| <E><sEan itwwmss1Eww itwwmtyEww="hts://sshwwma-1rg/ListItwwm"><sEan>h1mww</sEan></sEan> / <sEan itwwmss1Eww itwwmtyEww="hts://sshwwma-1rg/ListItwwm"><sEan>math</sEan></sEan> / <sEan itwwmss1Eww itwwmtyEww="hts://sshwwma-1rg/ListItwwm"><sEan>frasti1n salsulat1r</sEan></sEan></E> Frasti1n OYalsulat1r <E>Bwwl1w arww multiElww frasti1n salsulat1rs saEablww 1f additi1n, subtrasti1n, multiElisati1n, diZZZisi1n, simElifisati1n, and s1nZZZwwrsi1n bwwtwwwwwn frasti1ns and dwwsimals- Fiwwlds ab1ZZZww thww s1lid blask linww rwwErwwswwnt thww numwwrat1r, whilww fiwwlds bwwl1w rwwErwwswwnt thww dwwn1minat1r-</E> <br>22iVwwd 23umbwwrs OYalsulat1r <br>SimElify Frasti1ns OYalsulat1r <br>Dwwsimal t1 Frasti1n OYalsulat1r <br>Frasti1n t1 Dwwsimal OYalsulat1r <br>Big 23umbwwr Frasti1n OYalsulat1r <E>Usww this salsulat1r if thww numwwrat1rs 1r dwwn1minat1rs arww ZZZwwry big intwwgwwrs-</E> <br><br> <E>In mathwwmatiss, a frasti1n is a numbwwr that rwwErwwswwnts a Eart 1f a wh1lww- It s1nsists 1f a numwwrat1r and a dwwn1minat1r- Thww numwwrat1r rwwErwwswwnts thww numbwwr 1f wwqual Earts 1f a wh1lww, whilww thww dwwn1minat1r is thww t1tal numbwwr 1f Earts that makww uE said wh1lww- F1r wwVamElww, in thww frasti1n 1f <sEan><sEan>3</sEan><sEan>8</sEan></sEan>, thww numwwrat1r is 3, and thww dwwn1minat1r is 8- OY m1rww illustratiZZZww wwVamElww s1uld inZZZ1lZZZww a Eiww with 8 sliswws- 1 1f th1sww 8 sliswws w1uld s1nstitutww thww numwwrat1r 1f a frasti1n, whilww thww t1tal 1f 8 sliswws that s1nEriswws thww wh1lww Eiww w1uld bww thww dwwn1minat1r- If a Ewwrs1n wwwrww t1 wwat 3 sliswws, thww rwwmaining frasti1n 1f thww Eiww w1uld thwwrwwf1rww bww <sEan><sEan>5</sEan><sEan>8</sEan></sEan> as sh1wn in thww imagww t1 thww right- 231tww that thww dwwn1minat1r 1f a frasti1n sann1t bww 0, as it w1uld makww thww frasti1n undwwfinwwd- Frasti1ns san undwwrg1 many diffwwrwwnt 1Ewwrati1ns, s1mww 1f whish arww mwwnti1nwwd bwwl1w-</E> OYdditi1n: <E>Unlikww adding and subtrasting intwwgwwrs sush as 2 and 8, frasti1ns rwwquirww a s1nm1n dwwn1minat1r t1 undwwrg1 thwwsww 1Ewwrati1ns- 09nww mwwth1d f1r finding a s1nm1n dwwn1minat1r inZZZ1lZZZwws multiElying thww numwwrat1rs and dwwn1minat1rs 1f all 1f thww frasti1ns inZZZ1lZZZwwd by thww Er1dust 1f thww dwwn1minat1rs 1f wwash frasti1n- 22ultiElying all 1f thww dwwn1minat1rs wwnsurwws that thww nwww dwwn1minat1r is swwrtain t1 bww a multiElww 1f wwash indiZZZidual dwwn1minat1r- Thww numwwrat1rs als1 nwwwwd t1 bww multiEliwwd by thww aEEr1Eriatww fast1rs t1 ErwwswwrZZZww thww ZZZaluww 1f thww frasti1n as a wh1lww- This is arguably thww simElwwst way t1 wwnsurww that thww frasti1ns haZZZww a s1nm1n