Cockers arithmetick being a plain and familiar method suitable to the meanest capacity for the full understanding of that incomparable art as it is now taught by the ablest school-masters in city and countrey / composed by Edward Cocker ... ; perused and published by John Hawkins ...

CHAP. XIX. Reduction of Vulgar Fractions. (Book 19)

1. WHat a Vulgar fraction is, and its parts, and several kinds, hath been already shewed in the 19, 20▪ 21, 22, 23, 24 and 31 definitions of the first Chapter of this book, which the Learner is desired diligently to observe before he proceed.

2. To Reduce a Vulgar fraction (which discovereth the Principal knowledge of fractions, and therefore ought greatly to be Regarded) we shall discover plainly under these 8 several heads (or Rules) fol∣••owing, viz.

1. To Reduce a mixt number into an improper Fraction.

2. To Reduce a whole number into an Improper Fraction.

3. To Reduce an Improper fraction into its equivalent whole (or mixt) number.

4. To Reduce a Fraction into its lowest terms equivalent to the Fraction given.

5. To find the value of a Fraction in the known parts of Coin, Weight, Mea∣sure, &c.

6. To Reduce a Compound Fraction to a simple one of the same value.

7. To Reduce divers Fractions having unequal denominators, to fractions of the same value having equal denomination.

8. To Reduce a Fraction of one denomi∣nation to another of the same value.

I. To Reduce a mixt Number to an Improper fraction.

The Rule is* 1.1

Multiply the Integral part (or whole Number) by the denominator of the Fra∣ction, and to the product add the Nume∣rator, and that sum place over the deno∣minator, so this new fraction shall be e∣qual to the mixt number given. As for Example

1. Reduce 18 3/7 into an improper fra∣ction, multiply the whole number 18 by 7 the denominator, and 〈 math 〉〈 math 〉 to the product add the numerator 3 the sum is 129 which put over the denominator 7 and it makes 129/7 for the answer as per margent.

2. Reduce 183 5/12 to an improper fraction facit. 2201/1••.

3. Reduce 56 13/21 to an improper fracti∣on facit 1189/21.

II. To Reduce a whole number to an Improper fraction.

The Rule is

Multiply the given number,* 1.2 by the intended denominator and place the product for a numerator over it, as for example.

1. Let it be required to reduce 15 into a fraction whose denominator shall be 12.

To effect which I multiply 15 by the in∣tended denominator (12) the product is 180 which I place over 〈 math 〉〈 math 〉 12 as a numerator and it makes 180/12 which is equal to 15 which was required as per mar∣gent.

2. Reduce 36 into an improper fraction whose denominator shall be 20 facit 936/26.

3. Reduce 135 into an improper fracti∣on whose denominator shall be 16 facit 2160/16.

III. To Reduce an improper fracti∣on into its equivalent, whole or mixt number.

The rule is,

Divide the numerator by the denomina∣tor, and the quotient is the whole number equal to the given fraction, and if any thing remain put it for a numerator over the di∣visor, example.

1. Reduce 436/8 into its equivalent mixt number, divide the 〈 math 〉〈 math 〉 numerator 436 by the denominator 8 and the quotient is 54 and 4 remains which put for a nu∣merator, over the divisor 8 the answer 54 4/8 as per margent.

2. Reduce 3476/15 to a mixt number, facit 231 11/15.

3. Reduce 15576/136 to a mixt number, facit 114 72/136.

IIII. To Reduce a fraction into its lowest terms▪ equivalent to the fra∣ction given.

The Rule is.

1. If the numerator and denominator are even numbers take ½ of the one and half of the other as often as may be and when either of them falls out to be an odd num∣ber, then divide them by any number that you can discover will divide both numera∣tor and denominator without any remain∣der; and when you have thus proceeded as low as you can reduce ••hem then this new fraction so found out shall be the fra∣ction you desire, and will be in vallue equal

to the given fraction, example.

1. Let it be required to reduce 192/336 into its lowest terms.

192	96	48	24	12	4
336	168	84	42	21	7

First I take the half of the nume∣rator 192 and it is 96 then half of the deno∣minator and it is 168, so that now it is brought to 96/168 and next to 48/84, and by halv∣ing still to 24/42 and their half is 12/21 and now I can no longer half, it because 21 is an odd number, wherefore I try to divide them by 3, 4, 5, 6, &c. and I find 3 divides them both without any remainder, and brings them to 4/7 as per margent.

