of a unit is given to find the price of a quantity, how do you proceed? A. Multiply the price of one by the quantity, and the product will be the price of the quantity. If the price of one yard is 6 shillings, how would you obtain the price of five yards? A. Multiply 6 shillings, the price of one yard, by 5, the number of yards, and the product would be the price of 5 yards, because the price of 5 yards must be 5 times as much as the price of one yard. Where the price of one pound is given in farthings, how can you obtain the price of lcwt. or 112lb. ? A. By multiplying 2s. 4d. (the price of 1121b. at one farthing a pound) by the number of farthings which one pound cost. Why should that give it? A. Because 2 shillings 4 pence is the price of lewt., at one farthing a pound; and at two farthings it must be twice as much, at three farthings three times as much, and so on. When the price of 1lb. is given in pence, how may you obtain the price of lcwt. or 112lb.? A. By multiplying 9 shillings 4 pence, the cost of lewt. or 112lb., at one penny per pound, by the number of pence which one pound cost; because if lewt. cost, at one penny per pound, 9 shillings 4 pence, at 2 pence per pound it must cost double, and at 3 pence per pound thrice as much, and so on. What is compound division? A. It shows how often one number is contained in another of different denominations, or how often one number may be subtracted from another of different denominations, How do you proceed in the work? A. Place the divisor at the left hand of the dividend, and divide the left hand denomination the same as a sum in simple division, and if any thing remains, reduce it, by reduction, to the next lower denomination, and to the product add the next lower denomination of the dividend, then divide as before, and so continue to do, till the whole dividend be divided; the quotient figures will then be the answer in the same denominations as the dividend that produced them. Where the price of a quantity is given, how do you obtain the price of one yard, or one pound, &c.? A. Divide the price of the quantity by the quantity, and the quotient will be the price of a unit. Suppose the price of 4 yds. is given, how would you obtain the price of 1 yd.? A. By dividing the price of 4 yards by 4; the quotient will then be the price of one yard, because one yard must cost only one fourth part as much a$ 4 yards. Where you have the price of a quantity given, to find the price of some other quantity, how do you proceed? A. Divide the given price by the quantity of which it is the price, and the quotient will be the price of a unit, then multiply the price of one by the quantity which you wish to obtain the price of, and the product will be the price of the quantity required. How do you prove compound division? A. By compound multiplication; multiply the quotient by the divisor, and if the product equal the dividend the work is right. NOTE. The teacher should not wait to have the student complete the compound rules before he commences questioning him; but the questions should be asked as the student progresses in the rules. A little time should be spent every day by the teacher in questioning his class, and explaining the rules and their use in business. VULGAR FRACTIONS A vulgar fraction is a part, or the parts of a unit or integer, expressed by two numbers, one placed directly above the other with a line between; thus, signifies one fourth of 1, and signifies three eighths of one, &c. A vulgar fraction arises from unfinished division. The number above the line is called the numerator, and that below is called the denominator thus, 3 Numerator 8 Denominator The numerator, (which is the remainder in division,) shows the number of parts contained in the fraction; thus in the fraction, the 8 shows that an integer is divided into 8 parts, and 3, the numerator, shows that the fraction contains 3 of those parts expressed in the denominator. The denominator of a fraction, (which is the divisor in division,) shows into how many parts an integer or unit is divided. A fraction is expressed in its least or lowest terms, when expressed by the least numbers possible, thus, when reduced to its lowest terms, will be, and is equal to one third, or is the fraction expressed in its lowest terms. NOTE. The history and theory of vulgar fractions will be given in some other part of this work; we only introduce sufficient of fractions here for our present purpose. CASE I-To reduce fractions to their lowest terms. RULE. Divide the numerator and denominator of the given fraction by any number that will divide them without a remainder, and the quotient again in the same manner, and so continue to do till it appears that there is no number greater than one that will divide them; and the last quotients will express the given fraction in its lowest terms. DEM. We first set down our fraction, and divide the terms of the fraction by 6, saying 6 in 48, eight times, setting down the 8 for the numerator of a new fraction; we then divide the denominator by 6, saying 6 in 9 once, and 3 over, then 6 in 36, six times, which gives us 16 for the denominator; we then divide the new fraction by 4, which gives us, which we again divide by 2, which gives us, which is the same in value of 8. By this operation we do not alter the value of the fraction, because the numerator of the quotient bears the same proportion to the denominator of the quotient, in each place, that the numerator of the dividend bears to the denominator of the dividend, as our example plainly shows; neither does it make any difference what number we take for a divisor, if ‘it will only divide the terms of our fraction without a remainder, which is also shown in our example; the same result being produced by different divisors. 