Arithmetic, Part 1 · Section 24 – Section 29
The Golden Rule Reverſe.
Queſtion 1.
IF 12 workmen do any piece of worke in 8 moneths: how many workmen ſhall do the ſame in 2 moneths?
THE RƲLE.
Multiply the firſt term by the ſecond; and divide the Product by the third, the quotient is the number deſired.
Here 12 is not the firſt number, though it be firſt named; but the three numbers placed in order, ſtand thus, 8, 12, 2, for the middle terme muſt alwayes be of the ſame denomination with that which is required.
Now multiply 12 by 8, the product is 96, which divided by 2, the quotient is 48, which anſwers the queſtion, As in the example, …
For if 8 moneths require 12 men; then (a fourth part of 8) 2 moneths ſhall require (four times 12, that is) 48 men.
For here leſſe requires more, that is leſſe time, more hands; and therefore is it wrought by the Reverſe Rule.
Queſtion 2.
How many Ells of Tapeſtry will ſerve to hang a roome 3 yards high, 6 yards long; and 5 yards broad; not regarding Doors, Windowes, or Chimney, but as if there were no ſuch?
Firſt, multiply 6 by 3, the product is 18, which doubled (becauſe there are 2 ſides called lengths) is 36 yards for all the length.
Secondly (for the ſame reaſon) multiply 3 by twice 5, that is by 10, the product is 30 yards, for all the breadth; which added to 36, gives 66 yards equal to all the length and breadth in yards.
But now becauſe Ells, that is, Flemiſh Ells (for ſuch meaſure are hangings ſold by) is equal to 3 quarters of a yard, that is, their Ell is to our Yard as 3 to 4. Say therefore if 4 give 66, what 3? multiply 66 by 4, it produceth 264; then divide 264 by 3, the quotient is 88. Againe, multiply 88 by 4, and divide the product (which is 352) by 3, the quotient is 117, and 1 remaining, to which the diviſor 3 being applyed; the number juſtly anſwering the queſtion is 117 Ells, and one third part of an Ell.
Note 1.
Becauſe here we had to deal with things which had equal length and breadth, that is ſquare yards, and ſquare ells; therefore one multiplication and diviſion was not ſufficient to proportion this; but if inſtead of working by 4 and 3, we had done it by their ſquares which are 16 and 9, it might have been performed at once, thus, multiply 66 by 16, product is 1056, which divided by 9, the quotient is 117⅓, as before, but I began not with this way, for I ſuppoſed my reader ignorant of ſquares.
Note 2.
It might alſo have been done, by reducing all the terms into quarters of a yard at the firſt, and after the number is found, reducing them againe to ells; but becauſe it is more proper to worke thus, till fractions have been taught, I leave that, and proceed to another queſtion.
Queſtion 3.
If 1 cloſe would graze 21 horſes for 6 weeks; then (ſuppoſing no waſte to be made) how many horſes would it feed for 7 weeks?
Multiply 21 by 6 it produceth 126, which divided by 7, the quotient is 18. At that rate therefore it would keep 18 horſes for 7 weeks.
Queſtion 4.
If 1 cloſe will feed 18 horſes for 7 weeks, how long ſhall it feed 63 horſes.
Multiply (according to the rule) 18 by 7, the product is 126, which divided by 63, the quotient is 2, therefore 2 weeks it ſhall keep them.
The like way ſerves for hay, oates, or any other proviſion for Man or Beaſt; which may be of uſe in Garriſons, and ſuch like caſes where ſcarcity may be feared, to proportion either the mouthes to the meat, or meat to mouthes
I will ſay no more of this Rule; Neither will I treat of the double Rule of Three, as a rule by it ſelfe; but come to the Rule of five numbers, which is held an abridgment of the other.
The Golden Rule Compound of five Numbers.
Queſtion 1.
IF a hundred pound weight (that is 112 pound weight) carryed 120 miles coſt 14 s. how much ſhall three quarters of a hundred, (that is 84 pound) coſt, being carryed 40 miles?
THE RULE.
Multiply the three laſt numbers one into another, (that is) the third by the fourth, and that product by the fifth; the laſt product ſhall be the Dividend.
Again, Multiply the two firſt numbers together; the product ſhall be the diviſor. This Diviſion being made, the quotient will be the number of ſhillings deſired.
Example of the former Queſtion.
Firſt, place your numbers according to the tenor of the queſtion thus, …
Your numbers being placed in order, reduce the 13 s. into pence, and it is 168 d. then multiply 168 by 84, the product is 14112, which multiplied by 40 the later product, it produceth 564480, for the dividend.
Then multiply 112 by 120 it produceth 13440, for the Diviſor.
