The vvell spryng of sciences whiche teacheth the perfecte woorke and practise of arithmeticke, bothe in whole nombers and fractions, with suche easie and compendious instruction into the said arte, as hath not heretofore been by any sette out nor laboured. Beautified with moste necessary rules and questions, not onely profitable for marchauntes, but also for all artificers, as in the table doeth partlie appere: set forthe by Humfrey Baker citezeine of Lo[n]don.

¶ The seconde parte of Arithme∣ticke, whiche treateth of Fractions or bro∣ken nombers. (Book 2)

¶ The firste Chapiter treateth of Fractions, or broken nom∣bers, and the diffe∣rence thereof.

BRoken nomber is as muche as a parte or many partes of one, whereof there are two noumbers with a line betwene them bothe: That is to saie, the one whiche is aboue the line, is cal led the numerator. And the other vn∣derneath the line, is called the deno∣minator: as by example, three quar∣ters, whiche must bee set doune thus, 4: whereof. 3. whiche is the higher nō∣ber aboue the line, is called the nume∣rator, and. 4. whiche is vnder the line is called the denominator. And it is al waies conuenient, that the numera∣tor

be lesse in nomber, then the deno∣minator. For if the numerator, and the denominator were egall in value: then should thei represente a whole nomber, thus, as 1/1, 2/2, 3/3, whiche are whole nombers: by reason that the numerators of these, and al suche like maie bee diuided by their denomina∣tors, and their quotiētes will alwaies be but. 1. But in case that y^• numera∣tor do excede his denominator, then it is more then I whole: as 20/18, is more thā a whole nomber by 2/18, other diffinition dooeth not hereunto appertaine. Fur∣thermore it is to bée vnderstande, that the middest of all broken nombers, is the iuste halfe of. 1. whole, as 5/12, 7/14, 8/16, 9/18 and other like, are the halfes of one whole nomber, wherof doeth growe, and come forthe 2. progressions natu∣ral: the one progrediyng by augmen∣tyng, or encreasyng, as these.

½ ⅔ ¾ ⅘ ⅚ 6/7 ⅞ 8/9 9./10 &c.

And thei doe proceade infinitely, and wil neuer reche to make a whole nomber, thus 1/1. And the other pro∣gression, dooeth progrede by dimini∣shyng or decreasyng, as thus.

½ ⅓ ¼ ⅕ ⅙ 1/7 ⅛ 1/9 1/10 &c.

And these dooe proceade infinitely, and shall neuer come to make a. ○. whiche signifieth nothyng, but shall euer reiaine some certaine nomber whatsoeuer, whereby it doeth appere that broken nombers are infinite.

¶ The seconde Chapiter treateth of the reducyng or bringyng together of twoo nombers, or many bro∣ken dissemblyng, vnto one broken semblyng.

REduction, is as muche as to bryng together, or to put in sembleaunce, twoo or many noumbers dissem∣bling one from the other, in reducyng

them vnto a common denominator. For bicause the diuersitie and diffe∣rence of the broken numbers, doe come of the denominators part, or of diuers denominators, and for the vn∣derstanding hereof, there is a general rule whose operation is thus. Mul∣tiple the Denominators the one by the other, and so you shall haue a new denominator cōmon to al, the whiche denominator diuide by the perticuler denominators, and multiplye euerie quotiēt by his numerator and so you shall haue newe numerators, for the numbers whiche you woulde reduce, as appeareth by thys example follo∣wyng.

¶ Reduction in common denomination.

IF you wil reduce 3/3 and ⅘ toge∣ther, you must firste multiplie * 1.1 the. 2. denominators the one by the other, that is to saie 3. by 5. ma∣keth 15. which is your common deno∣minator,

that sette vnder the crosse, 〈 math 〉〈 math 〉 then deuide. 15. by the denominator 3 and you shal haue 5. whiche multiply by the numerator. 2. & you shall finde 10. sette that ouer the ⅔. and thei are 10/11 for the. ⅔. Afterwardes diuide. 15. by the denominator. 5. and thereof commeth 3. the whiche multiplie by the nume∣rator. 4. and you shall finde 12. whiche sette ouer the heade of the. ⅘. and thei make. 12/15. for the. ⅘: as appeareth more plainer in the margent.

If you will reduce ½, ⅔, ¾, ⅚, to∣gether, you must multiplie all the de∣nominators, the one by the other, that is to saie, 2. by. 3. maketh. 6. then. 6. by 4. and mounteth. 24. Laste of all. 24. by. 6. and thereof commeth. 144. for the common denominator. Then, for the firste diuide. 144. by the denomi∣nator. 2. and thereof commeth. 72. the whiche multiplie by the numerator 1.

and it is stil 72. set that ouer the ½ and it is 27/144, for the ½: Then deuide 144 by the second denominator 3. and ther∣of commeth 48: the whiche multiplie by the seconde numerator 2. and they are 96. whiche set ouer the ⅔ and they make 96/144, for the ⅔: Then diuide 144. by the thirde denominator 4. & thereof commeth 36. the whiche multiplie by the thirde numerator 3. and thei make 108. whiche sette ouer the ¾. and thei are 108/144 for the 5/4.

Finally diuide. 144. by the last de∣nominator. 6. and thereof commeth 24: the whiche multiplie by the laste numerator 5. and thereof commet 120 Whiche sette ouer the. ⅚. and thei are 120/144, for the ⅚, as appea∣reth here by pra∣tise.

¶ The example.

〈 math 〉〈 math 〉

¶ Reduction of broken nombers of broken.

IF you will reduce the broken * 1.2 of broken together, as thus, the ⅔ of ¼ of ⅘, you must mul∣tiplie the numerators the one by the other, to make one broken nō∣ber, of three broken nombers, that is to saie. 2. by. 1. maketh. 2. and then. 2. by 4. maketh 8. which is your numerator. Then 〈 math 〉〈 math 〉 multiplie the Denomi∣nators, the one by the o∣ther, that is to saie. 3. by 4. maketh. 12. and then

12. by. 5. maketh. 60. for your denomi∣nator, sette. 8. ouer. 60. with a line be∣twene them, and thei be. 1/60. whiche be∣yng abreuied, are. 2/15. and so muche are the ⅔. of. 5/4. of. ⅘. as appeareth in the margent.

¶ An other example of the same reduction, and of the se∣conde reduction.

IF you will reduce. ⅔. of. 5/4. of ⅘. the 5/4. of 5/7: and the ½. of the ½. of the. ⅔. of the. ⅓. Firste, it behoueth you of euery partie of the broken nōbers, to make of eche of them one broken, as by the third re∣duction is taughte: That is to saie, in multipliyng the numerators by nu∣merators, and denominators by de∣nominators: firste, for the firste parte, whiche is ⅔. of ¼. of ⅘, you muste as is said before, multiplie. 2 by. 1. and then by. 4. and you shall haue. 8. for the nu∣merator, likewise multiplie. 3. by. 4. and the producte by. 5. and you shall

haue. 60. for the denominator, so thei make. 8/60. whiche beyng abreuied, are 2/15 for the first parte, that is to saie, for the ⅔ of 〈◊〉〈◊〉 of ⅘, secondly for the ¾ of 5/7 multiplie likewise the numerator. 3. by. 5 maketh. 15. for the numerator, and multiplie. 4. by. 7. maketh. 28. for the denominator, and then thei bee. 15/28. for the seconde parte, that is to saie, for the ¾ of 5/7. Thirdly, for the ½ of ½ of ⅔ of ⅓. multiplie the numerators thone by the other, that is to saie. 1. by. 1. and then by. 2. and laste by. 1. and all ma∣keth but 2 for the numerator, likewise multiplie. 2. by. 2. maketh. 4. and 4. by 3. maketh. 12. and then. 12. by. 3. ma∣keth. 36. for the denominator, and thei are 2/36, whiche beyng abreuied maketh 1/18. for the thirde parte, that is to saie, for ½ of the ½. of ⅔ of ⅓. Last of al, take the 2/15 the 15/28. and the 1/18. and reduce them according to the order of the second re∣duction, and you shall finde 1000/7••60. for the 2/15. And 4050/••500 for the 15/28. And 426/7560 for 1/18: and thus are broken nombers of broken,

reduced, as appeareth by by practise. 〈 math 〉〈 math 〉

¶ Reduction of broken nombers, and the partes of bro∣ken together.