So I conclude 4/7 thus found to be equal in vallue to the given fractions 192/336.

2. What is 103••/1184 in its lowest terms? an∣swer 7/8.

3. What is 1342/1586 in its lowest terms? answer 11/13.

There is yet another way more excellent then the former to reduce a fraction into its lowest terms,* 1.3 and that is by find∣ing a common measurer, viz. the greatest number that will divide the numerator and denominator without any remainder, and by that means reduce a fra∣ction to its lowest terms at the first work, and to find out this common measurer, di∣vide

the denominator by the numerator, and if any thing remains divide your divisor thereby, and if any thing remains then divide your last divisor by it, do so untill you find nothing remains, then this last divisor shall be the greatest common measurer, which will divide both numerator denominator and reduce them into their lowest terms at one work.

Example.

4. Reduce 228/304 into its lowest terms by a common measurer. To effect which I divide the denominator 304 by the numera∣tor 228 and there remains 76, then I divide 228, (the first divisor) by 76 (the remain∣der) and it quotes 3 and remains nothing wherefore the last divisor 76 is the com∣mon measurer, by which I divide the nume∣rator of the given fraction, viz 228 it quotes 3 for a new numerator, then I divide the denominator 304 by 76 and it quotes 4 for a new denominator, so that now I have found ¾ equal to 228/304.

5. Reduce 6048/7392 into its lowest terms by a common measurer, facit 9/11.

6. Reduce 3081/20••82 into its lowest terms by a common measurer facit 13/6.

A Compendium.

Note that if the numerator and denomi∣nator of a fraction end each with a Cypher

or Cyphers then cut of as many cyphers from the one as from the other and the re∣maining figures will be a fraction of the same vallue, viz. 3400/710•• will be found to be reduced to 34/71 by cutting of the 2 cyphers from the numerator and denominator thus, 34/17|00/00 and 460/700 will be 46/70 thus 4••/70|0/0 &c.

V. To find the vallue of a fraction in the known parts of Coyn, weight, &c.

The rule is.

Multiply the numerator by the parts of the next inferiour denomination that are equal to an unit of the same denomination with the fraction, then divide that product by the denominator, and the quote gives you its value, in the same parts you multi∣plyed by▪ and if any thing remain multiply it by the parts of the next inferiour denomi∣nation, and divide as before, do so till you can bring it no lower and the several quoti∣ents will give you the vallue of the fraction as was required, and if any thing at last remain place it for a numerator over the former denominator, example.

1. What is the value of 27/29 l. Sterling? To Answer this question I multiply the nu∣merator 27 by 20 (the shi••••ings in a pound) the product is 540 which I divide by 29 (the denominator) and the quotient is 18 s. and there Remains 18 which I multiply by 12 pence and the product (216) I divide by the denominator 29 the quotient is 7 d. and 13 Remains, which I multiply by 4 farthings, the product is 52, which I still divide by 29, the quotient is 1 farthing, and there Remaineth 23, which I put for a Numerator over the denominator 29, so I find the vallue of 27/29 l. to be 18 s. 7 d. 1 qrs. 23/29 as per the following operation.

〈 math 〉〈 math 〉

2. What is the value of 11/15 l. Sterling? facit 14 s. 8 d.

3. What is the value of 28/137 l. Ster∣ling? facit 4 s. 1 d. 7/137.

4. What is 16/21 C. weight? facit 3 qrs. 1 l. 5 oz. 7/21.

5. What is 136/371 l. Troy weight? facit 4 oz. 7 p.w. 23 gr. 179/371.

6. What is 41/50 of a year? Answer 299 day. 7 hour. 12 min.

VI. To Reduce a Compound Fra∣ction to a simple one of the same value.

What a compound Fraction is, hath been shewed in Chap. 1. Definition 24, and to Reduce it to a simple Fraction of the same value.

The Rule is,

Multiply the Numerators continually and place the last product for a new Nume∣rator, then multiply the denominators continually, and place the last product for a new denominator. So this single Fraction shall be equal to the compound fraction given. Example.

