Arithmetic, Part 1 · Section 23
THE GOLDEN RULE, Or Rule of Three.
THis is the moſt uſeful and moſt eaſie Rule in Arithmetique, and deſerves a Golden name. It is when there are three numbers given or known to find a fourth in proportion with them.
But 4 numbers are in proportion, and called Proportional, when as the firſt is to the third, ſo is the ſecond to the fourth.
As if there were given, 3, 4, and 6, to find a fourth which may be to 4, as 6 to 3, that is double, and that fourth number is 8; And this is called Proportion direct: and the Rule whereby it is done, The Direct Rule.
There is alſo another proportion called Reciprocal; which is when as the firſt is to the third, ſo is the fourth to the ſecond: As 3, 4, 6, and 2, this is called The Reverſe Rule.
In direct proportion, the product of the two middle numbers multiplied together, is ever equal to the product of the firſt and laſt multiplied together; which ſerves not onelyonly for a Proofe, but a ground of the Rule, which Rule ſhall here follow: the Reverſe Rule being deferred till we have done with this.
The Rule Direct.
Multiply the ſecond terme (or number) by the third and divide the product by the firſt; the quotient ſhall be the fourth number deſired.
Example. Let the three numbers given be 2, 6 3, multiply 6 by 3, the product is 18, then divide 18 by 2, the quotient is 9, which is th fourth number in proportion with 2, 6, and 3.
For as 2 to 3, ſo 3 times 2, which is 6, is to 3 times … ▪ which is 9.
And ſo the product 18 divided by 2, and the quotient 9, cauſeth that the product of 2 into 9 ſhall be alſo 18, and conſequently if 2 be the firſt of the 4 proportional numbers, and 6 and 3 the two middlemoſt, then 9 is the laſt.
Otherwiſe.
Divide the ſecond by the firſt, and multiply the third by the quotient, the product ſhall be the fourth.
So if one divide 6 by 2, the quotient is 3, by which multiply 3, the product is 9, for the fourth number, as before, otherwiſe ſtill this Rule might be expreſſed; but where the firſt way is ſo ſhort and clear, there many other wayes would rather trouble then help the perſon that ſhould uſe them.
In the firſt way (which here we mean to uſe and no other) if the firſt number be 1, then the product of the ſecond and third gives the fourth, without any Diviſion. Or if the ſecond, or third number be 1, then there needs no multiplication, but dividing the greater of them by the firſt, the quotient (in whole numbers for yet we ſpeak of them) is the fourth number, which was ſought.
Note 1.
To know when to uſe the direct, or the Reverſe Rule, conſider, if more, require more; or if leſſe, require ſtill leſſe: then uſe the direct Rule, but if more require leſſe, or leſſe more, then uſe the Reverſe Rule, this will be eaſily underſtood when we come to examples.
Note 2.
To know how to place the three numbers when they are confuſedly given, remember that two of them are alwayes of one denomination, as both pounds, or both ſheep, or both yards, or acres; and the other number hath another denomination, now know that this ſingle number is ever the ſecond number in order.
And one of the other two, namely, that which hath ſome relation to this ſecond, is the firſt; and the other is the third number, whoſe relation is ſought for in the fourth, whence its plaine that the ſecond and fourth are alſo of the ſame denomination.
And having premiſed theſe things, let us now exemplifie the Rule in ſome queſtions.
Queſtion 1.
If 3 yards of cloth coſt 4 li. what ſhall 21 yards coſt?
Set the numbers in order as in the example, if 3 yards coſt 4 li. what 21 yards? Here you ſee that the firſt number and the third number are both of one denomination, viz. both yards, and the ſecond number is of another denomination, namely pounds, wherefore the fourth number which is ſought for, muſt be alſo pounds, therefore multiplying (according to the Rule before given) the ſecond number by the third, and dividing the product by the firſt, the quotient ſhall anſwer the queſtion.
Firſt, 21 multiplied by 4, (which is the third …mber multiplied by the ſecond) produceth 84, which divided by 3 the firſt number, the quotient is 28 li. and ſo much ſhall 21 yards coſt: for 28 is to 4, as 21 to 3, ſeeing each containes either 7 times.
And the work will ſtand thus, …
Queſtion 2.
If 4 men eat 2 pecks of corn in a week, how many pecks ſhall ſerve 100 men?
Place your numbers as here you ſee, then multiply 100 by 2, (that is the third number by the ſecond, and the product is 200, which divided by 4, the quotient is 50, for the number of pecks required.
…
Queſtion 3.
If 20 ſheep coſt 13 pound 13 ſhillings 4 pence, what is that for every ſheep?
Turn the ſhillings and pounds into pence, thus, …
| Multiply 12 s. by 12 the product is | 156 |
| And 13 li. by 240 (becauſe 240 pence make one pound) the product is | 3120 |
| To which adde the 4 d. | 4 |
| It makes in all | 3280 |
Then the queſtion will be, If 20 ſheep coſt 3280 pence, what ſhall one ſheep coſt?
…
By the rule before delivered, I ſhould multiply the ſecond number by the third, but in this example, the third number being 1, it doth not multiply; I therefore divide 3280 the ſecond number, by 20 the firſt number, and the motient 164, is the price of one ſheep in pence, which divided by 12, the quotient is 13 s. and 8 d. remaining, the price of every ſheep therefore is 13 s. 8 d.
Queſtion 4.
How many 10 inch tiles will pave a floore that contains 16 ſquare yards?
Firſt, remember there are 36 inches in one yard in length; which multiplied into 36, gives 1296, for the ſquare inches in one ſquare yard; multiply 1296 therefore by 16, thence comes 20736, the ſumme of all the 16 yards in inches.
Secondly, ſeeing every tile is 10 inches in length, and 10 in breadth, multiply 10 by 10, it produceth 100 for the ſquare inches in one tile; See the manner of worke.
…
Then by the golden Rule.
If 100 inches require 1 tile; what ſhall 207; 6 inches require?
…
Here becauſe 1 doth neither multiply nor divide (as hath been ſeveral times intimated) I therefore divide the third 20736, by the firſt 100, the quotient is 207, and 36 remaining.
So it appears that 207 is too little, and 208 too much to doe the worke: the juſt number being 207 36/100, we ſhall not trouble the reader with this till he know ſomething of Fractions.
Queſtion 5.
If 100 li. give 6 li. intereſt for a year, how much ſhall 750 li. give?
Multiply 750 by 6, the product is 4500, which divided by 100, the quotient is 45 li. for the thing required.
…
Queſtion 6.
If 750 li. give 45 li. intereſt for a year, what ſhall 100 li. give?
Multiply 45 by 100, the product is 4500, which divided by 750, the quotient is 6 li. for the intereſt of 100 li. for a year.
…
Many other queſtions might be added, but the rule is ſo plaine that it needs them not; and ſo general, that he which can reſolve one, may aſwel reſolve any other: And for that reaſon; and becauſe in all the Rules which follow, this rule will be conſtantly made uſe of, I will ſay no more of it here.