IF you will reduce ⅓. and the ½ of ⅓. together, to bryng them into one broken nomber, you muste firste sette doune the ⅓. and ½. as appeareth in the margent with a crosse be∣twene 〈 math 〉〈 math 〉 theim, and then multiply the twoo deno∣minators. The one by the other, that is to saie, 2. by 3. maketh. 6 set that vnder the crosse, then multiplie the firste numerator, one by the laste denominator twoo, and that maketh. 2. vnto the whiche

adde the laste numerator one, and thei be three, whiche set aboue your crosse, so you shall finde that the ⅕ and the ½ of ••/3. doe make 3/6. whiche beyng abre∣uied doeth make ⅓, which is as much as the. ⅓. and the. ½. of. ⅓. Likewise if you will reduce the ⅔, and the ¼. of ⅓. you must doe as before, set doune the ⅔ and ¼ with a crosse betwene thē, & then multiplie the twoo denominators, the one by the other, that is to saie. 3. by. 4 maketh. 12. whiche set vnder the crosse 〈 math 〉〈 math 〉 as you see in the margent, and then multiplie the firste numerator 2 by the last denominator. 4. and thereof com∣meth. 8. whereunto adde the laste nu∣merator. 1. and that maketh: 9. whiche sette ouer the crosse, so shall you finde that the ⅔. and the ¼ of ⅓ are worthe 9/••, whiche abreuied, dooe make ¼, as appeareth by exam∣ple in the margent.

¶ Reduction of whole nombers and broken together into a Fraction.

IF you will reduce whole nō∣ber * 1.3 with broken, you muste bryng the whole nomber in∣to brokē, as by this example maie appere: reduce 17. ⅝ into a brokē nomber, firste you must multiple the whole nomber. 17. by the denomina∣tor of the broken, whiche is. 8. in sai∣yng. 8. tymes. 17. doe masse. 136. vnto the whiche you muste adde the nume∣rator of. ⅚ whiche is. 5. and al amoun∣teth to. 141. whiche sette ouer. 8. with a line betwene theim, and thei will bee 141/8 so muche is. 17. ⅝. worthe in a frac∣tion, as it appeareth here by practise. 〈 math 〉〈 math 〉

In case you haue whole number and broken to bee reduced, with broken you muste bringe the whole number into his broken, in multiplyinge it by the denominator of the broken num∣ber going therwith, and adde there∣unto the numerator of the saide bro¦ken number, as in the laste example, and then reduce that broken number wyth the other broken, as here appe∣reth by this example. Reduce 10. ⅔ & 4/7 togither, first bring 10⅔ al into thirds, as by the syxt reduction, and you shal finde 〈◊〉〈◊〉, then reduce the 32/3 and 4/7 togi∣ther, by the fyrste reduction, and you shall finde 224/21 for the 32/3: and 〈◊〉〈◊〉 for 〈◊〉〈◊〉 as appereth here by practise. 〈 math 〉〈 math 〉

Also in case you haue in bothe partes of your Reduction, as well whole nū∣ber as broken, you must alwayes put the whole into their broken (as by the syxte reduction) of either part.

¶ Example.

If you wyll reduce 12. ¼ wyth 14. ⅔ to bringe them into one denomina∣tion, first bringe the 12. ¼ all into four∣thes, and you shall fynde 49/4: then like∣wise reduce 14. ⅔ all into thirdes, and you shall haue 4••/3, for the 14. ⅔, then reduce 〈◊〉〈◊〉 and 44/3 togither, by the order of the firste Reduction, and you shall fynde 〈◊〉〈◊〉 for the 〈◊〉〈◊〉. And 〈◊〉〈◊〉 for the 14. ⅔ as here by practise dothe plainlye ap∣pere. 〈 math 〉〈 math 〉

¶ The thirde Chapter treateth of ab∣breuiation of one greate broken number into a lesser broken.

ABbreuiation is asmuch as to set downe, or to write a brokē nūber by figures of lesse sig∣nification, & not diminishing y^• value thereof. The whiche to doe, there is a rule whose operation is thus, diuide the numerator and likewise the deno∣minator, by one whole number, the greatest y^t you maye fynde in the same broken number, and of the quotient of that numerator, make it the nume∣rator, and likewise of that of the de∣nominator, make it your denomina∣tor, as by example.

1. If you wyl abbreuiat 54/8••, you shal vnderstande that the greatest whole number that you maye take, by the which you maye diuide the numera∣tor & denominator is 27, which is the half of y^• numerator, & that is a whole number, for you can not take a whole

number oute of the denominator. 81. but that there will bee either more or lesse than a whole number, therefore if you diuide 54. by 27. you shall 〈 math 〉〈 math 〉 finde 2. for the nu∣merator, likewise if you diuide 81. by 27. you shall finde 3. for the denomi∣nator. then put. 2. ouer the 3. with a line betwene thē, and you shall find ⅔ and thus by this rule the 53/81 are a∣breuted vnto ⅔, as appeareth in the margent, and so is to be vnderstande all other.

¶ The forme & maner how to finde oute the greater number, by the which you mai wholy diuide, y^• nu∣merator & denominator (to thende y^t you may abreuiat them) is thus.

First, diuide the denominator by hys numerator, and if anye number doe remaine, let your diuisor be diui∣ded by the same number, and so you must continue vntill you haue so di∣uided y^• there maye nothing remaine, then is it to be vnderstande, that your last diuisor (wherat you did ende, and that o did remaine after your last di∣uision) is the greatest number, by the whiche you must abreuiat, as you did in the laste example, but in case that your last diuisor be 1. it is a token that the same nūber can not bee abreuied. Example, of 54/81 diuide 81. (whiche is the denomination) by 54. which is his numerator, and there resteth 27. then diuide 54. by 27. and there remaineth nothinge, wherefore your last diuisor 27. is the number, by the whiche you must abreuiat 54/81 as in the laste exam∣ple is specifyed.

¶ An other stile of abbreuiation.

2. Mediate the numerator, and al∣so the denominator of your fraction in case the noumbers be euen, that is to saye, take alwayes the halfe of the numerator, and likewise of the deno∣minator, and of the mediatiō or halfe of the numerator, make the nu∣merator, also of ½ the denominator, make your denominator, and so con∣tinue as often as you may in takinge alwayes the ½ of the numerator, and semblablie of the denominator, or els see if you may abbreuiate the num¦bers which doe remaine, by 3. by 4. by 5. 6. 7. 8. 9. or by 10. for you must ab∣breuiate them as often as you can by any of the saide numbers, and it is to bée noted, that with whatsoeuer num∣ber of these, you doe abbreuiate the Numerator of your Fraction, by the same you muste abbreuiate likewyse the Denominator, so continuynge vntil they can no more bee abbreuied. And it is to bee vnderstande that if the Numerator and the Denominator

be euen numbers, as you maye know when the fyrste fygure is an euen nūber, or a 〈◊〉〈◊〉, thē maye you perceaue if both the Numerator and the Deno∣minator may be abbreuied by 10. by 8 by 4. or by 2. although y^t some times they maye bee abbreuied by three.