Divide 5644800, by 13440, the quotient will be 42 pence; which is 3 s. 6 d. and anſwers the queſtion.
In this Rule the firſt number and fourth, alſo the ſecond and fifth; and alſo the third and ſixth, are of like denomination and nature.
Queſtion. 2.
If 100 li. for 6 moneths yield 3 li. intereſt, what ſhall 625 li. yeield for 36 moneths?
Place them, 100, 6, 3, 625, 36.
Multiply the three laſt, as before is ſhewed, the later product is 67500 for the dividend; And the 2 firſt multiplied make 600 the Diviſor, then divide 67500 by 600 (or 675 by 6, which is all one) the quotient will be 112 whole pounds; and 300 (or 3) remaining, which becauſe it is half the diviſor, fignifies the half of a pound; that is 10 ſhillings. So the anſwer to the queſtion is 112 li. 10 s.
…
Which might have been given in one denomination, namely 2250 ſhillings, if before the work the pounds had been turned into ſhillings, by multiplying them by 20, as hath been ſhewed before.
But ſince moſt queſtions, except ſuch as are ſtudied for the purpoſe, are apt to end in ſome fraction, I ſhall next treat of fractions.
OnelyOnly firſt, having ſpoken of the double Rule of Three, this may let you know, that all queſtions which are wrought at once by the compound rule of five; may be done at twice by the ſingle rule of three: and the doing of them ſo by two operations is is called the double rule.
As in our laſt queſtion there are two things conſiderable, the difference of money; and the difference of time.
Firſt, for the money,
Say, if 100 li. give 3 li. what 625 li.? anſwer 18 75/100 li.
Secondly, for the time;
Say if 6 li. give 18 75/100 li. what 36 mo. anſwer 112 50/100 li.
But this will be better underſtood anon; and then the Reader may uſe that which he likes beſt.
Of Fractions.
THe word Fraction ſignifies a breaking or breach of any intire thing into parts; and when a number is broken ſo, the parts (which muſt needs be every one leſſe then the whole; and the whole is accounted but One, or Ʋnity) being leſſe then Unity, are called fractions (that is fragments, or pieces) of Unity. Now the Unite, or intire number which is to be broken, may be any thing, as one pound, in reſpect of which, ſhillings and pence, and farthings are fractions; or one ſhilling, in reſpect of which, pence, and farthings are fractions; or one peny, in reſpect of which, farthings are fractions; and the like of weights and meaſures, or any other thing to be broken into parts.
In Fractions, we ſhall treat firſt of Numeration, then of Multiplication and Diviſion, then of Reduction; then laſtly, of Addition and Subtraction.
The reaſon of this Order will ſoon be ſeen; for Multiplication and Diviſion are here much eaſier then Addition, &c. and therefore ought to be learned before them.
Numeration.
NUmeration is nothing elſe but the way of writing Fractions; and that this may be done, we muſt conſider that any Unite, or Number repreſenting an Unite, may be broken into two parts equal; and then each of the parts is called one ſecond, or halfe; or it may be parted into three equal parts, and then each part is called one third; and two of them are called two thirds; and the like may be underſtood if it were parted into 4, 5, 6, 7, 8, 9, 20, 50, or 100, or how many ſoever.
Now to write theſe; doe thus,
Write one half Thus ½
Write one third Thus ⅓
Write one fourth Thus ¼
Write one fifth Thus ⅕
Write one ſixth Thus ⅙
Write one ſeventh Thus 1/7
Write one eighth Thus ⅛
Write one nine Thus 1/9
Write one tenth Thus 1/10
In every one of theſe 10 fractions, the Number below the line is called the Denominator, and it ſhews into how many parts the Unite is broken.
The Number above the line ſhews how many of thoſe parts are taken, or contained in the fraction, and is therefore called the Numerator: So in the fraction ⅗: the Denominator … ſhewes the Unite to be broken into 5 parts: and the Numerator 3 ſignifies 3 of ſuch parts to be contained in the fraction; which fraction therefore is called three fifths.
And here it is plaine, that as the Numerator is in proportion to the Denominator: ſo is the fraction to 1, for 3/3 or 5/5: or any the like, is equal to 1.
And therefore all fractions are quotients of leſſer numbers divided by greater, as 4/7 ſignifies 4 to be divided by 7, and as the dividend 4, is to the diviſor 7: ſo is the quotient 4/7 to Unity.
And therefore this line of ſeparation which is drawn between the dividend and diviſor, doth properly ſignifie diviſion.
