To find the new numerators, that is, how many 12ths each fraction is, we may take 2, 1, 3 and 4 of 12. Thus : of 12 = 9 of 126 of 128 72 = 718 138 = 133 157=1531 12 = 1 £= 1 = } z = t Ans., and 8. Reduce,, and to fractions having the least common denominator, and add them together. Ans. ++38=14=14, amount. 9. Reduce and to fractions of the least common denominator, and subtract one from the other. Ans. T's, difference. 10. What is the least number that 3, 5, 8 and 10 will measure? Aus. 120. 11. There are 3 pieces of cloth, one containing 7 yards, another 13 yards, and the other 15ğ yards; how many yards in the 3 pieces. Before adding, reduce the fractional parts to their least common denominator; this being done, we shall have, Adding together all the 24ths, viz. 18 + 20 21, we obtain 59, that is, 211. We write down the fraction under the other fractions, and reserve the 2 integers to be carried to the amount of the other integers, making in the whole 3741, Ans. 12. There was a piece of cloth containing 34 yards, from which were taken 123 yards; how much was there left? Ans. 37. 848=342 123 = 12! New numerators, which, We cannot take 16 twenty-fourths (1) from 9 twenty-fourths, (2) we must, therefore, borrow 1 integer, = 24 twenty-fourths, (,) which, with : Ans. 211 yds. makes ; we can now take from , and there will remain ; but, as we borrowed, so also we must carry 1 to the 12, which makes it 13, and 13 from 84 leaves 21. Ans. 211. 13. What is the amount of of 4 of a yard, of a yard,. and of 2 yards? Note. The compound fraction may be reduced to a sim ple fraction; thus, of; and of 2 =; then, ↑ → 1+3=138=1 yds., Ans. 62. From the foregoing examples we derive the following RULE:-To add or subtract fractions, add or subtract their numerators, when they have a common denominator; otherwise, they must first be reduced to a common denomi nator. Note. Compound fractions must be reduced to simple fractions before adding or subtracting. EXAMPLES FOR PRACTICE. 1. What is the amount of 4, 4 and 12? 2. A man bought a ticket, and sold of of the ticket had he left? 3. Add together, 8, 4, 6, and 14. 4. What is the difference between 14 f 5. From 1 take ₫. 6. From 3 take . 7. From 147 take 48. 8. From of take of + Ans. 17 of it; what part and 167? - Ans. 149 9. Add together 112, 311, and 1000. 10. Add together 14, 11, 43, and 1. 11. From take. From 7 take . 12. What is the difference between 1 and 4? and ? and ? and ?. § and 2 ? 13. How much is 1 ?? 2-4? 2—4? 24-4? 3 ― 1 -¿? 1 - #? Rem. 24 Rema. and? #? ? REDUCTION OF FRACTIONS. ナ ¶ 63. We have seen, (¶ 33,) that integers of one denomination may be reduced to integers of another denomination. It is evident, that fractions of one denomination, after the same manner, and by the same rules, may be reduced to fractions of another denomination; that is, fractions, like integers, may be brought into lower denominations by mul tiplication, and into higher denominations by division. To reduce higher into LOWER To reduce lower into HIGHER denominations. denominations. (RULE. See T34.) (RULE. See T 34.) 1. Reduce of a pound 2. Reduce of a penny to to pence, or the fraction of a the fraction of a pound. Note. Division is performpenny. Note. Let it be recollect- ed either by dividing the m ed, that a fraction is multiplied merator, or by multiplying the either by dividing its denomi- denominator. nator, or by multiplying its nu 12s.÷20 merator. 230 × 20 = s. X 12 = d. Ans. d. zo £. Ans.. Then, q. Ans. 5. Reduce of a guinea to the fraction of a penny. 4. Reduce of a farthing to the fraction of a pound. 2 q. ÷ 4 = √ d. ÷ 12 = 28% = 1828.20840=1280€. s. X 960 1280 7. Reduce of a guinea to the fraction of a pound. Or thus: of of = 1680=200£. Ans. Or thus: Denom. 