Arithmetic, Part 1 · Section 30 – Section 33
Reduction.
REductioon of Fractions is threefold.
1 To reduce one fraction (which is not already in the leaſt) to a leſſer Denomination.
2 To reduce many fractions of divers denominations, to one denomination.
3 To reduce any fraction from one denomination (as neer as may be) to any other denomination deſired.
For the firſt of theſe, divide both the Numerator, and the Denominator, by the greateſt common diviſor that you can think of; the two quotients being placed reſpectively in a fraction, that fraction ſhall be equal to the former fraction, and in leſſer termes.
So (in the 3 Example of Diviſion) to reduce 2880/360 to 8/1, divide 2880 by 360, quotient is 8, then divide 360 by 360, the quotient is 1, and the new fraction 8/1 is equal to the former fraction, 2880/360, and in leſſe termes; as you may ſee. But to find the greateſt common diviſor.
The Rule is,
Divide the greater terme by the leſſer (I mean by termes, the Numerator and Denominator) and by the remainer (if any be) divide the diviſor, and if any thing ſtill remains, by that divide the laſt diviſor, continuing this courſe till nothing remain greater then Ʋnity; that diviſor which is laſt of all is the greateſt common meaſure of both termes, by which both being divided, and the quotients placed like a fraction, that fraction ſhall be equal to the former fraction, and in the leaſt termes.
Example.
Reduce 148/16 to the leaſt termes; firſt divide 148 by 16, the quotient is 9, and 4 remaines: againe, divide 16 by 4, the quotient is 4, and nothing remaines; wherefore taking 4, (the laſt diviſor) for the greateſt common diviſor, by it divide 148, the quotient is 37, and by it divide 16, the quotient is 4. Theſe two laſt quotients placed orderly in a fraction, make 37/4, which is equal to 148/16, and in the leaſt terms, for no number greater then 1, will divide evenly both 37 and 4.
Other wayes there are of leſſening fractions, as dividing the termes (if they be even Numbers) by 2, and the quotients (if even) again by 2, or elſe by 3, o… any other Number that will divide them both evenly, that is, leave nothing remaining, but the former Rule being general and eaſie, ſhall ſerve for all.
Now ſecondly, to reduce many Denominations to one common denominator; let the fractions be ½, ¾, ⅘, ⅞, 9/10, to be reduced all to the denomination 3200.
THE RULE.
Multiply all the Denominators together: ſaying 2 times 4 is 8, and 8 times 5 is 40, and 40 times 8 is 320, and 320 times 10 is 3200, this laſt product 3200, ſhall be the common Denominator. Then to get Numerators for every one of them. Multiply every particular Numerator into all the Denominators except his own: As firſt, for the firſt, ſay 1 time 4 is 4, and 4 times 5 is 20, and 20 times 8 is 160, and 160 times 10 is 1600.
For the firſt Numerator; ſo the firſt fraction reduced is 1600/3200. Then for the ſecond Numerator: ſay, 3 times 2 is 6, and 6 times 5 is 30, and 30 times 8 is 240, and 240 times 10 is 2400. So the ſecond fraction reduced, is 2400/3200. After the ſame manner may the other three be reduced to 2560/3200 for the third: 2800/3200 for the fourth: and 2880/3200 for the laſt; theſe are ſeverally equal to the other, the firſt to the firſt, &c. as may be proved; thus,
Let the Unity be a pound Sterling, then
| s. | |||
| The ½ of it is | 10 | ||
| and ¾ is | 15 | ||
| and ⅘ is | 16 | d. | |
| and ⅞ is | 17 | 6 | |
| and 9/10 is | 18 | s. | d. |
| In all | 76 | 6 |
That is 3 whole Unites, and 16 s. 6 d. over, Turn 16 s. 6 d. all to ſix pences, it is 33, and becauſe 6 d. is the fortieth part of a pounds, therefore all the fractions are equal to 3 33/40.
Now adde the new Fractions (which being all of one denomination) may be added like whole Numbers: thus, …
Which divided by the Denominator 3200, the quotient is 3 2640/3200. Now 2640/3200, reduced to the leaſt termes, as hath been ſhewed how it may, will be 33/40, ſo the ſum of theſe alſo is 3 33/40. which is equal to the ſum of the fractions given to be reduced, and therefore they are equal in ſum, and might be thus proved equal ſeverally, that is, the firſt of them propounded, to the firſt reduced. Divide the Numerator 1600 by the Numerator 1, quotient is 1600. Alſo divide the Denominator 3200, by the Denominator 2, the quotient is alſo 1600, and ſo may any of the reſt be proved equal by the equality of quotients. But I leave it as plaine enough already.