And if they be odde numbers, then muste you consider if they maye bee abbreuied by 9. by 7 by. 5. or by 3: but when the first number, as well of the Numerator, as of the Denomina∣tor are euen numbers, then may you well knowe that suche numbers maye bee abbreuied be 2. as is afore∣saide. And if you adde the fygures of the Numerator togither, in su∣che manner as you doe in makynge the proofe by nyne in whole Num∣bers: that is, if you fynde 9. 〈◊〉〈◊〉 ap∣peareth that you maye abb〈…〉〈…〉 that number by 9. And lykewise by 3. and sometimes by 6. if you fynde 6. it maye bee abbreuied by 6. and

alwaies by 3. if you finde 3. it is a signe that you may abreuiate by 3. And by whatsoeuer nomber that you doe a∣breuiate the numerator, by the same must you abreuiate likewise the de∣nominator, and if the first figures of the same nomber be. 5. or 0. you maye abreuiate them by 5. but if the firste fygures be both 0. they may be abre∣uied by 10. in cutting awaye the twoo Cyphers thus, as 〈◊〉〈◊〉 whiche maketh 2/••, & sometimes by 100. thus, as 〈◊〉〈◊〉 in cutting away the foure ciphers af∣ter this sorte, 〈◊〉〈◊〉 and then the 100/200 doe make ½, and after this maner haue I set here diuers examples, althoughe that all numbers cannot be abreuied by this rule, that is to saye, all those whiche maye bee well abreuyed by the fyrste rule afore∣sayde.

〈 math 〉〈 math 〉

3. Furthermore you shall vnder∣stande that sometimes it happeneth, that all the fygures of the numera∣tor are egall vnto them of the deno∣minator, which when it so happeneth, you maye thē take one of them of the numerator, and also one of them of the denominator, and it shall bee a∣breuyed as 555/888, beinge abreuiated af∣ter this maner commeth to ⅝. And yet it happeneth sometimes, that two, or manye fygures of the numerator are proportioned vnto two, or many fy∣guree

of their denominators and the other fygures of the same number doe beholde the one the other in thys proportion? Then may you take twoo or many fygures, as well of the nu∣merator as of the denominator, and by this maner the same number shall bee abbreuied, as 4747/〈…〉〈…〉 whiche beinge abbreuied by this rule, do come to 〈…〉〈…〉.

4. Also it happeneth somtimes that you woulde abbreuiate one number vnto the semblaunce or likenesse of another. And for to knowe if the same maye by abbreuied, and also by what number it maye bee abbreuied, you must diuide the numerator of the one number, by the numerator of the o∣ther, and likewise the denominator of the one, by the denominator of the other, for in case that after euery di∣uision there doe remaine 0. and that the twoo quotiens be 〈◊〉〈◊〉 all, then is one of them the number by the whiche the saide fraction must be abbreuied, as by exāple of 11••/〈…〉〈…〉. I woulde knowe if

they maye be abbreuied vnto 5/9, and for to doe this, you must diuide 115. by 5. and you must diuide 207. by 9. and there will come into bothe the quoti∣ents 23. by the which it appeareth that this number may be abbreuied by 23. 〈 math 〉〈 math 〉

¶ The 4. Chapter treateth of the as∣sembling of two or many broken numbers togither, as by example.

FOr to adde broken numbers togither, there is a generall rule, which is thus, if the nū∣bers be vnlike y^t one to the other, you must reduce thē into a cōmō denomination, whiche after you haue reduced thē, you must then adde both y^• numerators togither, & set y^• product of the saide addition ouer the crosse, & diuide the same by the common deno∣minator, as by this exāple folowinge.

1. If you wyll adde ⅔ wyth ¾, you muste fyrst reduce the twoo fractions bothe into one denomination, accor∣dinge to the introduction of the fyrste reduction, that is to saye, in multi∣plyinge the denominator of the firste fraction whiche is 3, by the denomi∣nator of the other fraction whiche is foure, and they make 12. for your common de∣nominator, 〈 math 〉〈 math 〉 the which 12. set vnder the crosse, thē multiplie y^t fyrst nume∣rator 2. by the last deno∣minator 4. and thereof commeth 8. whiche sett o∣uer the ⅔, and then mul∣tiply y^• last numerator 3. by the fyrste denominator 3. and ther∣of commeth 9. whiche you must set o∣uer the ¾, then adde the numerator 8. with the numerator 9. & they make 17. which set ouer the crosse, and then your fraction wyll be 17/12 whiche is the addition of the ⅔ wyth ¾. And by∣cause

your numerator 17. is greater thā his denominator 12. therfore you must diuide 17. by 12. and thereof will come 1. and 5. remaining, which 5. are worth ••/12, and so muche are the ⅔ added with ¾ as doth appere.

¶ Addition in broken. numbers.

2. Also if you will adde ½, ⅔, ¾, ⅘, togither, you muste fyrste adde the 〈◊〉〈◊〉 and ⅔ together, accordynge to the doc∣trine of the laste rule, and you shall finde 7/6: then adde ¾ and ⅘ togither by the saide last chapter, and they make 〈◊〉〈◊〉. Then finally adde the 7/6 (whiche came of the ½ and ⅔ added together) with 31/20, & you shall fynde by the fore∣saide addition that they amounte vn∣to 326/120, Wherfore diuide 326. by 120. & therof cōmeth 2. and 86. remaineth whiche is 86/120 of one whole, & thei being abreuied do make 43/60: & thus the ½, ⅔, ¾, ⅘, added together doe amount to 2. 4••/••••,

as here vnder doth appere. 〈 math 〉〈 math 〉

¶ Addition of broken num∣bers of broken.

3. Furthermore, if you will adde the broken numbers of broken togi∣ther, as to adde the ⅔ of ¾ of ⅘ with the ⅚ of ½ of 〈◊〉〈◊〉: first you must reduce the nū∣bers accordyng to the order of y^• fourth reduction, in multiplying the nume∣rator of the fyrst 3. fractions, the one by the other, and of the producte make your numerator, & likewise you must multiplie y^• de nominators of the fore∣said thre fractions, the one by the other

and of the product make your deno∣minator, and you shal finde 24/60 for the first three broken numbers, which be∣ing abbreuied do make ⅖, then reduce the other 3. fractiōs, by the said fourth reduction, in multiplyinge the nume∣rators by numerators, & denomina∣tors, by denominators, as you did by the first 3. broken numbers, & you shal finde 25/96 then must you adde y^• ⅖ whiche came of the fyrste 3. broken numbers, & 25/56 whiche are of the last 3. fractions, both togither, by the instruction of y^• first additiō & you shall find 317/480 whiche cannot be abreuied, but is the product of y^• addition: so muche are the ⅔ of ¾ of ⅘ added with the ⅚ of 〈◊〉〈◊〉 of ⅝ as hereafter by practise doth euidently appere. 〈 math 〉〈 math 〉

¶ Addition of broken number the partes of broken together.

4. Likewise if you will addde the ½, and the ½ of ⅓ with the ⅘ and ••/4 of 〈◊〉〈◊〉, you must reduce the 11/32 by the fyft reduc∣tion and therof cōmeth 3/6 for the 〈◊〉〈◊〉 & 〈◊〉〈◊〉, of one of the saide thirdes, then reduce the ⅘ and ¼ by the saide fift reduction, and thereof commeth 〈◊〉〈◊〉.

Last of all adde the ⅚ and 17/20 togither according to the firste rule of addition, and you shall fynde 20••/〈…〉〈…〉 which beinge diuided bringeth 1. & 〈◊〉〈◊〉 part remai∣ning, whiche abrouied maketh 41/60 and thus you doe perceaue that the 2/1 & 〈◊〉〈◊〉 added with the ⅘ and ¼ doe amounte vnto 1. 〈◊〉〈◊〉 as hereafter by practise doth plainly appere. 〈 math 〉〈 math 〉

¶ Addition of whole nomber and broken together.