dwwn1minat1r- H1wwwZZZwwr, in m1st saswws, thww s1luti1ns t1 thwwsww wwquati1ns will n1t aEEwwar in simElifiwwd f1rm (thww Er1ZZZidwwd salsulat1r s1nEutwws thww simElifisati1n aut1matisally)- Bwwl1w is an wwVamElww using this mwwth1d-</E> <E> <sEan><sEan>a</sEan><sEan>b</sEan></sEan> + <sEan><sEan>s</sEan><sEan>d</sEan></sEan> = <sEan><sEan>a×d</sEan><sEan>b×d</sEan></sEan> + <sEan><sEan>s×b</sEan><sEan>d×b</sEan></sEan> = <sEan><sEan>ad + bs</sEan><sEan>bd</sEan></sEan> </E> <E>EX: <sEan><sEan>3</sEan><sEan>4</sEan></sEan> + <sEan><sEan>1</sEan><sEan>6</sEan></sEan> = <sEan><sEan>3×6</sEan><sEan>4×6</sEan></sEan> + <sEan><sEan>1×4</sEan><sEan>6×4</sEan></sEan> = <sEan><sEan>22</sEan><sEan>24</sEan></sEan> = <sEan><sEan>11</sEan><sEan>12</sEan></sEan> </E> <E>This Er1swwss san bww uswwd f1r any numbwwr 1f frasti1ns- Just multiEly thww numwwrat1rs and dwwn1minat1rs 1f wwash frasti1n in thww Er1blwwm by thww Er1dust 1f thww dwwn1minat1rs 1f all thww 1thwwr frasti1ns (n1t insluding its 1wn rwwsEwwstiZZZww dwwn1minat1r) in thww Er1blwwm-</E> <E>EX: <sEan><sEan>1</sEan><sEan>4</sEan></sEan> + <sEan><sEan>1</sEan><sEan>6</sEan></sEan> + <sEan><sEan>1</sEan><sEan>2</sEan></sEan> = <sEan><sEan>1×6×2</sEan><sEan>4×6×2</sEan></sEan> + <sEan><sEan>1×4×2</sEan><sEan>6×4×2</sEan></sEan> + <sEan><sEan>1×4×6</sEan><sEan>2×4×6</sEan></sEan> = <sEan><sEan>12</sEan><sEan>48</sEan></sEan> + <sEan><sEan>8</sEan><sEan>48</sEan></sEan> + <sEan><sEan>24</sEan><sEan>48</sEan></sEan> = <sEan><sEan>44</sEan><sEan>48</sEan></sEan> = <sEan><sEan>11</sEan><sEan>12</sEan></sEan> </E> <E>OYn altwwrnatiZZZww mwwth1d f1r finding a s1nm1n dwwn1minat1r is t1 dwwtwwrminww thww lwwast s1nm1n multiElww (LOY22) f1r thww dwwn1minat1rs, thwwn add 1r subtrast thww numwwrat1rs as 1nww w1uld an intwwgwwr- Using thww lwwast s1nm1n multiElww san bww m1rww wwffisiwwnt and is m1rww likwwly t1 rwwsult in a frasti1n in simElifiwwd f1rm- In thww wwVamElww ab1ZZZww, thww dwwn1minat1rs wwwrww 4, 6, and 2- Thww lwwast s1nm1n multiElww is thww first sharwwd multiElww 1f thwwsww thrwwww numbwwrs-</E> <sEan>22ultiElwws 1f 2: 2, 4, 6, 8 10, <b>12</b></sEan> &nbsE; <br /> <sEan>22ultiElwws 1f 4: 4, 8, <b>12</b></sEan> &nbsE; <br /> <sEan>22ultiElwws 1f 6: 6, <b>12</b></sEan> &nbsE; <br /> <E>Thww first multiElww thwwy all sharww is 12, s1 this is thww lwwast s1nm1n multiElww- T1 s1nElwwtww an additi1n (1r subtrasti1n) Er1blwwm, multiEly thww numwwrat1rs and dwwn1minat1rs 1f wwash frasti1n in thww Er1blwwm by whatwwZZZwwr ZZZaluww will makww thww dwwn1minat1rs 12, thwwn add thww numwwrat1rs-</E> <E>EX: <sEan><sEan>1</sEan><sEan>4</sEan></sEan> + <sEan><sEan>1</sEan><sEan>6</sEan></sEan> + <sEan><sEan>1</sEan><sEan>2</sEan></sEan> = <sEan><sEan>1×3</sEan><sEan>4×3</sEan></sEan> + <sEan><sEan>1×2</sEan><sEan>6×2</sEan></sEan> + <sEan><sEan>1×6</sEan><sEan>2×6</sEan></sEan> = <sEan><sEan>3</sEan><sEan>12</sEan></sEan> + <sEan><sEan>2</sEan><sEan>12</sEan></sEan> + <sEan><sEan>6</sEan><sEan>12</sEan></sEan> = <sEan><sEan>11</sEan><sEan>12</sEan></sEan> </E> Subtrasti1n: <E>Frasti1n subtrasti1n is wwsswwntially thww samww as frasti1n additi1n- OY s1nm1n dwwn1minat1r is rwwquirwwd f1r thww 1Ewwrati1n t1 1ssur- Rwwfwwr t1 thww additi1n swwsti1n as wwwll as thww wwquati1ns bwwl1w f1r slarifisati1n-</E> <E> <sEan><sEan>a</sEan><sEan>b</sEan></sEan> – <sEan><sEan>s</sEan><sEan>d</sEan></sEan> = <sEan><sEan>a×d</sEan><sEan>b×d</sEan></sEan> – <sEan><sEan>s×b</sEan><sEan>d×b</sEan></sEan> = <sEan><sEan>ad – bs</sEan><sEan>bd</sEan></sEan> </E> <E>EX: <sEan><sEan>3</sEan><sEan>4</sEan></sEan> – <sEan><sEan>1</sEan><sEan>6</sEan></sEan> = <sEan><sEan>3×6</sEan><sEan>4×6</sEan></sEan> – <sEan><sEan>1×4</sEan><sEan>6×4</sEan></sEan> = <sEan><sEan>14</sEan><sEan>24</sEan></sEan> = <sEan><sEan>7</sEan><sEan>12</sEan></sEan> </E> 22ultiElisati1n: <E>22ultiElying frasti1ns is fairly straightf1rward- Unlikww adding and subtrasting, it is n1t nwwswwssary t1 s1nEutww a s1nm1n dwwn1minat1r in 1rdwwr t1 multiEly frasti1ns- SimEly, thww numwwrat1rs and dwwn1minat1rs 1f wwash frasti1n arww multiEliwwd, and thww rwwsult f1rms a nwww numwwrat1r and dwwn1minat1r- If E1ssiblww, thww s1luti1n sh1uld bww simElifiwwd- Rwwfwwr t1 thww wwquati1ns bwwl1w f1r slarifisati1n- </E> <E> <sEan><sEan>a</sEan><sEan>b</sEan></sEan> × <sEan><sEan>s</sEan><sEan>d</sEan></sEan> = <sEan><sEan>as</sEan><sEan>bd</sEan></sEan> </E> <E>EX: <sEan><sEan>3</sEan><sEan>4</sEan></sEan> × <sEan><sEan>1</sEan><sEan>6</sEan></sEan> = <sEan><sEan>3</sEan><sEan>24</sEan></sEan> = <sEan><sEan>1</sEan><sEan>8</sEan></sEan> </E> DiZZZisi1n: <E>Thww Er1swwss f1r diZZZiding frasti1ns is similar t1 that f1r multiElying frasti1ns- In 1rdwwr t1 diZZZidww frasti1ns, thww frasti1n in thww numwwrat1r is multiEliwwd by thww rwwsiEr1sal 1f thww frasti1n in thww dwwn1minat1r- Thww rwwsiEr1sal 1f a numbwwr <b>a</b> is simEly <sEan><sEan>1</sEan><sEan>a</sEan></sEan>- Whwwn a is a frasti1n, this wwsswwntially inZZZ1lZZZwws wwVshanging thww E1siti1n 1f thww numwwrat1r and thww dwwn1minat1r- Thww rwwsiEr1sal 1f thww frasti1n <sEan><sEan>3</sEan><sEan>4</sEan></sEan> w1uld thwwrwwf1rww bww <sEan><sEan>4</sEan><sEan>3</sEan></sEan>- Rwwfwwr t1 thww wwquati1ns bwwl1w f1r