Thirdly, Any fraction being given, to change the denomination to any other more requiſite, retaining ſtill (as neer as may be) the ſame value.
THE RƲLE.
Multiply the Numerator given, by the Denominator required, and divide the product by the Denominator given; the quotient ſhall be the Numerator required.
Example.
Let the Fraction given be 7/13 of a pound Sterling, what is that in twentieth parts or ſhillings? Multiply 7 by 20, the product is 140, which divided by 13, the quotient 10 10/13, that is, 10 s. and 10/13 of a ſhilling: which may be brought to pence thus, multiply 10 by 12, product is 120, which divided by 13, quotient 9 3/13 d. And againe, multiply 3 by 4, the product is 12, which divided by 13, quotient is 12/13 of a farthing, ſo ſeven thirteenths of a pound is 10 s. 9 d. and almoſt a farthing.
But he which is reſolved to have it in the ſmalleſt coine, may do it at firſt work; for ſeeing a farthing is the 960 part of a pound, multiply 7 by 960, they produce 6720, which divided by 13, the quotient is 516 farthings, and 12/13 of a farthing: theſe farthings may be turned to ſhillings, dividing by 48, or to pence by 4.
This Rule though it be brief and plain; is of great uſe in Arithmetick; either for turning Natural and ſurd Fractions into Decimals; or any other deſired Denomination, with ſuch facility and ſpeed as may be wiſhed.
Fractions of Fractions.
In reduction of Fractions, ſome make another, or more parts, as fractions of fractions for one: that is, when there is a part of a fraction, or a part of a part of a fraction, &c. to be valued in one fraction.
THE RULE.
Multiply all the Numerators together, the laſt product ſhall be the Numerator deſired: then multiply all the Denominators together, and this laſt product ſhall be the Denominator ſought.
Example.
Let the Fractions of Fractions propounded, be ⅘ of ¾ of ½, for ſo they are uſually written; and let the Numerator be multiplyed: ſaying, 4 times 3 is 12, and 12 times 1 is 12, the Numerator therefore required is 12: then for the Denominator, ſay, 5 times 4 is 20, and 20 times 2 is 40, for the Denominator required; and 12/40 is equal to ⅘ of ¾ of ½.
Proof.
Let the Unite be 40 s. one fifth of 40 is 8, and therefore ⅘ is 32, of which one fourth is 8, and ¾ is 24, of which one half is 12, and therefore 12/40 is the juſt ſum of all the Fractions, This needs no further exemplifying.
Addition.
TO adde many fractions into one ſum, conſider whether they be of one Denomination or divers; if of one, then adde all the Numerators together into one ſumme, that ſumme is the new Numerator: and the Denominator, in this caſe is not altered.
Example.
Let the fractions to be added be 2/4, 4/4, 5/4, ¼, Adde the Numerators: ſaying 2 and 4 is 6, and 6 and 5 is 11, and 11 and 1 is 12. So the ſumme of them all is 12/4, that is 3 Unites.
As, let the Unite be 20 s, one fourth is 5 s. and 2/4 is 10 s. and 4/4 is 20 s. which added to 10 s. is 30 s. then 5/4 is 25 s. which added to thirty ſhilling gives 55 s. And laſtly, ¼ is 5 s. which added to 55 s. makes 60 s. that is 3 times 20 s. that is 3 l. or 3 Unites.
But if the fractions to be added, be of divers denominations, as let them be 3/2, ¾, ⅘, ⅞, then (by the reduction afore-going) they muſt be turned all into one denomination, and then they will be 320/480, 360/480, 384/480, and 420/480, and may be added like thoſe before: thus, …
So the ſumme of all is 1484/480, or 371/120, that is, 3 11/120, which if it be money, and the Unite 1 l. it is then 3 l. 1 s. and 10 d. as may be tryed thus: Firſt, ⅔ of a pound, is 13 s. and 4 d. and ¾ is 15 s. and ⅘ is 16 s. Laſtly, ⅞ is 17 s. 6 d. Theſe all added together, the ſumme if 3 l. 1 s. 10 d.
Subtraction.
IN Subtraction of one fraction from another, if they be both of one denomination; it is done by taking the Numerator of one from the Numerator of the other, the remaine is the new Numerator, and the Denominator the ſame as before.
So if 2/9 be ſubtracted from 8/9. the remaine is, 6/9 the like of all others.
But if they be not of one Denomination, they muſt firſt be reduced to be ſo; then that which is ſaid before is ſufficient.
The golden Rule in fractions is the ſame as in whole Numbers, I will give you but one inſtance.