5. Also if you will adde. 12. ⅘. with 20. ⅚, you maie (if you will) adde. 1••. and. 20. together, and thei make. 32. and then adde the twoo broken nom∣bers together, that is to saie ⅘, and ⅚, by the order of the firste addition, and thei make 49/〈…〉〈…〉: therefore diuide 49. by 〈◊〉〈◊〉, and thereof commeth 1. and 〈◊〉〈◊〉 par∣tes remaine, whiche. 1. you must adde vnto the. ••2 and the whole additiō wil be. 33 〈◊〉〈◊〉, or otherwise, you maie reducte 12. ⅘. into the likenesse of a fraction by the sixt reduction, and thei will bee 〈◊〉〈◊〉, and likewise by thesame reduction, re∣duce. 2〈…〉〈…〉, and thei be 〈◊〉〈◊〉, then adde 〈◊〉〈◊〉 with the 〈◊〉〈◊〉, by the firste addition, and you shall finde. 〈◊〉〈◊〉. Therefore diuide ••••09. by. ••0, and thereof commeth 〈◊〉〈◊〉.

as before, and as by practise of the∣same bothe the waies, doeth here vn∣der appeare. 〈 math 〉〈 math 〉

¶ The .v. Chapiter treateth of Sub∣straction in broken nombers.

IF you will substracte ⅔| from 2/4 you must firste reduce bothe the fractions into a common denomination by the fyrst re∣duction, and you shall finde 8/12 for the ⅔, and 9/12 for the ¾. Therefore abate the numerator 8. from the numerator 9. and there remaineth 1/12 as may appere here by practise.

〈 math 〉〈 math 〉

2. But if you haue a broken nōber, to bee substracted from a whole nom∣ber, you must borow one of the whole nomber, and resolue it into a fraction of like denomination, as is the frac∣tion, which you would abate from the same whole nomber, and then abate the saied fraction there from, and you shall finde what doeth remaine, as by this example. If you abate ⅘ from. 8. you must borowe out of the said. 8. and resolue it into fiftes like vnto the frac∣tion, because it is 4. fiftes, that. 1. will bee 5. fiftes thus 5/5. therefore abate ⅘. from 5/5. and there will remaine ⅘, and substract that. 1. whiche you borowed from 8. and there doeth remain. 7. and also the ⅕. Thus the ⅘ being substrac∣ted frō. 8. doeth leaue. 7. ⅕, as by prac∣tise

doeth plainly appere 〈 math 〉〈 math 〉

3. If you will substract broken nom∣ber, from whole nomber, and broken beyng together: thus, as if you would substract ¾. from. 6. ⅚, you maie by the first substraction, abate ¾. from ⅚, and there will remaine 1/12, and the 6. doeth still remaine whole, because that 5/4. beyng abated from. 6. ⅚. leaueth. 6. 1/12. as appeareth by practise. 〈 math 〉〈 math 〉

Likewise if you will abate ⅔, from 14. ⅖, you muste firste reduce. 14. ⅖. all into fiftes by the. 6. reduction, and thei

bee 〈◊〉〈◊〉, then reduce ⅔. into a common denomination, by the firste reduction, and you shall finde 10/15. for the ⅔: and 21••/15 for the 70/5: then substracte the numera∣tor. 10. of the firste fraction, from the numerator. 216. of the seconde fractiō, and there remaineth 206/15. Therfore di∣uide 206. by. 15. and thereof commeth 13.11/15, and so muche remain of this sub∣straction, as maie appeare. 〈 math 〉〈 math 〉

4. If you will substracte whole nō∣ber and broken, from whole and bro∣ken,

as thus, if you will substract. 9¼. from. 20. ½. you must reduce. 9. ¼. into fowerthes, and likewise the. 20 ½. in∣to halfes by the sixt reductiō: and you shall find 37/4 for the. 9. ¼. And 42/2. for the 20. ••/2. Then reduce 57/4 and 31/2 into one denominatiō, accordyng vnto the first reduction and you shall finde 74/8 for the 37/4, and ••64/•• for the 41/2 thā abate the nu∣merator of ••65/8 and there remaineth 90/8 then diuide 90. by 8. and thereof commeth 11. ½ whiche is the remaine of this substraction. 〈 math 〉〈 math 〉

¶ Substraction of broken nombers of broken.

5. If you will substracte, the ½ of ⅔ of ⅗. from the ⅚. of ¾. of ⅞, you muste firste bryng the ½. of ⅔. of ⅗. into one fraction by the. 3. reduction, and the ⅚ of ¾ of ⅞ likewise into one fraction by the same reduction, and you shal finde 6/30. for the firste. 3. broken nombers, whiche beyng abreuied dooe make. ⅕: and for the other 3. broken nombers, you shall finde 105/192: whiche beyng like∣wise abreuted dooe make 35/64. then you shall substracte ⅕. from. 35/64. by the in∣struction of the firste substraction, in reducyng bothe the fractions into a common denomination, as before is dooen, and you shall finde remainyng 〈◊〉〈◊〉/3••0, as maie appeare by example. 〈 math 〉〈 math 〉

¶ The sixt Chapiter is of multi∣plication in broken nombers.

FIrste, for to multiplie in broken nomber, there is a rule, whiche is thus, multiplie the numera∣tor of the one fraction, by the numerator of the other. And then diuide that fraction if you maie, or els abreuiate it, and you haue do••en: but if there be whole nomber and broken together, you muste reduce the whole nombers into broken, and adde ther∣vnto the numerator of his brokē, and then multiplie, as is before saied, as also hereafter by examples shal more plainly appeare.

1. If you will multiplie ⅔ by ¾, you muste multiplie the numerator. 2. by the numerator. 3. and therof commeth 6. for the numerator. Likewise mul∣tiplie the denominators thone by the other, that is to saie. 3. by. 4. and there∣of

commeth. 12. for the denominator, so that this multiplication commeth to 6/12, whiche beyng abreuied doe make ½. and so muche amounteth the mul∣tiplicatiō of the ⅔. by ¾ as by practise. 〈 math 〉〈 math 〉

2. Likewise, if you will multiplie a broken nomber by whole nomber, or whole nomber by broken, whiche is all one, as ⅘. by. 18 or els. 18. by ⅘, you must set. 1. vnder. 18. thus. 18/1: and then multiplie. 18. by the numerator 4. and therof commeth. 72. the whiche diuide by the denominator. 5. and thereof cō∣meth. 14. ⅖. for the whole multiplica∣tion, or otherwise abate from. 18. his ••/5. parte, whithe is. 3. ⅗, and there re∣maineth. 14. ⅖. as aboue. 〈 math 〉〈 math 〉

3. Also if you wil multiplie a whole nomber, by whole nomber & brokē, or els whole nomber & brokē by a whole nōber, whiche is all one. As by exam∣ple. If you will multiplie. 15. by. 16. ¾. or els. 16. ¾ by. 15 First reduce. 16 ¾ all into fourthes, in multipliyng 16. by y^e deneminator of ¾ whiche is 4. & therof commeth 64 whereunto adde the nu∣merator. 3. & it maketh 67/4, whiche mul∣tiplie by 15/1, accordyng vnto thinstruc∣tion of the laste example, and you shall find y^e product of this multiplicatiō to be 251 ¼ as by practise doth here appere 〈 math 〉〈 math 〉

4. And if you will multiple a bro∣ken nomber, by whole nomber and broken, or els whole nomber and broken by a broken. As by example, if you will multiplie ¼. by. 18. ⅔, or els 18 ⅔. by ¼, whiche is all one: you muste reduce the whole noumber into his broken by the sixt reduction. And you shall finde, 56/3, whiche you shall mul∣tiplie by the ¼, after the doctrine of the first multiplication, that is to saie: in multipliyng the numerator. 56. by the Numerator of ¼, which is 1. And it is still 56. because 1. doth neither mul∣tiplie nor deuide. And likewise you muste multiplie the Denominator. 3. by the Denominatour. 4. and it ma∣keth. 12. then diuide. 56. by. 12. and thereof commeth. 4. ⅔. And so muche

amounnteth the multiplication of the 〈◊〉〈◊〉. ⅔. multiplied by ¼, as by example. 〈 math 〉〈 math 〉

5. If you will multiplie whole nō∣ber and broken, with whole and bro∣ken, you muste firste put either whole nomber into his broken, accordyng to the instruction of the sixte reduction, and 〈◊〉〈◊〉 multiplie the one numera∣〈◊〉〈◊〉 the other, and of the producte make your numerator. And likewise multiplie the denominators, the one by the other, and thereof make the de∣nominator, then diuide the numera∣tor, by the denominator, and the quo∣tient shalbe the encrease of this mul∣tiplication.

