An introduction of the first grounds or rudiments of arithmetick plainly explaining the five common parts of that most useful and necessary art, in whole numbers & fractions, with their use in reduction, and the rule of three direct. Reverse. Double. By way of question and answer; for the ease of the teacher, and benefit of the learner. Composed not only for general good, but also for fitting youth for trade. / By W. Jackson student in arithmetick.

Reduction.

Q. Now tell me what is Reduction?

A. Reduction is a changing of numbers or fractions out of one form or denomination, in∣to another.

Q. Why are they so reduced?

A. Either for more ease in working, or for the more easie estimation of the value of two or more fractions, either com∣pared one with another, or ad∣ding them together, or sub∣stracting one from another.

Q. How are fractions of several denominations reduced to one de∣nomination?

A. First multiply the deno∣minators together, and set the product for a common denomi∣nator;

then multiply the nume∣rator of the first, by the deno∣minator of the second, and set the product for a new numera∣tor for the first fraction; and multiply the numerator of the second, by the denominator of the first, and set the product for the new numerator of the se∣cond fraction, and so are both those fractions brought into one denomination.

Q. Give an Example hereof?

A. Two thirds and three quarters, being so reduced, make 8/12 for 2/3, and 9/12 for 3/4, which yet still retain their first value, but are now both of one deno∣mination.

Q. You have shewed how to re∣duce two Fractions into one deno∣mination, but what if there bee three or more?

A. Then I must multiply all the denominators together, and set the product down so many times as there bee fractions, for a common denominator to them; and then multiply the numerator of the first, by the denominator of the second, and the product by the denominator of the third, and that by the de∣nominator of the fourth, if I have so many, and so forward, and the product is a new nu∣merator for the first fraction; then multiply the numerator of the second, by the denomi∣nator of the first, and the pro∣duct by the denominators of the third and fourth, and so forward, if you have so many, and set the product for a new numerator for the second fra∣ction, and multiply your third

numerator by the first and se∣cond denominators, and the product by the denominator of the fourth, if you have to many, and that product is your third numerator; then if you have so many, multiply the numerator of the fourth by the other three denominators, the product is a new numerator for the fourth fraction, &c.

Q. Must this order then bee ob∣served still, when you have many fractions?

A. Yea, alwaies multiply all the denominators together for a new denominator, and one numerator by all the other de∣nominators, except its own, the product is a new numerator for that fraction whose nume∣rator was taken to multiply by.

Q. Is there any other form of reducing to one denomination?

A. Yea, several varieties.

Q. What is one way?

A. This is one, when you have found a new denominator as above, then divide the same by the denominator of any of your fractions, and multiply the quotient by the numerator of the same, and the product shall bee a new numerator for that fraction, &c.

Q. What is another way?

A. If the lesser denominator will by any multiplication make the greater, then note the multiplier, and by it multiply the numerator over the lesser denominator, and in place of the lesser put the greater deno∣minator, and so it is done with∣out any of the other fractions.

Q. What other sort of Redu∣ction is there?

A. A second sort is when fractions of fractions are to bee reduced to one denomination.

Q. How is that done?

A. By multiplying the nu∣merators each into other, and setting the product for a new numerator, and in like sort mul∣tiply all the denominators each into other, and take that pro∣duct for a new denominator, and then they express it in the parts of a simple fraction.

Q. What if I have a mixt num∣ber of unites, and parts to bee re∣duced into fraction form?

A. Multiply the unites or whole number by the denomi∣nator of the fraction, and there∣to add the numerator of the fraction, and set the offcome

above the line over the said de∣nominator.

Q And how reduce you such an improper fraction into its unites and parts?

A. I must divide the nume∣rator by the denominator, and the quotient shews how many unites it contains, and the re∣mainder, if any bee, is the nu∣merator of a fraction, over and above the said unites in the quotient, to which the divisor is denominator.

Q. How is a whole number re∣duced into the form of a fraction?

A. By multiplying it by that number, which you would have denominator to it.

Q. What is next in Reduction of Fractions?

A. To reduce a fraction into its smallest or least t••rms.

Q. What bee the terms of a Fraction?

A. The terms bee the nume∣rator and denominator whereby it is exprest.

Q. What mean you by greatness and smalness of terms?

A. By great, I mean, when a fraction is exprest in great num∣bers, as 480/960 which in its smal∣lest terms is (1/2) one half.

Q. How are such reduced into their smallest terms?

A. If they be both even num∣bers by halfing them both so of∣ten as you can, but if they come to bee odd, or either of them odd, then by dividing them by 3, 5, 7, 9, &c. which will divide them both, without any re∣mainder, and take the last numbers for the terms of the fraction.

Q. But is there no way to disco∣ver what number would reduce a fraction into its smallest terms, but by halfing or parting in that sort?

A. Yes, thus, divide the de∣nominator by the numerator, and if any thing remain, divide the numerator by it, and if yet any thing remain, divide the last divisor by it, and so do till nothing remain, and with your last divisor, which leaves no re∣mainder, divide the numerator of the fraction, and the quotient is a new numerator, and divide the denominator in like sort by it, and the quotient is a new de∣nominator.

Q. What if no number will di∣vide them evenly, till it come to one?

A. Then the fraction is in its

smallest terms already.

Q. How reduce you fractions of one denomination, into another denomination?

A. I multiply the numerator by the denominator, into which I would reduce it, and divide the product by the first denomi∣nator, and the quotient is the new numerator.

Q. Give an Example of this.

A. If 3/4 bee to bee turned into twelfth parts, I multiply 12 by 3 comes 36, which I divide by 4, the quotient is 9, so it is 9/12, e∣qual to 3/4.