2. Reduce to its lowest terms. Ans. 1 10. Abbreviate 754 as much as possible. Ans. 3. Ans. 1. Ans.. Ans.. Ans. . Ans. 2. Ans. 14. Ans.. 11. Express 44 by the least number of figures possible. 12. Reduce to its lowest expression. Ans. 3. CASE II. To find the value or quantity of a fraction in the known parts of an integer, that is, in the inferiour denomination of the integer. RULE.-Multiply the numerator by the common parts of the integer, (the same as in reduction descending,) and divide the product by the denominator, and if you have a remainder, reduce it to the next inferiour denomination, and again divide the product as before, and so on, till the work is finished, or till you have reduced it to the lowest denomination. EXAMPLES: 1. What is the value of of a pound sterling? Ans. 12s. DEM.--The denominator, 5, shows that a pound is divided into 5 parts, and the numerator, 3, shows how many of those parts our fraction contains. It is evident, if £1 or 20s. is divided into 5 parts, that one 33 20 5)60(12 5 10 10 part must be 4 shillings because 4 shillings is the fifth part of £1 or 20s.; but our numerator expresses three parts, consequently the value of our fraction must be 12 shillings, because three times 4 shillings make 12 shillings. Again, to instruct the student so that he shall find no difficulty in this rule, we give him our reasoning in a manner, if possible, more plain than in the preceding demonstration. The student must bear in mind that 3, the numerator, arises from unfinished division, where it had the name of remainder, and the dividend, of which the 3 is a part, must have been pounds, and 5, the denominator, was the divisor; it is then plain, when we wish to continue the division down, and express this fraction in the inferiour denomination, we must reduce the remainder to the next inferiour denomination, and divide by our divisor which is the denominator of our fraction, because a fraction is formed by placing the divisor below the remainder. 2. What is the value of of a pound sterling? £3 20 4)60(15s. Ans. 4 20 20 0 Ans. 15s. DEMONSTRATION. Here our denominator shows that £1 is divided into 4 parts, each part then must be 5 shillings; and our numerator shows, that our fraction contains three of those parts, which must be 15 shillings, because three times 5 shillings make 15 shillings. ་ Ans. 12s. Ans. 4 d. 3. What is the value of of a pound sterling? 7. Reduce of a cwt. to its proper quantity. Ans. 9oz. Ans. 12oz. Ans. 3gr. 3lb. 1oz. 12 dr. proper distance, in inferiour Ans. 6fur. 26pol. 3yds. 2ft. 8. Reduce of a mile to its denominations. 9. Express of an acre in the inferiour denominations of Ans. 2 roods, 20 poles. the integer. 10. Express of a hogshead in an inferiour denomination, or inferiour denominations of the integer. Ans. 54 gallons. 11. Reduce of a tun to its proper quantity. Ans. 15cwt. 12. Reduce of a pound Sterling to its proper value.. Ans. 7s. 3d L 13. Express of a day in the inferiour denominations of a Ans. 16h. 36m. 55 sec. qay. CASE III-To reduce inferiour denominations to the fraction of some superiour denomination retaining the same value: RULE Reduce the given sum to the lowest denomination mentioned for a numerator; then reduce the unit to the same denomination, for a denominator, which will be the fraction required. EXAMPLES. 1. Reduce 3s. 4d. to the fraction of a pound? s. d. 3 4 the given sum: 12 1 40 numerator. 1 a unit or integral part. 20 20 Ans. or £. DEM. We first reduce 3s. 4a. to pence for a numerator, we then reduce £1 or 20 shillings to pence for a denominator; and we find that a pound reduced to pence is divided into 240 parts, and our numerator contains forty similar parts, because both numerator and denominator express pence, but the denominator expresses the pence in a pound, and when we reduce the terms of the fraction to its lowest terms, it stands one sixth of a pound, and it is plain, that it should be one sixth, because if we multiply 40, the numerator, by six, the product will be 240, a number equal to the denominator; or if we multiply 3 shillings 4 pence by 6, the product will be 20 shillings, a sum equal to our denominator; and our denominator, 240, shows that a unit or, £1, is divided into 240 parts, and our numerator shows that our fraction contains 40 of those parts. 12 240 Denominator. 5) Ans. £. 2. Reduce 13s. and 4d, to the fraction of a pound. 3. Reduce 6 furlongs, 16 poles, to the fraction of a mile. 4. Reduce 4 pence to the fraction of a shilling. 6. Reduce 10s. 6d. to the fraction of a pound. Ans. . 3 Ans. |