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REduction, is as much as to bring togither, or to put in sēblaūce 2. or ma∣nye nūbers dissembling one from the other, in reducynge
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∣tiplie the Denominators the one by the other, and so you shal haue a new denominator common to all, ye which denominator diuide by the perticuler denominators, and multiplye euery quotiēt by his numerator and so you shall haue newe numerators, for the numbers which you woulde reduce, as appeareth by thys example follo∣wyng.
[unspec I] IF you wyll reduce ⅔ and ⅘ togi∣ther, you must fyrst multiplie ye two denominators the one by ye other, that is to say 3. by 5. maketh 15. which is your commō denominator,
that set vnder the crosse, then diuide 15. by the denomi∣nator 3. & you shall haue 5. which mul∣tiply by the nume∣rator 2. and you shall finde 10. set that ouer the ⅔ and they are 10/11, for the ⅔. Afterwardes diuide 15. by the deno∣minator 5. and therof commeth 3. the which multiply by the numerator 4. and you shall finde 12. which set ouer the head of the ⅘ and they make 12/15 for the ⅘: as appeareth more plainer in the margent. 〈 math 〉〈 math 〉
2. If you will reduce ½, ⅔, ¾, ⅚, to∣gither, you must multiplie all the de∣nominators the one by ye other, that is to say, 2. by 3. maketh 6, then 6. by 4. and mounteth 24. Last of all 24. by 6. and thereof commeth 144. for the common denominator. Then, for the fyrst diuide 144. by the denomi∣nator 2. and thereof commeth 72. the which multiplye by the numerator 1.
and it is still 72. set yt ouer the ½ and it is 72/144, for the ½: Then diuide 144. by the seconde denominator 3. & ther∣of commeth 48: the which multiplie by the seconde numerator 2. and they are 96. which set ouer the ⅔ and they make 96/144, for the ⅔: Thē diuide 144. by the thirde denominator 4. & therof commeth 36. the which multiplye by the thyrde numerator 3. & they make 108. which set ouer the ¾ and they are 108/144 for the ¾.
Finally diuide 144. by the last de∣nominator 6, & thereof commeth 24: The which multiply by ye last nume∣rator 5. & therof commeth 120. Which set ouer the ⅚ and they are 120/144, for the ⅚, as appeareth here by prac∣tise.
〈 math 〉〈 math 〉
IF you wyl reduce ye broken of bro∣kē togither, as thus, the ⅔ of 7/4 of ⅘, [unspec 3] you must multiplye the numerators the one by the other to make one bro∣ken number of the thre broken num∣bers, that is to saye 2. by 1. maketh 2. and then 2. by 4. maketh 8. which is your numerator. Then
| 8 | ||
| ⅔ | ¼ | ⅘ |
| 60. |
12. by 5. maketh 60. for your denomi∣nator, set 8. ouer 60. wt a line betwene them, and they be 1/60 which being ab∣breuied are 2/15 and so much are ye ⅔ of ⅙ of ⅘ as appereth in the margent.
[unspec 4] IF you will reduce ⅔ of, ¼, of ⅘, ye ¾, of 5/7: And the ½, of the ½, of the ⅔ of ⅓. First it behoueth you of euery par∣tie of ye brokē nūbers to make of eche of thē one broken, as by the third re∣ductiō is taught: yt is to say, in multi∣plying ye numerators by numerators & denominators by denominators: first, for ye fyrst part which is ⅔ of ¼ of ⅘, you must as is said before, multi∣plie 2. by 1. and then by 4. & you shall haue 8. for the numerator lykewise multiplie 3. by 4. and the product by 5. and you shall haue 60. for the deno∣minator, so they make, 1/60 which be∣ing
abreuied are 2/15 for ye first part, that is to say, for the ⅔ of ¼ of ⅘, se∣condely for the ¾ of 5/7 multiplye lyke∣wise the numerator 3. by 5. maketh 15. for ye numerator, & multiplie 4. by 7. maketh 28. for the denominator, and then they be 15/29 for ye seconde part that is to say for the ¾ of 5/7. Thirdely for the ½ of ½ of ⅔ of ⅓ multiply the nu∣merators the one by the other, that is to say, 1. by 1. and then by 2. and last by 1. and al maketh but 2. for ye nume∣rator, likewyse multiplie 2. by 2. ma∣keth 4. and 4. by 3. maketh 12. and then 12. by 3. maketh 36. for denomi∣nator, and they are 2/36, which beyng abreuied maketh 1/18, for the thyrde part, ye is to say for ½ of the ½ of ⅔ of ⅓. Last of all take the 2/15 the 15/28 and ye 1/18 & reduce them according to the order of the seconde reduction, and you shall finde 1008/7560 for the 2/15. And 4050/7500 for ye 15/28. And 420/7560 for ye 1/18: and thus are bro∣ken numbers of broken, reduced as appereth by practise.
