8. Reduce of a nail to the fraction of an ell English. Ans. reg. Reduce of a penny to the fraction of a pound. Ans. 576. 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. 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, x12x4=48; and by Case 1st, 48, Ans. 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.§. Statement, 1. 20. 12 360 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 Ans. . of a penny to the fraction of a farthing. 6. Reduce 60000 of a mile to the fraction of a barley corn. of a cwt. to the fraction of a pound Avoirdu 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)4 0 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 inations. of a pound sterling to its value in lower denom Solution of a pound=20 of a shilling, and 20 of a shilling=24of a penny-4 d. Ans. 3. Reduce of a pound Troy to its integral value. Ans. 9 oz. 4. Reduce 5. Reduce Ans. 54 m. of a day to its integral value. 9 of an hour to its value in ΤΟ Ans. 1 h. 20 m. 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 gr., 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, 40.4; and the same canceled is, 118=1}, Ans. 110 40. 4 4. Reduce 12 ounces to the fraction of a pound Avoirdupois. Ans. 2. 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. 16. 7. Reduce 3 qr. 2 na. to the fraction of an ell English. Ans. 10. 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. 13. Ans. 16. 3 11. Reduce 6 qt. to the fraction of a bushel. 48° Ans. 25. 14' 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. PERFORMED. 7x5x11=385, the common denominator. 6×5×11=330, the numerator for, which therefore equals 330 4x7x11=308, the numerator for, therefore, =308. 8×5×7=280, the numerator for, therefore, 8 The required fractions, therefore, are 330, 308, 380. 280 2. Reduce,,,, to a common denominator. 120, 120, 120, and 12. 60 40 24 Ans. 3. Reduce 8, 2, 1, and 2, to a common denominator. Ans. 168 72 84 63 12, and, to a common denominator. Ans. 252 252 252, 252′ 5. Reduce 32 810 1620 8,,, and, to a common denominator. Ans. 1080 1440 486 1620 1620' 1620' 6. Reduce,,, and to a common denominator. Ans. At the commencement of this rule, the scholar was instructed relative to the peculiar form and nature of fractions, and made acquainted with certain principles of universal application. In the course of the preceding eight cases, he has been shown the various changes of which fractions are susceptible, while their value remains unaffected. His attention will now be directed to those operations by which their value is affected. ADDITION OF FRACTIONS. If the scholar will turn back to Simple Addition, he will there find it stated, that numbers, or quantities of the same kind only, can be reduced to a single number or quantity by adding. The same is true of fractions. It is obvious that of a shilling, and of a penny, make neither of a shilling nor of a penny. But of a shilling makes 12 of a penny; and to this, we can add of a penny, and the amount will be 13 of a penny. Therefore, before we can add fractions, they must be reduced to the same denomination. (See Case 5th.) 3 |