4 4 q. in 1 a. 16 12 d. in 1 s. 192 20 s. in 1 £. 3840 Then, 1200£. Ane. 6. Reduce of a penny to the fraction of a guinea. 8. Reduce of a pound to the fraction of a guinea. 10. Reduce of a guinea Consult ¶ 34, ex. 11. 9. Reduce of a moidore, at 36 s. to the fraction of a guinea. to the fraction of a moidore. 12. Reduce & of an ounce the fraction of a pound Troy. 11. Reduce of a pound, Troy, to the fraction of an to ounce. L* L 13. Reduce of a pound, 14. Reduce of an ounce avoirdupois, to the fraction of to the fraction of a pound an ounce. avoirdupois. 15. A man has g of a 16. A man has of a pint hogshead of wine; what part of wine; what part is that of is that of a pint? a hogshead? 18. A cucumber grew to the length of 1 foot 4 inches of a foot; what part is that of a inile? 17. A cucumber grew to the length of of a mile; what part is that of a foot? 19. Reduce of of a 20. 29 of a shilling is 3 of pound to the fraction of a shil- what fraction of a pound? ling. 21. Reduce of of 3 pounds to the fraction of a penny. 22. 180 of a penny is of what fraction of 3 pounds? 180 of a penny is of what part of 3 pounds? 4 of a penny is off of how many pounds? 13! 64. It will frequently be It will frequently be rerequired to find the value of a quired to reduce integers to fraction, that is, to reduce a the fraction of a greater defraction to integers of less de-nomination. · less denominations. 1. What is the value of of a pound? In other words, Reduce of a pound to shillings and pence. 2. Reduce 13 s. 4 d. to the fraction of a pound. d 1 3 24 13 s. 4 d. is 160 pence; there are 240 pence in a of a pound is 413 shii- pound; therefore, 13 s. 4 d. is lings; it is evident from of 13 160 of a pound. That a shilling may be obtained is,-Reduce the given sum or some pence; of a shilling is quantity to the least denomina4d. That is,-Multiply tion mentioned in it, for a nuthe numerator by that number merator; then reduce an intewhich will reduce it to the next ger of that greater denominaless denomination, and divide tion (to a fraction of which it the product by the denominator; is required to reduce the givif there be a remainder, multiply en sum or quantity) to the and divide as before, and so on; same denomination, for a denomi the several quotients, placed one nator, and they will form the after another, in their order, fraction required. will be the answer. EXAMPLES FOR PRACTICE. EXAMPLES FOR PRACTICE 3. What is the value of ! 4. Reduce 4 d. 2 q. to the of a shilling? fraction of a shilling.. OPERATION. OPERATION. Numer. 3 12 Denom.8)36(4 d. 2 q. Ans. 32 4 4 16(2 q. 5. What is the value of of a pound Troy? 7. What is the value of of a pound avoirdupois ? 9. t of a month is how many days, hours, and minutes? is 11. Reduce of a mile to its proper quantity. 13. Reduce of an acre to its proper quantity. 15. What is the value of 4 d. 24. 4 18 Numer. pence, &c.? 17. What is the value of 1% of a yard? 19. What is the value of of a ton ? 6. Reduce 7 oz. 4 pwt. to the fraction of a pound Troy. 8. Reduce 8 oz. 143 dr. to the fraction of a pound avoirdupois. Note. Both the numerator and the denominator must be reduced to 9ths of a dr. 10. 3 weeks, 1 d. 9 h. 36 m. what fraction of a month? 12. Reduce 4 fur. 125 yds. 2 ft. 1 in. 24 bar. to the frac tion of a mile. 14. Reduce 1 rood 30 poles to the fraction of an acre. of a dollar in shillings, the fraction of a dollar. 16. Reduce 5 s. 7 d. to 18. Reduce 2 ft. 8 in. 1 b. to the fraction of a yard. 20. Reduce 4 cwt. 2 qr. 12 lb. 14 oz. 121 dr. to the frac tion of a ton. Note. Let the pupil be required to reverse and prove the following examples: 21. What is the value of of a guinea? |