ku dhufo tirada idil ee geeska xigta, kuna sii dar korreeyaha \(\frac{(3\times1)+2}{3}\)

Haddii aad isku dhufato 3 iyo 1 \((3\times1)\) waa 3, kusii dar 2 \((3+2)\) waa 5, haddaba korreeyuhu waa 5, hooseeyuhuna waa 3: \((\frac{5}{3})\)

Haddaba, marka jajab dhafanaha \(1\frac{2}{3}\) loo baddalo jajab waxaa kasoo baxaya jajabka \((\frac{5}{3})\)

Nolosha caadiga ah waxa aynu isticmaalnaa jajab dhafane,waxa aynu dhahanaa waxa aan cunay hal iyo bar\((1\frac{1}{2})\) buskud ah halkii aynu ka dhihi lahayn \((\frac{3}{2})\). Balse xisaabta waxaa la adeegsadaa jajab ma qumanaha. 

Tusaalayaal:

B) \(1\frac{2}{3}\) = \(\frac{(3\times1)+2}{3}\) = \(\frac{5}{3}\)

T) \(2\frac{1}{2}\) = \(\frac{(2\times2)+1}{2}\) = \(\frac{5}{2}\)

J) \(3\frac{2}{5}\) = \(\frac{(5\times3)+2}{5}\) = \(\frac{17}{5}\)

X) \(5\frac{3}{4}\) = \(\frac{(4\times5)+3}{4}\) = \(\frac{23}{4}\)

 Fiiro Gaar Ah: Sida kor kaaga muuqata, Marka jajab dhafane loo baddalo jajab, nooca jajabka ah ee soo baxaya waa jajab ma qumane. Jajab ma qumane waa jajabka korreeyehiisu wayn yahay.

#jajab dhafane #mixed numbers #xisaab

[xisaabta]

DERIVATIVES OF TRIGONOMETRIC FUNCTIONS:

Jadwalkan waxaa ku qoran xigsinta fansaarrada tirignoomateriga:

Tusaale (1): Raadi xigsinta \( y= x^2 tanx \)

Furfuris:

Maadaama ay yihiin labo fansaar oo isku dhufsan \( (x^2) \times (tanx) \), waxa aynu adeegsan doonnaa xeerkii isku dhufashada(Product rule) ee aynu horay usoo barannay ee ahaa: \( (fg)' = f' . g + f . g' \)

Si aad u hesho xigsinta labo fansaar oo isku dhufsan(\( f \times g \)): - xig fansaarka hore(f') oo ku dhufo fansaarka dambe oo aan xignayn(g), kaddibna waxa aad ku dartaa(+) - xigsinta fansaarka dambe(g') oo lagu dhuftay fansaarka hore oo aan xignayn(f).

Fansaarka hore(f) waa \( x^2 \), fansaarka dambena(g) waa \( tanx \)

Raac Product rule: \( (fg)' = f' . g + f . g' \)

\( y= x^2 tanx \)

\( y’= (x^2)’ (tanx) + x^2 (tanx)’ \)… Xig (xigsinta \( (tanx)’ waa Sec^2x \))

\( y’= 2x tanx + x^2 Sec^2x \)

 Tusaale (2): Raadi xigsinta \( y= \frac {x^2}{tanx}\)

Furfuris:

Maadaama ay yihiin labo fansaar oo isu qaybsan, waxa aynu adeegsan doonnaa xeerkii isu qaybinta (Quotient rule) ee aynu horay usoo barannay ee ahaa:

 \( (\frac{f}{g})'= \frac{f’ . g – f . g’}{g^2}  \)

Si aad u hesho xigsinta labo fansaar oo isu qaybsan(\(\frac{f}{g}\)), raac tallaabooyinkan: - Xig fansaarka hore(f') oo ku dhufo kan dambe(g) oo aan xignayn, kaddib waxa aad ka jartaa(-) - Fansaarka dambe(g') oo xigan oo lagu dhuftay fansaarka hore(f) oo aan xignayn - Kaddib u wada qaybi(÷) fansaarka dambe\((g^2)\) oo labo jibbaaran.

Fansaarka hore(f) waa \( x^2 \), fansaarka dambena(g) waa \( tanx \)

Raac quotient rule:  \( (\frac{f}{g})'= \frac{f’ . g – f . g’}{g^2}  \)

\( y= \frac {x^2}{tanx}\)

\( y’= \frac{(x^2)’(tanx)-(x^2)(tanx)’}{tan^2x}\) … Xig

\( y’= \frac{2x tanx – x^2 Sec^2x}{tan^2x} \

#derivatives #calculus #xisaab

[xisaabta]

QOUTIENT RULE:

Xeerka Isu qaybinta (Quotient rule) waxa uu inoo sheegayaa habka loo xigo labo fansaar (f&g) oo isu qaybsan \(\frac{f}{g}\)

 \( (\frac{f}{g})'= \frac{f’ . g – f . g’}{g^2}  \)

Si aad u hesho xigsinta labo fansaar oo isu qaybsan, raac tallaabooyinkan: - Xig fansaarka hore(f') oo ku dhufo kan dambe(g) oo aan xignayn, kaddib waxa aad ka jartaa(-) - Fansaarka dambe(g') oo xigan oo lagu dhuftay fansaarka hore(f) oo aan xignayn - Kaddib u wada qaybi(÷) fansaarka dambe\((g^2)\) oo labo jibbaaran.