¶ Example.

If you would multiplie. 12. ⅘. b〈…〉〈…〉. ¾: firste by the sixte reduction, the. 12. ⅘

will make 64/5, and the. 6. ¾. will make 27/4 then multiplie the numerator. 64. by the numerator. 27. and thereof com∣meth. 1728. for the numerator. And then you must multiplie the denomi∣nator. 5. by the denominator. 4. and thei doe make. 20. then diuide. 1728. by. 20. and thereof commeth. 86. ⅖. for the whole multiplicatiō, as by exāple. 〈 math 〉〈 math 〉

6. If you will multiplie one bro∣ken nomber by many broken nom∣bers, thus: As to multiplie ⅔. by 5/7. and by 4/9: you muste multiplie the nume∣rators of all the fractions, the one by

the other, and of the product make the numerator, that is to saie: 2. by. 5. and thei be. 10. then. 10. by. 4. and thei bee 40. for the numerator. Likewise you must multiplie the denominators, the one by the other, that is to saie. 3. by. 7. maketh. 21. then. 21. by. 9. maketh. 189 for the denominator: then sette. 40. o∣uer the. 189. with a line betwene thē, and thei make 40/189. And so muche a∣mounteth the whole multiplication of the ⅔, multiplied by 3/7 and 4/9, as by example folowyng. And thus is to be vnderstande of all suche like. 〈 math 〉〈 math 〉

¶ The .vii. Chapiter treateth of di∣uision in broken nombers.

NOte, that in diuision of bro∣ken nombers, you muste sette

your diuisor downe firste, nexte vn∣to the lefte hande, and the diuidende or nomber, whiche is to bee diuided alwaies towarde the right hande.

And then multiplie crosse wise, that is to saie, the numeratour of youre Diuisour by the Denominator of the Diuidende, and the producte shalbee the Denominatour, whiche after∣warde shall bee your Diuisour. And likewise you muste multiplie the De∣nominatour of your firste noumber, that is to saie of your Diuisour: By the Numeratour of the Diuidende, whiche afterwarde shall bee the Di∣uidende, and that muste bee sette o∣uer the Crosse, and the Denomina∣tour vnder the Crosse, then shall you diuide the Numeratour by the Deno∣minatour, if any maie bee diuided, if not, you muste abreuiate theim, as hereafter by examples shall more pla∣inly appeare.

1. If you will diuide ¾. by ⅔, you muste sette the Diuisour (whiche is

⅖) nexte to the lefte hande, and the di∣uidende ¾. towarde your right hande, with a crosse betwene them: as maie appere by this example in the margente. Then 〈 math 〉〈 math 〉 you shall multiplie the numer at or of the ⅔, whi∣che is. 2. by the denomi∣nator of the ¾. which is 4. and thereof commeth. 8. which shal∣be your newe diuison: set that. 8. vnder the 〈…〉〈…〉osse, as the denominator, then multiplie the numerator of the diui∣dende, that is to saie, of the ¾ whiche is 3. by the denominator of the diuisour, that is to wit, of the. ⅔. whiche is. 3. set that ouer the crosse, and it is. 9. for the numerator, whiche shalbe now the di∣uidende, or nōber to be diuided. Then finally, you shall diuide. 9. by. 8. and thereof commeth into the quotient. 1. 2/8 and so often times is ⅔. conteined in ¾, as dooeth appeare before in the mar∣gente. But in case you would diuide ⅔. by. ¾, you muste likewise sette your

diuisor ¾ nexte to your left hande, as is before said. And then procede, as is aboue declared, & you shall finde that ⅔ diuided by ¾ bringeth into y^e quotiēt 2/9, whiche can not be diuided nor ab∣breuied, wherfore it appereth that ⅔ diuided by ¾ bringeth but 2/9 of one vni∣tie into the quotient as doth appere. 〈 math 〉〈 math 〉

2. Likewise if you wil diuide a bro∣ken number by a whole nomber, or els a whole number by a broken, as to diuide ¾ by 13. you shall put 1. vnder 13. and it wil be 13/1 which is your diuisor, set y^t toward 〈 math 〉〈 math 〉 your left hande, and then multiply 13. by 4. accordīg to the first diuision, & ther∣of commeth 52. for the de∣nominator,

set that vnder y^e crosse & multiply 3. by 1. which is 3. for the nu∣merator, that set ouer the crosse, and it is 3/52 as appeareth in the margent. But if you will diuide 13. by ¾ then set the ¾ nexte your left hand and put one vnder 13. as in the last example, & it is ••/3 set y^t toward your right hande thus, as appeareth in the margent 〈 math 〉〈 math 〉, and then worke according to y^e doc∣trine of the first diuision, & you shall finde that 13. being diuided by ¾ bringeth into y^e quo∣tient 52/4, then diuide 52. by 3. and therof commeth 17. ••/3, and so oftentimes is ¾ conteined in 13. as doth ap∣pere. 〈 math 〉〈 math 〉

3. And if you wil diuide whole nū∣ber by whole number and broken, or els whole nūber and broken by whole number, as to diuide 20. by 5. ⅚, you shall reduce 5. ⅚ into his broken by y^e sixt reduction, & it maketh 35/•• for your

diuisor, then put 1. vnder 20. And it wyll bee 20/1, then shall you multiply 〈 math 〉〈 math 〉 35. by 1. and 20. by 6. as is taught in the other diuisi∣ons, and you shall finde 220/3: then di∣uide 12. by 35. and you shall fynde in your quotient 3. 3/•• & so many tymes is 5. ••/6 conteined in 20. as in the mar∣gent doth appere.

But if you will diuide 5. 3/6 by 20. you must diuide 35. by 120. which you can not, wherefore you shall abbre∣uiate 35/120, and thereof commeth. 7/24.

4. If you will diuide a broken num∣ber by whole number and broken, or els a whole number and broken, by a broken number. As to diuide ¾ by 13. ⅔, you muste reduce 13. ⅔, into hys broken, by the syxte reduction

And they be 41/3 for your diuisor, then multiplye 41. by 4. & they make 164. for your denominator, likewise mul∣tiplye 3. by 3. and they make 9. for the numerator, and then will your sūme be 9/104. But yf you will diuide 13. ⅔ by ¾ then you must diuide 164. by 9. and you shall fynde 18. 2/9.

5. If you will diuide whole num∣ber and broken, by whole number & broken, as to diuide 7. ¾ by 13. ⅔ you must reduce the whole numbers in∣to their broken, by the doctrine of the sixt reduction, & you shall fynde 31/4 for the 7. ¾, & 41/3 for the 13. ⅔. Then sett downe 41/3 to∣warde 〈 math 〉〈 math 〉 the left hande by∣cause it is your diuisor, and the 31/4 towarde the right hande, and multi∣plie 41. by 4. for your denominator, and thereof commeth 164. Likewise multiply 31. by 3. for your numerator, and it amounteth to 93. the whiche diuision will bee thus 〈◊〉〈◊〉 as before

doth appere.

But if you will diuide 13⅔. by 7. ¾ you muste contrariwyse to the other example, diuide 164. by 93. and you shall fynde in the quotient 1. 71/93.