Hitherto we have ſpoken onelyonly of ſuch fractions as are leſſe then 1, and thoſe are called Proper Fractions: but there are alſo 2½, 3¾, 5 1/7, 6⅗, and the like mixed Numbers; which ſo written ſignifie two and an half, 3 and 3 quarters, five and a ſeventh, 6 and 3 fifths. Theſe by multiplying the whole Numbers, by the Denominator, and to the product adding the Numerators reſpectively are turned to 5/2, 15/4, 36/7, 33/5, which are called Improper fractions, becauſe every one of them contains more then Unity.
Theſe, nevertheleſſe may be multiplyed, divided, added, or ſubtracted in the ſame way as are proper fractions. And this ſhall ſerve for Numeration of Fractions.
Multiplication.
THE RULE.
MƲltiply all the Numerators together, the laſt product ſhall be the Numerator of the product required: Likewiſe multiply all the Denominators together, the laſt product ſhall be the Denominator of the product ſought.
Example 1.
If ⅗ be multiplyed by 4/9, multiply the Numerator 3 by the Denominator 4, the product is 12, for the Numerator of the new product. Alſo multiplying the Denominator 5, by the Denominator 9, they produce 45, for the Denominator of the deſired product, ſo that product which was required, is 12/45.
Example 2.
If ½, ¾, ⅘, 5/9, and 3/11 were to be multiplyed all together, begin with the Numerators, ſaying, once 3 is 3, and 3 times 4 is 12, and 12 times 5 is 60, and 60 times 5 is 180, for the Numerator: Then multiply the Denominators: ſaying 2 times 4 is 8, and 8 times 5 is 40, and 40 times 9 is 360, and 360 times 11 is 3960, for the new Denominator. So that the product of all theſe is 180/3960, that is equal to 1/22, as ſhall be ſeen hereafter in Reduction.
And thus it appears that proper fractions being leſſe then one, are ſtill made leſſe by multiplying; as here the product 1/22 is much leſſe then 3/11, which is the leaſt Multiplyer, and the reaſons hereof is plain, for ſeeing Multiplication is but the taking of a Number, a certaine number of times, if that number of times be more then 1, then the Number to be taken is increaſed by being taken more then once; but if the Number of times be 1, it is not increaſed, nor diminiſhed, but is ſtill the ſame; Laſtly, If that number of times be leſſe then 1, as ½, the number not being taken once, but half of once, produceth a number leſſe by half; that is the half of the number to be taken; and the like reaſon is of all others.
Example 3.
Multiply the mixt Numbers 3½, 4⅓, and 5¾: Firſt, (as hath been ſhewn already) turn them to improper fractions: thus, firſt, ſay 2 times 3 is 6, and 1 is 7. So the firſt is 7/2. Secondly, 3 times 4 is 12, and 1 is 13: ſo the ſecond is 13/3. Laſtly, 4 times 5 is 20, and 3 is 23: ſo the laſt is 23/4 Now the fractions to be multiplyed are 7/2, 13/3, and 23/4; Firſt, for a new Numerator, ſay, 7 times 13 is 91, and 91 times 23 is 2093, for a new Numerator.
Then ſay, 2 times 3 is 6, and 6 times 4 is 24. So the new Denominator is 23.
And the product of all theſe fractions is 2093/24, that is, if real diviſion be made, 87 5/24.
Diviſion.
DIviſion of one fraction by another, is but croſſe multiplication of them; that is, the Numerator of the one, by the Denominator of the other, and hereby the proportion of one fraction to another is ſeen.
Example 1.
Divide ¾ by 6/8, to do it, ſet them thus: and multiply as the croſſe leads, ſaying, 3 times 8 is 24, for a new Numerator, and 6 times 4 is alſo 24, for a new Denominator; ſo the quotient is 24/24, that is 1, which ſhewes the fractions to be equal one to another.
Example 2.
In diviſion it is to be remembred that the Numerator of the quotient ever ariſeth of the Numerator of the Dividend; And the Denominator of the quotient comes of the Denominator of the Dividend, each being croſſe multiplied, as before.
If a Fraction be to be divided by a whole number, multiply the Denominator by that number, the product gives the new Denominator, and the Numerator remaines the ſame. So if ¼ be divided by 9, ſay 9 times 4 is 36. So the quotient is 1/36.
Or if ¼ were to be multiplyed by 9, the product (by multiplying the Numerator by 9), will be 9/4: that is, 2¼.
Example 3.
For 320/8 is equal to 40, and 45/9 equal to 5, but 40 contains 5 eight times.
And ſo in the ſecond example, it may be proved, that as 27 to 20, ſo is ⅗ to 4/9. For firſt multiply the two middlemoſt, then 20 times ⅗ is 60/5, that is 12.
Secondly, multiply the firſt and laſt, and then 27 times 4/9 is 108/9: that is alſo 12.
Wherefore by that which hath been ſaid in the Golden Rule, the four Numbers 27, 20, 3/5, 4/2, are proportional.