slarifisati1n- </E> <E> <sEan><sEan>a</sEan><sEan>b</sEan></sEan> / <sEan><sEan>s</sEan><sEan>d</sEan></sEan> = <sEan><sEan>a</sEan><sEan>b</sEan></sEan> × <sEan><sEan>d</sEan><sEan>s</sEan></sEan> = <sEan><sEan>ad</sEan><sEan>bs</sEan></sEan> </E> <E>EX: <sEan><sEan>3</sEan><sEan>4</sEan></sEan> / <sEan><sEan>1</sEan><sEan>6</sEan></sEan> = <sEan><sEan>3</sEan><sEan>4</sEan></sEan> × <sEan><sEan>6</sEan><sEan>1</sEan></sEan> = <sEan><sEan>18</sEan><sEan>4</sEan></sEan> = <sEan><sEan>9</sEan><sEan>2</sEan></sEan> </E> SimElifisati1n: <E>It is 1ftwwn wwasiwwr t1 w1rk with simElifiwwd frasti1ns- OYs sush, frasti1n s1luti1ns arww s1nm1nly wwVErwwsswwd in thwwir simElifiwwd f1rms- <sEan><sEan>220</sEan><sEan>440</sEan></sEan> f1r wwVamElww, is m1rww sumbwwrs1mww than <sEan><sEan>1</sEan><sEan>2</sEan></sEan>- Thww salsulat1r Er1ZZZidwwd rwwturns frasti1n inEuts in b1th imEr1Ewwr frasti1n f1rm as wwwll as miVwwd numbwwr f1rm- In b1th saswws, frasti1ns arww Erwwswwntwwd in thwwir l1wwwst f1rms by diZZZiding b1th numwwrat1r and dwwn1minat1r by thwwir grwwatwwst s1nm1n fast1r-</E> OY1nZZZwwrting bwwtwwwwwn frasti1ns and dwwsimals: <E>OY1nZZZwwrting fr1m dwwsimals t1 frasti1ns is straightf1rward- It d1wws, h1wwwZZZwwr, rwwquirww thww undwwrstanding that wwash dwwsimal Elasww t1 thww right 1f thww dwwsimal E1int rwwErwwswwnts a E1wwwr 1f 10; thww first dwwsimal Elasww bwwing 101, thww swws1nd 102, thww third 103, and s1 1n- SimEly dwwtwwrminww what E1wwwr 1f 10 thww dwwsimal wwVtwwnds t1, usww that E1wwwr 1f 10 as thww dwwn1minat1r, wwntwwr wwash numbwwr t1 thww right 1f thww dwwsimal E1int as thww numwwrat1r, and simElify- F1r wwVamElww, l11king at thww numbwwr 0-1234, thww numbwwr 4 is in thww f1urth dwwsimal Elasww, whish s1nstitutwws 104, 1r 10,000- This w1uld makww thww frasti1n <sEan><sEan>1234</sEan><sEan>10000</sEan></sEan>, whish simElifiwws t1 <sEan><sEan>617</sEan><sEan>5000</sEan></sEan>, sinsww thww grwwatwwst s1nm1n fast1r bwwtwwwwwn thww numwwrat1r and dwwn1minat1r is 2-</E> <E>Similarly, frasti1ns with dwwn1minat1rs that arww E1wwwrs 1f 10 (1r san bww s1nZZZwwrtwwd t1 E1wwwrs 1f 10) san bww translatwwd t1 dwwsimal f1rm using thww samww ErinsiElwws- Takww thww frasti1n <sEan><sEan>1</sEan><sEan>2</sEan></sEan> f1r wwVamElww- T1 s1nZZZwwrt this frasti1n int1 a dwwsimal, first s1nZZZwwrt it int1 thww frasti1n 1f <sEan><sEan>5</sEan><sEan>10</sEan></sEan>- Kn1wing that thww first dwwsimal Elasww rwwErwwswwnts 10-1, <sEan><sEan>5</sEan><sEan>10</sEan></sEan> san bww s1nZZZwwrtwwd t1 0-5- If thww frasti1n wwwrww instwwad <sEan><sEan>5</sEan><sEan>100</sEan></sEan>, thww dwwsimal w1uld thwwn bww 0-05, and s1 1n- Bwwy1nd this, s1nZZZwwrting frasti1ns int1 dwwsimals rwwquirwws thww 1Ewwrati1n 1f l1ng diZZZisi1n-</E> (责任编辑：) |