6. The broken numbers of bro∣ken, muste be diuided in suche maner as broken numbers are, & there is no difference, sauing onely that of many broken numbers you must make but two broken numbers, that is to saye y^e diuisor, and the diuidend, or number that is to be diuided, example. If you will diuide the ¾ of ⅗ of ½, by the 2/2 of 4/7. For the fyrst, the ¾ of ⅗ of ½ are 9/4. by the thirde reduction: and the ⅔ of 4/7 are by the same Reductiō 8/21, then haue you 8/21 for your 〈 math 〉〈 math 〉 diuisor, & 9/40 for your nū∣ber to bee diuided, then multiply 8. by 40. which maketh 320. set that vn∣der the crosse and multiply 9. by 21: & thereof cōmeth 189. which set ouer the crosse for the numerator, and they

make 189/120 for this diuision as doth ap∣pere.

But if you woulde diuide 1/21 by 9/200. you must worke contrary to the laste example, that is to saye, you must di∣uide 320. by 189. And therof commeth in the quotient 1. 131/189.

¶ The eyght Chapter treateth of du∣plation, triplation, and quadrupla∣tion of all broken numbers.

IF you wyll double any broken number, you shall diuide y^e same by½: likewise if you wyll triple any fraction you muste diuide it by ⅔. And for to quadruple any broken nū∣ber, you shall diuide it by ¼, and so is to be vnderstande of all other.

Example of duplation.

IF you will double ⅜ you shal diuide 3/•• by ½, and thereof com∣meth 6/8, which being ab∣breuied 〈 math 〉〈 math 〉 are ¾: as by ex∣ample.

Or otherwise, in case the denomina∣tor of any fraction bee an euen num∣ber, you may take halfe the sayde de∣nominator, without anye other ope∣ration, and the numerator to abyde still y^e numerator, vnto the said halfe of the denominator of the Fractiō, as by the other exāple before rehearsed: that is to say of 〈◊〉〈◊〉, take ½ of 8. which is 4. and that is the denominator, and 3. remaineth stil numerator to 4. and it maketh ¾ and so of all other. But in case the denominator bee an odde nomber, that is to say, not euen, then you may multiply the numerator by 2. or els double y^e numerator, whiche is al one thing; and that fraction shall bee doubled. Example, if you will double ••/5 you must onely multiply the numerator 3. by 2. & they be 6. whiche maketh that fraction to be 〈◊〉〈◊〉, the which 6. being diuided by 5. bringeth 1. ⅕ and so much is the double of 3/••.

Example of Triplation.

If you wil triple ⅗ you muste diuide ••/5 by ••/3 and thereof commeth 9/5 whiche beinge diuided bringeth 1⅘, or other∣wise, bicause the denominator is an odde number you maye multiplie the numerator 3. by 3. and therfore com∣meth 9. which maketh 〈◊〉〈◊〉 as before.

Example of quadruplacion.

If you will quadruple ⅘, you shall diuide ⅘ by ¼ and thereof commeth ⅘ which 16. being diuided by 5. bringeth 3••/5, or otherwise, bicause the denomi∣nator of the fraction is an odde nūber, you shall multiplie the numerator of the 〈◊〉〈◊〉 that is to say 4. by 4. and therof commeth 16. the whiche diuide by 5. and you shall finde 3. ••/5 as before, and this sufficeth for duplacion, tripiacion and quadruplacion.

¶The 9. Chapter treateth of the proofes of broken numbers. And first of Reduction.

IF you doe abbreuiate y^• broken nū∣bers whiche bee reduced, you shall retourne them into their firste estate: as by example, if you reduce ••/5 wyth ⅘ you shall fynde 10/〈…〉〈…〉 and 12/〈…〉〈…〉, then ab∣breuiate 10/15 and you shall fynde ⅔, ab∣breuiate likewise 〈◊〉〈◊〉 and thereof com∣meth ⅘ as before.

The proofe of Abbreuiation.

IF you doe multiplie that number whiche you haue abbreuied by that or those numbers, by the whiche you haue abbreuied them, you shall re∣turne them againe into their firste e∣state. Example, if you wyll abbre∣uiate 32/48 by 16. in takyng y^• 〈◊〉〈◊〉 part both of the numerator, and also of the de∣nominator, you shall finde ⅔, y^• proofe is thus, you must multiplye bothe the numerator & denominator of 2/•• by 16. that is to say, three by 16. maketh 48. for the denominator, & 2. by 16. maketh 32. for the numerator, then set the nu∣merator 32. ouer the denominator 48

and they be 32/48 as before.

The proofe of Addition.

If you doe substract one of the nū∣bers, or manye of them (which you haue added) from the totall summe, there shal remaine the other, or others; Example: if you do adde 1/•• with ¼ you shall fynde 7/12. The proofe is, is if you substract ⅓ from 7/12 you shall fynde re∣maining the other number whiche is 2/4, or els yf you doe substract ¼ frō 〈◊〉〈◊〉 there will remains the other nomber, which is 1/••.

The proofe of Substraction.

If you do adds that number whiche remaineth, with the number whyche you did substract, you shall fynde the totall summe, oute of the which you made y^• abateinēt: or otherwise, if you adde the twoo lesser nōbers togither, you shall finde the greater. Examplet

if you doe abate or substracte ¼ from ⅓ there will remaine ••/12. The profe is thus: you muste adde 1/12 & ¼ togither, and you shall fynde ••/3, whiche is the greatest nomber.

The proofe of multiplication.

If you diuide the producte of the whole multiplication, by the multi∣plicator, you shal fynde in your quo∣tient, the multiplicande or nomber by the which you haue multiplyed: or els if you diuide the totall sōme which is come of the multiplication, by the multiplicande: you shall finde in the quotiēt the multiplicator. Example, if you multiply 〈◊〉〈◊〉 by ⅘, the product of this multiplication will bee 8/15. The proofe is thus: you shal diuide 8/15 by y^• multiplicator ⅘, and therof cōmeth ⅔. Or els diuide 8/15 by ⅔ & you shall finde the ⅘ which is the multiplicator.

The proofe of Diuision.

If you doe multiplye the quotient by the diuisor, you shall finde the num¦ber which you did diuide, y^t is to say, your diuidende. Example: if you di∣uide ⅔ by ¾, your quotient will bee 2/9 y^• proofe is thus, you must multiply 8/9 by ¾, and thereof commeth 24/36 whiche be∣ing abbreuiated art ⅔ whiche is your diuidende, & by this maner all whole numbers haue their proofes as well as broken numbers.

¶ The tenth Chapter treateth of cer∣taine questions done by broken numbers. And first by Reduction.

FInde twoo numbers, where of the 2/•• of the one number may bee egal vnto the 〈◊〉〈◊〉 of the other. Aun∣swere: you shall reduce 2/7 & ⅜ crossewise, and you shall finde 16. ouer the ⅔ and 21. ouer the ⅜, which are the two num∣bers that you seeke: for the ⅜ of 16. are 6. and so are the 2/•• of 21. lykewise 6. wherefore you may perceiue that the

the ⅜ of 16. which are 6. are egall vnto the 2/7 of 21. whiche is also 6.

2. Finde twoo numbers, wherof y^• 2/•• of the one may be double to the ¼ of the other. Aunswere: double ¼ & you shal haue 2/4, which being abbreuiated is ½: thē reduce ⅔ & ½ crossewise, & you shall finde 4. ouer the ⅔ & three ouer the ½ which are the two numbers that you seeke. For the 2/1 of 3. which is 2. is double vnto the ¼ of 4. which is but 1.

3. Finde two numbers whereof the ⅔ and the ¼ of the one, maye bee egall vnto the ¼ & ⅕ of the other. Aunswere: Adde the ⅓ and ¼ togither, and they make 7/12 then adde ¼ and ••/5 togither, & they are 6/20, then reduce 7/17 & 9/20 crosse∣wise, & you shall haue 140. ouer the 7/12 & 108. ouer the 9/20, whiche are the two nūbers that you seke. For 63. whiche are the 7/12 of 108. are also the 9/20 of 140.