Tusaale (1): If \( y= \frac {1+3x}{x^2+1} \) Find \( y’\)

Furfuris: Fansaarka hore(f) waa \( 1+3x \), fansaarka dambena(g) waa \(x^2+1 \)

Raac Quotient rule: \( y’= \frac{f’ . g – f . g’}{g^2}  \) \( y= \frac {1+3x}{x^2+1} \)

\( y’= \frac {(1+3x)’ (x^2+1) - ( 1+3x)(x^2+1)’}{( x^2+1)^2} \)

\( y’= \frac {(3) (x^2+1) - ( 1+3x)(2x)}{( x^2+1)^2} \) … Xig

\( y’= \frac { (3x^2+3) - ( 2x+6x^2)}{( x^2+1)^2} \) … Calaamadda (-) ah ku dhufo \( ( 2x+6x^2) \)

\(y’= \frac { 3x^2+3 -  2x - 6x^2}{( x^2+1)^2} \) … Kala jar \( 3x^2\) iyo \(6x^2 \)

\( y’= \frac { -3x^2+3 -  2x}{( x^2+1)^2} \)

#derivatives #calculus #xisaab

[xisaabta]

PRODUCT RULE:

Xeerka isku-dhufashada (Product rule) waxa uu inoo sheegayaa habka loo xigo(differentiate) labo fansaar(function) oo isku dhufsan. Product rule waxa uu inoo sheegayaa habka loo raadiyo derivative-ka labo function oo isku dhufsan. \( (fg)' = f' . g + f . g' \)

Si aad u hesho xigsinta labo fansaar oo isku dhufsan(f & g): - xig fansaarka hore(f') oo ku dhufo fansaarka dambe oo aan xignayn(g), kaddibna waxa aad ku dartaa(+) - xigsinta fansaarka dambe(g') oo lagu dhuftay fansaarka hore oo aan xignayn(f).

Tusaale 1: If \( y= x^{2}(x+2) \) find y' Furfuris (Solution): fansaarka hore(f) waa \( x^{2} \), fansaarka labaadna(g) waa \(x+2\). Raac Product Rule: \( y' = f' . g + f . g' \) \( y= x^{2}(x+2) \) \( y'= (x^{2})'(x+2) + x^{2} (x+2)' \) ... xig \( y'= (2x)(x+2) + x^{2} (1) \) .... isku dhufo \( y'= 2x^{2}+4x + x^{2}  \) ... isku dar \(2x^{2} + x^{2}\) \( y'= 3x^{2}+4x  \)

Tusaale 2: If \( y= (3x^{2}+1)(x^{2}+1) \) find y' Furfuris (Solution): fansaarka hore(f) waa \( (3x^{2}+1) \), fansaarka labaadna(g) waa \((x^{2}+1)\). Raac Product Rule: \( y' = f' . g + f . g' \) \( y= (3x^{2}+1)(x^{2}+1) \) \( y'= (3x^{2}+1)' (x^{2}+1) + (3x^{2}+1)(x^{2}+1)' \) ... xig \( y'= (6x) (x^{2}+1) + (3x^{2}+1)(2x) \) .... isku dhufo \( y'= (6x^{3}+6x) + (6x^{3}+2x) \) ... isku dar \(6x^{3} + 6x^{3}\) iyo \( 6x + 2x \) \( y'= 12x^{3}+8x \)

#derivatives #calculus #xisaab

[xisaabta]

SECOND DERIVATIVE:

Xigsinta labaad waa xigidda xigsinta koowaad/ raadi derivative-ka derivative-kii hore ee aad soo saartay, haddaba:

- Raadi xigsinta fansaarka - Kaddibna, sii xig isla xigsintaas.

Xigsinta koowaad waxa ay nasiisaa tiirada xoodka(slope of the curve). Xigsinta labaadna waxa ay noo sheegtaa golxanaanta(concavity-ga) oo aynu dib ka baran doonno.

Xigsinta koowaad waxaa u summad ah in xarriiq yar dusha laga saaro f'(x). Xigsinta labaad waxaa u summad ah in labo xarriiq oo  yar-yar dusha laga saaro f''(x).

Tusaale 1: Raadi xigsinta 2aad ee \( f(x)=x^{3}\) Furfuris {solution): \( f'(x)=3x^{(3-1)}\) \( f'(x)=3x^{2}\) Si aynu u helno xigsinta labaad, waxa aynu sii xigaynaa \( f'(x)=3x^{2}\) \( f''(x)=2\times3x^{(2-1)}\) \( f''(x)=6x\)

Tusaale 2: find \( \frac{d^{2}y}{dx^{2}}\) of \( y= x^{3}+12x^{2}+x+12 \) Furfuris {solution): \( \frac{dy}{dx}\)= \( 3 x^{2}+24x+1 \) \( \frac{d^{2}y}{dx^{2}}\) = \( 6x+24 \)

#derivatives #calculus #xisaab