the denominator, the operation will be, this, by Case 1st, is reduced to, required. Hence of a penny equals 5 5 × 12×20=1440' and which is the fraction. of a pound. The canceling principle may however be successfully applied in the solution of sums of this character. RULE FOR CANCELING.-Place the numerator of the given fraction above a horizontal line, and its denominator below it; then place also BELOW the line, such numbers as are necessary to reduce the denomination given to that required. Cancel, &c. as before. 5 6. 12. 20' We will solve the above example by this rule also. Statement, The scholar should compare the statement with the rule, to see that he understands its application. The above * statement canceled, ; and 6 × 12 × 4-288, the denomi6. 12. 20 4 nator as before, and nothing remains as a numerator; therefore, as before, of a pound is the answer. (See Note 1st, Case 4th.) 2. Reduce of a farthing to the fraction of a shilling. By the common rule, which, by Case 1st, equals 3 3 4X4X12=192' Ans. By the rule for canceling, 3 celed, 1-4. 4. 4. 18=64, Ans. 4 3. Reduce of a penny to the fraction of a pound. Statement, 4. Reduce of a gallon to the fraction of a hogshead. Ans. 117 2 6 Statement, 7. 63° lons to hogsheads. The 63 below the line reduces the gal 5. Reduce of an ounce. Troy to the fraction of a pound. Ans. 6. Reduce 19 of a minute to the fraction of a day. Ans. 7. Reduce of a pound Avoirdupois to the fraction of a 8. Reduce of a nail to the fraction of an ell English. Ans. 180. 9. Reduce of a penny to the fraction of a pound. Ans. 10. Reduce of an hour to the fraction of a year. Ans. 9490 Secondly. To reduce fractions of high denominations to equivalent fractions of lower denominations. RULE.-Multiply the numerator by such numbers as are required to reduce the given quantity from the given to the required denomination, and then by Case 1st reduce the result to its lowest terms. 96 Ex. 1. Reduce of a shilling to the fraction of a farthing. To reduce shillings to farthings, we must multiply by 12 and 4; therefore, × 12×4=48; and by Case 1st, 18=1, Ans. 96 4 RULE FOR CANCELING.-Place the numerator of the given fraction above a horizontal line, and the denominator below, as before; then place above the line such numbers as are necessary to reduce the denomination given to that required. Cancel, &c. as before. 1. 12. 4 The above sum solved by this rule. Statement, 96 The scholar will carefully observe the difference between the statement here, and the one given for reducing low denominations to high. 2. Reduce of a pound to the fraction of a penny. Ans.§. 1. 20. 12 Statement, 360 3. Reduce of a pound Troy to the fraction of a pwt. Ans. 20, or 23 pwt. Statement, 1. 12. 20 108 4. Reduce of a pound Troy to the fraction of an ounce. Ans.. 5. Reduce of a penny to the fraction of a farthing. Ans. . 6. Reduce 560000 of a mile to the fraction of a barley corn. of a cwt. to the fraction of a pound Avoirdu 3 of an ell English to the fraction of a nail. of a year to the fraction of an hour. Ans. CASE 6th.-TO REDUCE FRACTIONS OF A HIGHER DENOMINA TION TO THEIR VALUE IN WHOLE NUMBERS OF A LOWER DENOMINATION. RULE. Reduce the fraction to its next lower denomination by multiplying the numerator by the requisite number, and divide the product by the denominator; the quotient thus obtained will be a whole number of the lower denomination, and the remainder, if any, may be reduced and divided as before. This process may be continued till nothing remains, or till it is reduced to the lowest denomination. Ex. 1. Reduce of a pound sterling to its value in shillings, and pence. OPERATION. 20 Div. by denom. 3)40 shillings. 1 3 s. and 1 s. remains, which equals 12 d., and 12 d.