4. Finde two numbers, wherof the ••/2 the and the ¼ of the one of them, maye by egall vnto the ⅕ the ⅙ and 1/7 of the other number. Aunswere: first you

must adde ½, ⅓, and ¼ togither, & they make 13/12: then adde ⅕, ⅙ and 1/7 togi∣ther, & they make 207/210. Then reduce 13/12 and 107/210 crossewise, as by the fyrste question of reduction, and you shall finde 2730. ouer the 13/12 and 1284. ouer the 107/210, whiche are the twoo nombers that you seeke: for 1391 which is the ½ the ⅓ and ¼ of 1284. is lyke to the 11/56 & ¼ of 2730, which is also 1391.

5. Finde three nombers, whereof the ⅖ of the first, the 4/7 of the seconde, & the 4/9 of the thirde, maye be egall the one to the other. Aunswere: set downe the 23/57 and 〈◊〉〈◊〉, and then multiplie the Denominator of the ⅖ that is to saye 5. by the Numerators of the other twoo Fractions, that is to saye, by the Nu∣merator of 3/7, and by the Numerator of 4/9, whiche is 3. and 4. And thereof commeth 60. for your fyrste nomber, then shall you multiplye the Deno∣minator of the 3/7 whiche is 7. by the Numerators of ⅕ and 4/9, that is to say by 2. and 4. and thereof commeth 56.

for the seconde number: Then multi∣plie the Denominator of 4/9, that is 9. by the Numerators of ⅖ and ••/7 that is by 2. and by 3. and thereof commeth 54. for the third nomber

And thus the ⅖ of 60. which is 24. is li∣kewise the 3/7 of 56. whiche is the se∣cond nomber and the 4/9 of 54. whiche is the thirde nomber.

6. Finde three nombers, of whiche the fyrste and the seconde maye bee in suche proporcion as ½ and ⅓, and the se∣conde and thirde in suche proportion as ¼ and ⅕. Aunswere: reduce ½ and ⅓ crossewise, and you shall haue 3. ouer the ½ and 2. ouer the ⅓, then reduce ¼ and ⅕ in lyke maner, and you shall fynde 5. ouer the ¼ and 4. ouer the 3/••. Then say by the Rule of three, •••• 5. do gyue me 4. what shall twoo gyue me, whiche is the seconde proportio∣nall, multiply the seconde nomber 4. by the thyrde nomber twoo, and ther∣of commeth eyght, the whiche diuide

by the first number 5. and therof com∣meth 1. ⅗ for the third proportionall, and you shal fynde that 3. 2. 1. ⅗ are the three nombers proportionall that I demaunde, or els 15. 10. & 8. in whole numbers.

1. What number is that, vnto the whiche if you doe adde 13. the whole amounteth to 31. Aunswere: rebate 13. from 31. and there wyll remaine 18. which is the number that you seeke.

2. What number is that, vnto the which if you adde ⅖ the addition wyll be ⅚. Answere: abate ⅖ from ⅚, and there will remaine 13/30, whiche is the number that you desire.

3. What number is that, whereun∣to if you adde 7. ⅔, the whole additiō will be 12. ¼. Aunswere: abate 7. ⅔ frō 12. ¼. & the remaine will be 4 7/12 which is the number y^t you desire to know.

4. What number is that, where∣into if you adde the ¾ of it selfe, that is to say, of the nūber that your séeke, the whole addition may be ⅚.

Aunswere: Here followeth a generall rule for all suche like questions. First, of. 3. whiche is the numeratour of. ¾. make still the numeratour, and like∣wise of. 3. and. 4 together, whiche is bothe the numerator, and the deno∣minator of the. ¾. make your denomi∣natour, so you shall finde. 3/7, then take the 3/7 of ⅚, whiche is 15/42, or 5/14; and sub∣stracte theim from ⅚, and there will remaine 10/21, whiche is the nomber that you seke.

5. What nomber is that, vnto the whiche if you adde his owne ⅔. that is to saie ⅔. of it self, the whole addition shall bee. 20. Aunswere: do••e as in the laste question: of the numeratour of ⅔ that is to saie, of. 2. make stil your nu∣merator. And likewise of the nume∣rator. 2. and the denominator. 3. of the ⅔, make of them bothe, your denomi∣nator, and you shall finde ⅖, then take the ⅖. of. 20. whiche are. 8. And abate them from. 20. and there will remain 12. whiche is the nomber that you de∣sire,

and so is to bee dooen of all suche like reasons.

1. What nomber is that, from the whiche if you dooe abate. 17. the reste maie bee. 19. Aunswere: adde. 17. and 19. together, and you shall finde. 36. whiche is the nomber that you seke.

2. What nomber is that, from the whiche if you abate ⅗, the rest maie bée ⅛. Aunswere: adde ⅗ and ⅛ together, and you shall finde 29/40, whiche is the nomber that you demaunde.

3. What nomber is that, from the whiche if you deduct. 13. ½ the rest maie be. 5. 5/7. Aunswere: adde. 13. ½ and. 5. 5/7. together, and there of commeth. 19. 1¾, whiche is the nomber that you seke.

4. What nomber is that, from the whiche if you substracte his ⅖, the reste maie bee. 12. Aunswere: and a rule for * 1.4 suche like reasons, that is to saie, from the denominator of 5/2. whiche is. 5. a∣bate. 2. whiche is his numerator, and there resteth. 3. for the denominator, and thus of ⅖. you haue you made ⅖,

then take the 〈◊〉〈◊〉 of. 12. whiche are 8: and adde them vnto. 12. and therof cōmeth 20. for the nomber whiche you desire.

5. What nomber is that, from the whiche if you doe abate his ¾, the reste maie bee 8/••. Aunswere: from the deno∣minator of ¾, whiche is. 4. substracte his numerator. 3. and there resteth. 1. Thus of ¾. you haue made. 3/1. Then multiplie 5/•• by 8/9, and therof commeth 2. ⅔, the whiche adde vnto 8/9, and you shall haue. 3. 5/9, whiche is the nomber that you seke.

6. What nomber is that, from the whiche if ye abate his ⅘, the rest maie be. 12. ⅔. Aunswere: Doe as you did in the laste question, and you shall finde that the ⅘. will bee 4/1. And therefore multiplie. 12. ⅔ by 4/1, and thereof com∣meth. 50 ⅔, the whiche adde vnto. 12 ⅔, and you shall finde. 93. 〈◊〉〈◊〉, for the nom∣ber that you demaunde. And thus of all like questions.

1. What nomber is that, which be∣yng multiplied by 13. the whole. Mul∣••iplication

shal m••••nte to. 221. Aun∣swere: 〈…〉〈…〉de. 221. by. 13. and thereof comnieth. 17. whiche is the nomber that you seeke.

2. What nomber is that, whiche beyng multiplied by. 15. the whole multiplication wil amount to ¾. Aun∣swere: diuide ¼. by 15/1. and thereof com∣meth 1/20. whiche is the noumber, that you seeke.

3. What nomber is that, whiche beeyng multiplied by. 21. the whole multiplication will bee: 16. ⅘. Aun∣swere: diuide. 16. ⅘. by. 21/5, and you shall finde ⅘, whiche is the nomber that you demaund••.

4. What nomber is that, whiche beyng multiplied by ¾. the multipli∣plication will amounte to. 18. Aun∣swere: diuide 18/1 by ¾, and there of com∣meth. 24. whiche is the nomber that you desire to knowe.

5. What nomber is that, whiche if it bee multiplied by ⅔. the whole mul∣tiplication will bee. ••/4. Aunswere: di∣uide

¼ by ⅔. and the quotient will bee ⅜, whiche is the nomber that you re∣quire to knowe.

