difference is unity or 1, and the last term the number of things proposed to be varied together: and the product will be the number of changes or variations required. EXAMPLES. (1) Six gentlemen that were travelling, met together by chance at a certain inn upon the road, where they were so pleased with their landlord, and each other's company, that in a frolic they made a contract to stay at that place so long as they, together with their landlord, could sit every day in a different order or position at dinner. Quere, the time they staid. (2) I demand the number of changes that may be rung on 12 bells. Also, in what time they may all be rung, allowing 3 seconds to every round, and 365 days 6 hours to the year. (3) An accomptant told a gentleman, who had constantly 8 persons at his table, that he would gladly make a ninth, and was willing to give 20 guineas for his board, so long as he could place the said company at dinner differently from any one day before. This being accepted, what did his entertainment cost him per year? END OF BOOK II. 140 PART II. XXXVII. VULGAR FRACTIONS. A FRACTION is a part or parts of something considered as a unit or integer, and consists of two parts or quantities, one written over the other, with a line between them, as 4, 3, ,&c. The number placed below the line is called the denominator of the fraction, because it denominates or shows how many parts the unit is broken or divided into; and the number above the line is called the numerator, because it enumerates or shows how many of those parts are contained in the fraction. A vulgar fraction is either proper, improper, compound, or mixed. A proper fraction is such whose numerator is less than the denominator, as 4, 3, 3, 3, &c. An improper fraction is when the numerator is equal to, or greater than, its denominator, as 2, 18 ΤΣ 247, &c. A compound fraction is the fraction of a fraction, and known by the word of, as 3 of 3 of 4, &c. A mixed number is composed of a whole number and a fraction, as 44, 12, 142, &c.. XXXVIII. REDUCTION OF VULGAR FRACTIONS. Case 1. To reduce a vulgar fraction to its lowest terms. RULE. Divide the greater term by the less, and that divisor by the remainder following, till nothing remain: then by the last remainder divide both parts of the fraction, and the quotients will give the fraction required. If the remainder be 1, the fraction is already in its least terms. 1. When the numerator and denominator each of them end with ciphers, strike off an equal number of ciphers in both, and the remaining figures will be a fraction of the same value, which reduce to its lowest terms. 2. When you discern any number will equally divide both numerator and denominator, you may abbreviate the fraction thereby. (8) Reduce, 324' T44' 96 , and, to their lowest terms. Case 2. To reduce a compound fraction to a single one. RULE. Multiply all the numerators together for a new numerator, and the denominators for a new denominator. Reduce the new fraction to its lowest terms by the last case. When it can be done, you may cancel the fractions, by dividing the numerator and denominator of any two terms by the same number, and use the quotient instead thereof. EXAMPLES. TO (9) Reduce 4 of of to a single fraction. (10) Reduce of 2 of 4 to a single fraction. (11) Reduce 3 of of to a single fraction. Case 3. To reduce whole or mixed numbers into an improper fraction. RULE. 1. If the whole number have no assigned denominator, a unity subscribed underneath must be the denominator. 2. If the whole number have an assigned denominator, multiply the whole number by the assigned denominator, and the product will be the numerator to the assigned denomina tor. 3. If the whole number have a fraction annexed, multiply the whole number by the denominator of the fraction, and to the product add the numerator for a new numerator, which place over the denominator. EXAMPLES. (12) Reduce 12, 27, and 176, to fractions. (13) Reduce 27 into a fraction, whose denominator shall be 12. (14) Reduce 42 to an improper fraction. Case 4. To reduce an improper fraction into its equivalent or proper terms. RULE. Divide the numerator by the denominator: the quotient gives the whole number, and under the remainder (if any) subscribe the denominator. EXAMPLES. (18) Reduce to its proper terms. Case 5. To reduce fractions of different denominations to fractions of equal value, that shall have one common denomi nator. RULE. Multiply each numerator, taken separately into all the denominators but its own, and the products will be the new numerators: then multiply all the denominators into one another for a common denominator. EXAMPLES. (22) Reduce, I, and to a common denominator. (23) Reduce, 4, 4, and & of 2, to a common denominator. (24) Reduce, 1, 4, 3, and 4, to a common denominator. (25) Reduce, 7, 4, and 4 of 3, to a common denominator. Case 6. To reduce fractions of one denomination to another, retaining the same value.. RULE. 1. If the fraction given is to be brought from a less to a greater denomination, multiply the denominator by the parts contained in the several denominations between it and that you would reduce it to, for a new denominator; which, placed under the given numerator, will give the new fraction, which reduce to its lowest terms. 2. If the fraction given is to be brought from a greater to a less denomination, then multiply the numerator in the same manner as you did before the denominator, and place over the given denominator, and it will give the new fraction, which also reduce to its lowest terms. (26) Reduce (27) Reduce (28) Reduce (29) Reduce Τ EXAMPLES. of a shilling to the fraction of a guinea. (30) Reduce 4 of a guinea to the fraction of a farthing. of a dwt. to the fraction of a lb. Troy. (31) Reduce (32) Reduce pois. of a cwt. to the fraction of a lb. avoirdu (33) Reduce of a drachm to the fraction of a cwt. (34) Reduce (35) Reduce (36) Reduce of a lb. Troy to the fraction of a dwt. of a league to the fraction of a pole. |