÷3-4 d. Therefore, 13 s. 4 d. is the required number. It is evident that of a pound sterling is 20 times as many thirds of a shilling; viz. 40=13 shillings; and of a shilling is 12 thirds of a penny, that is, 4d. Hence, † of a pound is 13 s. 4 d. 2. Reduce of a pound sterling to its value in lower denominations. Solution of a pound-28 of a shilling, and 28 of a shil ling=240 of a penny 4 d. Ans. = 3. Reduce of a pound Troy to its integral value. Ans. 9 oz. of a day to its integral value. Ans. 1 h. 20 m. of an hour to its value in whole numbers. of a hogshead to its value in whole numbers. 6. Reduce Ans. 2 qt. 1 pt. 1 gi. 7. Reduce 8. Reduce of a quart to its integral value. Ans. 1 pt. 1 gi. of an ell English to a whole number. Ans. of a yard to a whole number. Ans. 3} qr. of an ell French to a whole number. Ans. CASE 7th.-TO REDUCE THE LOWER DENOMINATIONS OF A COMPOUND NUMBER TO FRACTIONS OF A HIGHER DE NOMINATION. RULE.-Reduce the given quantity to the lowest denomination in that quantity, for a numerator of the fraction; and then reduce a unit of the higher denomination to the same denomination with the numerator for the denominator of the fraction. This fraction reduced to its lowest terms by Case 1st, will be the one required. Ex. 1. Reduce 2 qr. 2 na. to the fraction of a yard. Operation, 2 qr. 2 na.= 10 nails, the numerator; and 1 yd. =4 qr., and 4 qr.=16 nails, the denominator; therefore, 18 is the fraction, which equals &, Ans. The operation may be much abbreviated by canceling; for which the following will be found a convenient rule. RULE FOR CANCELING.-Reduce the given quantity to the lowest denomination mentioned, (if it consist of different denominations,) and place it above a horizontal line; and beneath the same line place the numbers required to reduce this denomination to the required denomination. Cancel, multiply, &c., and the terms of the required fraction will be obtained. Ex. 2. Reduce 3s. 4 d. to the fraction of a pound. Operation, 3 s. 4 d. -40 d., and pence are reduced to pounds by dividing by 12 and 20. Therefore, Nothing remains above the line, the numerator of the required fraction therefore is 1; and 6 remains below the line, and consequently is the denominator; therefore, is the fraction required, and 3 s. 4 d. is of a pound. 3. Reduce 2 roods and 30 rods to the fraction of an acre. 2 roods and 3 rods 110 rods; and rods are reduced to acres, by dividing by 40 and by 4; the statement therefore is, ; and the same canceled is, 110 40.4 40. 4 110. 1, Ans. 4. Reduce 12 ounces to the fraction of a pound Avoirdupois. Ans. 4. 53 5. Reduce 26 gal. 2 qt. to the fraction of a hogshead. Ans. 126. 6. Reduce 3 fur. 20 rods to the fraction of a mile. Ans. 7. 7. Reduce 3 qr. 2 na. to the fraction of an ell English. Ans. Ans. . Ans. 8. Reduce 2 dr. 2 sc. to the fraction of an ounce. 9. Reduce 2 qr. 24 lb. to the fraction of a cwt. 10. Reduce 6 oz. 10pwt. to the fraction of a pound Troy. Ans.. 31 ៩. Ans. 16 3 11. Reduce 6 qt. to the fraction of a bushel. 48 9 Ans. 25. CASE 8th.-TO REDUCE FRACTIONS HAVING DIFFERENT DENOMINATORS TO EQUIVALENT FRACTIONS HAVING A COMMON DENOMINATOR. RULE.-Multiply all the denominators together for a new denominator, and each numerator into all the denominators except its own, for a new numerator to each fraction. The several numerators placed over the common denominator will give the required fractions. Note. The fractions should be reduced to their lowest terms before multiplying. If the scholar looks carefully into the nature of this rule, he will see that it is only multiplying numerators and denominators by the same numbers; and he has already learned, that this does not affect the value of the fraction. Ex. 1. Reduce §,, and, to a common denominator. |