6. What noumber is that, whiche beyng multiplied by ⅝, the product of that multiplication will bee. 16. ⅔. Aunswere: diuide. 16. by ⅔. by 〈◊〉〈◊〉. and thereof commeth. 26. ⅔, whiche is the nomber that you seeke.

¶ Here ensueth other necessarie que∣stions, whiche are wrought by multiplication in bro∣ken nombers.

I Demaunde howe muche the ⅝ of. 20. shillynges are worth or what are the ⅝. of. 20. shil∣lynges. Aunswere: you muste multiplie ⅝ by 1••/5, and the product will bee 100/8, therefore diuide. 100. by. 8. and thereof commeth. 12. ½, whiche is to saie. 12. shillynges. 6. pence, and so muche are the 5/••. of. 20. shillynges worthe.

2. I demaunde what the ¼ of ⅚ of 〈◊〉〈◊〉 pounde of money are worthe, that is to sa••e, of. 20. shillynges. Aunswere: multiplie. ¾. by ⅚. And thereof com∣meth 5/2. Then take the 5/•• of 20. shillin∣ges, as in the last questiō goyng before and you shall finde. 12. shillynges. 6. pence, and so muche are the ¾ of ⅚, of 20 shillynges worthe.

3. I demaunde what the ⅔ of. 8. pence ••/2 are worth. Aunswere: multiplie. 8. ••/2 by ⅔, or els ⅔ by. 8. & ••/2 whiche is all one, and you shall finde 34/6. Then diuide 34 by. 6. and your quotient will bee fiue pence ⅔, and so muche are the. 〈◊〉〈◊〉. of. 8. pence 〈◊〉〈◊〉 〈…〉〈…〉the.

4. What are the ¾. of. 14. pence 〈◊〉〈◊〉, Aunswers multiplie. 14. ⅗ by ¾, and therof commeth 219/20. Therefore diuide 219. by. 20. and your quotiente will bee. 10. pence 19/20. and so muche are the ••/4 of. 14. ••/5.

5. How many 〈◊〉〈◊〉 of sworth 〈◊〉〈◊〉 ••••••contain••r in. 7. ⅔. Aunswer multiplie, 7. ⅔ by 4/•• (because one whole

containeth. 4. quarters) and thereof commeth, 30. ⅔, and so many quarters are in the. 7. ⅔. that is to saie. 30. quar∣ters, and ⅔ of a quarter.

6. Howe many thirdes are in ¾ and ••/2, that is to saie in. 3, quarters, and ••/2 of one quarter, whiche are ⅞ by the 〈◊〉〈◊〉 reduction. Aunswere: multiplie 7/•• by 〈◊〉〈◊〉 (for because that in one whol•• are cō∣tained. 3. thirdes) and thereof wil come ⅔ and ⅝ of a third, and so many thirdes are in ¾ and ½ or in ⅞, whiche is al one

¶ Question doen by diuisi∣on in broken nomber.

WHat nomber is that, which beyng diuided by. 17. the quotiente will bee. 13. Aun∣swere: multiplie. 17. by. 13. And thereof commeth. 221. whiche is the nomber that you seke.

2. What nomber is that, which be∣yng diuided by ¾, the quotient will be 21. Aunswere: multiplie 23/1. by ••/4 and therof cōmeth 63/4. Then diuide. 〈◊〉〈◊〉 by 4

and thereof commeth. 15. ¾, whiche is the nomber that you seke.

3. What nomber is that, which be∣yng diuided by. ⅛, the quotiente will bee ⅔. Aunswere: multiplie 2/1. by ⅛. and thereof commeth 2/24, whiche bee∣yng abreuiated are 1/12. for the nomber, whiche you require.

4. What nomber is that, whiche beyng diuided by ⅘. the quotient will be. 16. ⅔? Aunswere: multiplie. 16. ⅔, by ⅘, and thereof commeth 200/15. There∣fore diuide. 200. by. 15. and thereof cō∣meth. 13. ••/3, whiche is the nomber that you desire to finde.

5. What nomber is that, which be∣yng diuided by. 13. ⅓, the quotient will bee. 20. Aunswere: multiplie 〈◊〉〈◊〉, by. 13 〈◊〉〈◊〉 and thereof commeth 900/3, then diuide 800. by. 3. and thereof commeth. 266 〈◊〉〈◊〉 for the nomber, whiche you seeke,

6. What nomber is that, whiche it it hee diuided, by. 12. ½, the quotiente wil be 7/••. Answere: multiplie 7/•• by. 12 ½ and thereof commeth 175/〈…〉〈…〉, then diuide

175 by. 16. and thereof commeth. 10 15/•••• for the nomber whiche you desire.

¶ Other necessarie questions dooen by diuision in broken nomber.

I Demaunde what parte. 30. is of. 70. Aunswer: diuide. 30. by 70. whiche you can not, for thei are 30/7••, but abreuiate them and thei are 3/7. Thus. 30. are the 3/7. of. 70.

2. I demaunde what parte. 10. is of 16 ⅔. Aunswere: diuide. 10/〈…〉〈…〉. by. 16 ⅔, and thereof commeth 30/50, whiche beyng a∣breuiated are 〈◊〉〈◊〉. And thus 10. is found to bee ⅗ of. 16. 2/••.

3. More, what parte is. 25. of. 5/••. Aunswere: diuide 5/••. by. 25/10, and there∣of commeth 5/20••, whiche beyng abre∣niated is. 1/40. And thus. 5/••. is but the 1/40. of. 25.

4. More, ⅚ what parte are thei of ⅞ Aunswere: diuide ⅚ by ⅞, and you shal

finde 4••/42 whiche abreuiated are 20/21.

5. More, ⅘ what parte are thei of. 13. 〈◊〉〈◊〉. Aunswer: diuide ⅘ by. 13. ⅓. and you shall finde 12/200, whiche beyng abbre∣uiated are ••/50. And thus 4/1 are the 3/50. of 13. ⅓.

6. More. 12. ½. what parte are thei of. Aunswere: diuide. 12. ½. by 30/••, and you shall finde. ••5/60, whiche beyng abre∣uiated are ••/12, and thus. 12. ½, are the 5/12 of. 30.

7. More. 16. ••/3, what parte are thei of. 57. 1/7. Aunswere: diuide. 16. ⅔ by. 57 ••/7. and therof commeth 550/1200, whiche be∣yng abreuiated are 7/24, and thus. 16. ⅔. are the. 7/24 of. 57. 1/7.

8. More, ••/4 and ⅔ of ••/4, or three quar∣ters, and ⅖ of one quarter, what part are thei of. 1. Aunswere: reduce 5/4, and the ⅔ of ••/4, into one broken by the first reduction, and you shall finde 11/12. And thus the ••/4, and ••/3 of ¼ are the 11/12 of one whole.

9. More, of what nomber are. 9. the 〈◊〉〈◊〉. Aunswere: diuide. 9. by 3/3, and thereof

commeth. 13. ••/2, whiche is the nomber whereof. 9. are the 〈◊〉〈◊〉.

10. More, of what nomber are ⅘ the ••/4. Aunswere: diuide ⅖ by ¾, and thereof commeth 8/15, whiche is the nōber wher¦of 2/•• are the ¾ of the same nomber.

11. More, of what nomber are. 5. ¾. the ••/7. Aunswere: diuide. 15. ¾ by 3/••, and you shall finde. 13. 5/12, whiche is the nō∣ber whereof. 5, ••/4 are the 3/7.

12. More. 9. 〈◊〉〈◊〉 what parte are thei of. 33. Aunswere: diuide. 9. 2/•• by 33. ½. And thereof com∣meth 18/20••: and thus 9. 〈◊〉〈◊〉 are the 58/201 of 33. ½ as ap∣pereth.

Notes

• * 1.1

  1.
• * 1.2

  3.
• * 1.3

  6.
• * 1.4

  A rule.