1. Reduce ⅔ of ⅗ of ⅝ to a Simple Fraction.

Multiply the Numerators 2, 3, and 5

together, they make 30, for a new Nume∣rator; then I multiply the denominators 3, 5, and 8 together and their product is 120 for a denominator, so the simple Fracti∣on is 30/120 and cutting off the Cyphers it is 3/••12 equal to ¼.

〈 math 〉〈 math 〉

2. What is 7/10 of 5/9 of 4/7 of 11/12 An∣swer 1540/7560 or 154/756 or 77/373.

3. What is 11/12 of 13/14 of 21/29 Answer 3003/4872.

By this you may know how to find the value of a Compound Fraction, viz. first Reduce it to a simple one, and then find out his value by the 5 Rule foregoing.

4. What is the value of ¾ of ⅚ of 9/10 of a pound Answer 11 s. 3 d.

VII. To Reduce Fractions of un∣equal denominators to Fractions of the same value, having equal Deno∣minators.

The Rule is,

Multiply all the denominators together, and the product shall be the Common de∣nominator? Then multiply each Numera∣tor into all the denominators except its own, and the last product put for a Nu∣merator over the denominator found out as before; So this new Fraction is equal to that fraction whose Numerator you mul∣tiplyed into the Denominators. Do so by all the Numerators given, and you have your desire. Example,

1. Redu•••• ••/4 ⅘ ⅚ and ⅞ into a com∣mon Denomination.

Multiply the Denominators 4, 5, 6, and 8 together continually, and the product is 960 for the common Denominator; then multiply the Numerator 3 into the Deno∣minators 5, 6, and 8, and the product is 720, which is a Numerator to 960 (found as before) so 720/96•• is equal to the first fra∣ction

¾, then I proceed to find a new Nu∣merator to the second fraction, viz. 4, and I multiply 4 (into all the denominators ex∣cept its own; viz.) into 4, 6, and 8, which produceth 768/960 equal to ⅘, then multiply the Numerator 5, into the denomina∣tors 4, 5, and 8, the product is 800/960 equal to ⅚. Then multiply the Numerator 7 into the denominators 4, 5, and 6, the pro∣duct is 840/960 equal to ⅞ and the work is done, so that for ¾ ⅘ ⅚ and ⅞ I have 720/960, 768/960 800/960 and 840/960.

2. Reduce 1114/1223 and 19/21 into a common denominator, faciunt 5313/5796 3528/5796 and 5244/5796.

VIII. To Reduce a fraction of one Denomination to another.

1. This is either Ascending, or Descend∣ing. Ascending when a fraction of a smal∣ler is brought to a greater Denomination; and Descending when a fraction of a great∣er Denomination is brought lower.

2. When a fraction is to be brought from a lesser to a greater Denomination, then make of it a Compound fraction, by comparing it with the intermediate Deno∣minations between it, and that you would

have it Reduced to, then (by the 6 Rule foregoing) Reduce your compound to a simple fraction, and the work is done. Ex∣ample.

Quest. 1. It is Required to know what part of a pound sterling 5/7 of a peny is?

To Resolve this, I consider that 1 d. is 1/12 of a shilling, and a shilling is 1/20 of a pound; wherefore 5/7 d. is 5/7 of 1/12 of 1/20 of a pound, which by the said 6th. Rule I find to be 7/1680 l.

Quest. 2. What part of a pound Troy weight is ⅘ of a peny weight? Answer ⅘ of 1/20 of 1/12 l. equal to 4/1200 l. Troy.

3. VVhen a fraction is to be brought from a greater to a lesser denomination, then multiply th•• Numerator by the parts con∣tained in the several denominations betwixt it, and that you would reduce it to, then place the last product over the denomina∣tor of the given fraction. Example,

Quest. 3. I would reduce ⅗ l. to the Fraction of a peny? to do which I multi∣ply the Numerator 3 by 20 and 12 the pro∣duct is 720 which I put over the denomi∣nator 5 it makes 720/5 of a peny, equal to ⅗ l.

Quest. 4. VVhat parts of an ounce Troy is ⅚ l.? Answer 60/••6 oz.

Notes

• * 1.1

  vide Chap. 1. defin. 31.
• * 1.2

  Vide Ch. 1. defin. 23.
• * 1.3

  Vide Ought. cla. Math. cap. 7.