A Treatise on Arithmetic, Combining Analysis and Synthesis

(a) In Ex. 1, 2qr. are equal to 3d.; .. 5d. and 2qr. are equal

RULE 2.-Divide the number of the lowest denomination given by the number required to reduce it to the next higher denomination, and annex the fractional quotient so obtained to the given number of that higher denomination; divide the mixed number so formed by the number required to reduce it to the NEXT higher denomination, annex the quotient to the given number of that denominaton, and so proceed as far as necessary.

NOTE 1.-This rule is frequently preferable to the 1st, because it enables us to use smaller numbers and gives the result in lower terms.

3. Reduce 2r. 5rd. 10yd. Oft. 108in. to the fraction of an acre.

Ans. 8

4. Reduce 4oz. 6dwt. 93gr. to the fraction of a pound.

Ans.

15.

NOTE 2.-In Example 4, by Rule 1, reduce 4oz. 6dwt. 93 gr. to fifths of a grain for a numerator, and 1lb. to fifths of a grain for a denominator. How shall it be done by Rule 2? Which mode is preferable? Why?

5. Reduce 73 to the fraction of a pound.

6. Reduce 2oz. 44dr. to the fraction of a pound.

7. Reduce 1fur. 4ch. 11li. 624in. to the fraction of a mile.

8. Reduce 7 cord feet, 9 cubic feet and 1036 inches to the fraction of a cord.

9. Reduce 1qr. 1na. in. to the fraction of a yard.

10. Reduce 2qt. 137gi. to the fraction of a gallon.

11. Reduce 2pk. 1qt. 1-75pt. to the fraction of a bushel.

12. Reduce 11.m. 3wk. 3d. 4h. 17m. 8 sec. to the fraction of

a Julian year.

13. Reduce 1s. 10° to the fraction of a circumference.

14. Reduce 1m. 2fur. 11 rd. 2yd. 1ft. 24b. c. to the fraction of a league.

15. Reduce 8ft. 576in. to the fraction of a ton.

16. Reduce 256d. 20h. 26m. 40sec. to the fraction of a civil

year (365 days).

17. Reduce 113 73 29 18gr. to the fraction of a pound.

18. Reduce 8s. 25° 30′ 20′′ to the fraction of a circumference.

CASE 14.

138. If numbers of the same kind are added together, their sum will be of the same kind as the numbers added; thus, 3 books 4 books = 7 books; 3 hats + 4 hats = 7 hats ; and for a like reason,

+ = 3; 13 +13 = 13, etc., etc.

(a) Numbers of different kinds cannot be united by addition; thus, 3 hats + 4 books are neither 7 hats nor 7 books; so + are neither nor ; but numbers that are unlike may sometimes be made alike by reduction, and then added; thus,

3+1=33+28 (133) = 43.

(b) Again, 2bush. +3pk. are neither 5bush. nor 5pk.; but 2bush. Spk., and then 8pk. +3pk. = 11pk.; so bush. + #pk. are neither bush. nor #pk.; but bush. = $pk. (134), and then pk. + pk. : = pk. Hence,

To add fractions,

RULE.-Reduce the fractions, if necessary, first to the same denomination, then to a common denominator; after which, write the sum of the numerators over the common denominator.

10. Add 2, 42, 42, 72, 44 and together.

11. Add together, and

3+12+18=18+39 +18 (133, a)==143, Ans.

6 48

12. Add together 3, 4, 71 and .

8+1+1+3=3388 +3388 +3588 +1888 (133) =

1485 3960

15. What is the sum of 1⁄2, 1, 3 and 11⁄2? 16. What is the sum of 3, 12, 1, § and †? 17. What is the sum of 22, 11, 33 and 22?

18. What is the sum of

25 8

13383, Ans.

39609

of, of § off

and 1⁄2 of ? Ans. 17.

fs.+jd.2d + 3d.=1§d. + 1d.=11d.=211d., Ans. ;

or, fs.+jd.fs. + 36s.=13860s. + 180S. = 14s., 2d Ans. = 1st Ans.

23. Add gal., qt. and pt. together.

NOTE. The answer to the 23d example may be the fraction of a gallon, of a quart or of a pint; thus,

1st, gal.qt. + pt. gal. + gal. + gal. = 140gal. +gal.+gal. = 138gal.gal., 1st Ans.

2d, gal.+qt. +pt.qt. + sqt + fqt. = 168qt. +12qt. +qt.=178qt. 17qt., 2d Ans.

3d, gal.qt.+ pt. pt. + 3pt.+4pt. 288pt. +pt.

+pt.=533,37pt., 3d Ans.

24. Add together of a ton and of a hundred weight. 25. Add together bush. pk. and 4qt.

26. wk.d. + 4h. + 3m. = what fraction of a week? 3d Ans. 867h.437h.

day? hour? minute?

27. Add together lb. 433 and 9.

(d) Mixed numbers may be added by first reducing them to improper fractions and then to a common denominator, etc.; but it will be easier and .. better to add the fractional parts and then add that sum to the sum of the integral parts.