| 8 | 15 | 2 | |||
| ⅔¼⅘ | ¾ 5/7 | ½½⅔⅓ | |||
| 1008 | 4050 | 420 | |||
| 60 | 28 | 36 | 2/15 | 15/28 | 1/18 |
| 760 |
[unspec 5] IF you wyll reduce ⅓ and ye ½ of ⅓ to∣gither to bring them into one bro∣ken number, you must first set down the ⅓ and ½ as appereth in the marget 〈 math 〉〈 math 〉 with a crosse betweene them, & then multiplie the twoo deno∣minators the one by the other, that is to say, 2. by 3. maketh 6. set that vnder the crosse, then multiplie the fyrst Numerator, one by the laste denominator twoo, and that maketh 2. vnto the whych adde the laste numerator one, and they be three, whych set aboue your
crosse, so you shall finde that the ⅓ and the ½ of ⅓ doe make ⅓ which being ab∣breuied dothe make ⅓, which is as much as the ⅓ and the ½ of ⅓. Likewise if you wyll reduce the ⅔ and ye ¼ of ⅓, you must do as before, set downe the ⅔ and ¼ with a crosse betweene them, multiplie the two denominators the one by ye other, that is to say, 3. by 4. maketh 12. which set vnder ye crosse as you see in the margent 〈 math 〉〈 math 〉 and thē multiply the first numerator 2. by ye last denominator 4. and thereof com∣meth 8. whereunto adde the last nu∣merator 1. and that maketh 9. which set ouer the Crosse, so shall you finde that the ⅖ and the ¼ of ⅓ are worth 9/12, which abbreuied doe make ¼, as appeareth by example in the margent.
[unspec 6] IF you will reduce whole number with broken, you muste bring the whole number into broken, as by this exāple may appere: reduce 17. ⅝ into a broken number, first you must multiplye the whole number 17. by the denominator of the brokē, which is eight in saying eight times 17. doe make 136. vnto the which you must adde the numerator of ⅝ which is 5. and all amounteth to 141. which set ouer 8. with a line betweene them, & thei wil be 141/8 so much is 17. ⅛ worth in a Fraction as appeareth here by practise.
| 17. | 141. | |||
| 8. | 5. | |||
| 136. | 17. | 8. | maketh | 141./8. |
| 5. | ||||
| 141. |
In case you haue whole number and broken to bee reduced, with broken you muste bring the whole number into his broken, in multiplying 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 brokē number wyth the other broken, as here appe∣reth by this example. Reduce 10. ⅔ & 4/7 togither, first bring 10 ⅔ al into thirds, as by the syxt reduction, and you shal finde 32/3, 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 12/21 for 4/7, as appereth here by practise.
〈 math 〉〈 math 〉
Also in case you haue in both partes of your Reduction, as well whole number as brokē, you must alwayes put the whole into their broken (as by the syxte reduction) of either part.
If you wyll reduce 12. ¼ with 14. ⅔ to bring them into one denomina∣tion, fyrst bring the 12. ¼ all into four∣thes, and you shall fynde 49/4: then likewise reduce 14. ⅔ all into thirdes, and you shall haue 44/3, for the 14. ⅔, then reduce 49/4 and 44/3 togither, by the order of the fyrst Reduction, and you shall fynde 147/12 for the 49/4. And 170/12 for ye 14. ⅔ as here by practise doth plainly appere.
〈 math 〉〈 math 〉