If ¾ of a yard of tape coſt ½ of a peny, what ſhall one inch, that is, 1/36 of a yard coſt?
Multiply the ſecond by the third, the product is 1/72, which divided by ¾, the quotient is 4/216 of a peny, for the price of 1/36 of a yard.
Otherwiſe,
Seeing ¾ of a yard may be turned to 27 inches; Say, if 27 coſt ½, what 1? divide ½ by 27, it makes 1/54 for the anſwer: which is equal to 4/216, and in the leaſt termes.
And whereſoever this may be done, to have the firſt and third Numbers fractions of one denomination, the beſt way is to work with their Numerators, not regarding their denominator at all: as if ⅔ coſt ¾, what 7/3? inſtead thereof write, if 2 coſt ¾, what 7? multiply ¾ by 7, it produceth 21/4, which divided by 2, the quotient is 21/8, and that is the anſwer in the leaſt terms.
And all this while it ſhould have been noted that the fractions are ever written in a ſmaller figure then the whole Numbers.
The Rule of Fellowſhip.
THis Rule is uſefull for Merchants, and all ſuch as trade in Companies, with a joint ſtock; and muſt ſhare a proportional part of the gaines, or loſſe; every one according to his ſtock which he laid in.
The Rule is two-fold, with equal time; or with unequal time.
That which is with equal time, is commonly called, The Rule of Fellowſhip without time.
Of this we will firſt ſpeak.
THE RULE.
As the whole joint ſtock, is to all the gain or loſſe; So is each mans particular ſtock, to his part of the gaine, or loſſe.
Example 1.
Two Merchants A. and B. buy 700 l. a yeare Land for ever, (when money is at 8, per cent.) for 14000 l. of which A. paid 8000 l. and B. 6000 l. after 5 yeares (money being fallen to 6 per Cent.) they ſell it for 18700 l. ſo there is gained 4700 l. how much of this muſt A. have?
Firſt for A.
Say, if 14000 gaine 4700, what 8000? anſwer, 2685 10000/14000.
Then for B.
If 14000, gaine 4700, what 6000? anſwer 2014 4000/14000. As by the following operation doth appeare.
(I) For A.
…
(II) For B.
…
Here note, that this work might have been much abreviated, if from each of the three numbers you had cut off two Cyphers towards the right hand, as hath been formerly ſhewed in the Compendium of Multiplication and Diviſion.
Now for the proof hereof,
| If you adde | 2685 10000/14000. | |
| which is the ſum that A. gained; | To—2014 4000/14000. | The ſumme of them is 4700. |
Which is equal to the total gain.
And according to the proportion of theſe two Numbers: that is, as 8 to 6, or 4 to 3. So they ought to have parted the yearly rent alſo, all the time they received it: that is, A. ought to have 400 l. yearly; and B. 300 l.
Example 2.
A. B. and C. joyne their moneys to make a ſtock of 25000 l. of which A. laid in 10000 l. B. 8000 l. and C. put in 7000 l. with this (after a certain time in trading) they gained 7500 l. how muſt this be parted?
Firſt for A.
Say, if 25000 gaine 7500, what 10000?
Or ſhorter, if 25 get ½, what 10? Multiply 7½ by 10, it produceth 75, which divided by 25, the quotient is …, that is, (reſtoring the 3 Cyphers) 3000 l. for A.
Then for B.
Say, if 25000 gaine 7500, what 8000?
Or ſhorter, if 250 get 75, what 80.
Multiply and divide as the Golden Rule requires, and to the quotient, reſtore the 2 Cyphers, then it will be 2400 l. for B.
Laſtly, for C.
Say, if 250 give 75, what 70? anſwer 21, to which put the 2 Cyphers, it makes 2100 for C.
And theſe three 3000, 2400, and 2100, being added together, make 7500. And have that proportion as the particular ſtocks had: and therefore the work is right.
(I) for A.
(II) for B.
…
(III) for C.
…
And if inſtead of gaining 7500 l. whereby every one is ſuppoſed to have his ſtock, and a part of the gaines; they had loſt 7500 l. then their particular ſtocks had not been due to them, but ſo much as would be left after their proportional parts of the loſſe were abated.
Example 3.
A. B and C. with a joynt ſtock of 25000 l. gaine 7500: of which A. gets 3000. B. 2400. C. 2100, what was their ſtock?
This is but the Converſe of the former, therefore ſay, if 5700 require 25000, what doth 3000 require? 10000 for A. and ſo work for the other two.
Many examples are of little uſe, (except to load the Readers memory) where the Rule is ſo ſhort and plain, I will therefore adde no more to this part of the Rule, but immediately come to the Rule of Fellowſhip with time.