a mixed number, reduce the fractional part of it to the next lower denomination, as before, and so proceed as far as desirable.

NOTE. If, at any time, the reduced fraction is proper, there will be no whole number of that denomination.

2. Reduce £ to whole numbers of lower denominations. 13£ (= 13s. × 20) = fs.41%s.; 1es. (=td. × 12)= d., a proper fraction; d. (=3qr. × 4)=3qr.; . }}£= 48. Od. 3qr., Ans.

3. Reduce of an acre to lower denominations.

Ans. 1r. 17rd. 18yd. 1ft. 50ĝin.

4. Reduce of a furlong to rods, yards, etc.

5. Reduce

6. Reduce

of a week to days, etc.

Aus. 18rd. 3yd. 2ft.

of a rod, Long Measure, to yards, etc.

7. Reduce 1883 of a circumference to signs, etc.

8. Reduce

9. Reduce 10. Reduce 11. Reduce

of a ton to hundred weights, etc.

b to ounces, drams, scruples, etc.

368

circ. to signs, degrees, etc.
of a civil year (365 days) to days, etc.

12. What is the value of 76% of a pound Troy?

13. What is the value of

14. What is the value of 15. What is the value Weight?

16. Reduce

of a bushel?

of a gallon?

of

24

of a pound, Apothecaries'

of a mile to furlongs, chains, etc.

17. Reduce of a cord to cord feet, cubic feet, etc.

18. Reduce

of a yard to quarters, nails, etc.

PROBLEM 13.

151. To reduce whole numbers of lower denomina

tions to the fraction of a higher denomination.

Ex. 1. One farthing is what part of a penny?

Since 4 farthings make a penny, 1 farthing is

Ans.

of a penny.

2. Six pence and 1 farthing are what part of a shilling? 6d.1qr.25qr; and Is.=48qr.; .. 6d. and 1qr. = s., Ans.

To determine what part one thing is of another, considered as a unit or whole thing, the part is always made the numerator of a fraction, and the unit or whole thing is put for the denominator; thus, the fraction & expresses the part that 3 miles is of 5 miles. Before the comparison can be made, the part and the whole must be of the same kind or denomination; thus, 3 pecks is not of 5 bushels, but, reducing the 5 bushels to 20 pecks, we have 3 pecks equal to of 20 pecks, i. e. 3 of 5 bushels. Hence,

20

RULE 1. Reduce the given quantity to the lowest denomination it contains, for a numerator; and reduce a unit of the higher denomination to the same denomination as the numerator, for a denominator.

3. Reduce 6rd. 5ft. 9in. to the fraction of a furlong.

6rd. 5ft. 9in. 1257in. and 1fur. = 7920in.

.. 6rd. 5ft. 9in.fur.fur., Ans.

4. Reduce 7oz. 4dwt. to the fraction of a pound.

Ans. 3.

5. Reduce 9 rods, 1 foot, and 6 inches to the fraction of a furlong.

=

9rd. 1ft. 6in.= 1800in. and 1fur.= = 7920in.;

.. 9rd. 1ft. Gin.4888fur.fur., Ans. (a) In Ex. 5, 6in. = ft.; 14ft.yd.rd. and 94rd. rd.fur., Ans., as by Rule 1. Hence,

RULE 2. Divide the number of the lowest denomination given by the number required to reduce it to the next higher denomina、 tion, 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 denomination, 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.

151. Rule for reducing the lower denominations of a compound number to a fraction of a higher denomination? Explanation? Principle? Second rule for reducing integers of lower denominations to the fraction of a higher denomi nation? Explanation? Why preferable to Rule 1?

6. Reduce 1r. 2sq. rd. 20sq. yd. 1sq. ft. 72sq. in. to the fraction of an acre.

Ans. 1.

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

NOTE 2. In Example 7, by Rule 1, reduce 4oz. 6dwt. 93gr. 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?

8. Reduce 1pk. 3qt. 1pt. to the fraction of a bushel.

9. Reduce 6s. 20° 20′ 30′′ to the fraction of a circumference. 10. Reduce 1m. 2fur. 11rd. 2yd. 1ft. 24b. c. to the fraction of a league.

11. Reduce 1qr. 2na.in. to the fraction of a yard.

12. Reduce 3wk. 6d. 9h. 27m. to the fraction of a Julian year. 13. Reduce 1qt. 1pt. 14gi. to the fraction of a gallon. 14. Reduce 4 cord feet, 12 cubic feet, and 13823 cubic inches to the fraction of a cord. Ans. 3.

15 Reduce 3oz. 4dr. 1sc. 10gr. to the fraction of a pound. 16. Reduce 4fur. 5ch. 2rd. 20li. to the fraction of a mile. 17. Reduce 11cwt. 11lb. 1oz. 123dr. to the fraction of a ton. 18. Reduce 3 bushels, 1 peck, 4 quarts, and 1 pint to the fraction of a bushel. Ans. 24.

NOTE 3. Sometimes, as in Ex. 18, the number called the part is greater than the unit with which it is compared; sometimes it is equal to the unit.

PROBLEM 14.

152. 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+=1; 1+1=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 3+ are neither nor ; but numbers that are unlike may sometimes be made alike by reduction, and then added; thus, 3+3=33+38 (Art. 147)=1}•

(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. (Art. 148), and then pk. +3pk.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 new numerators over the common denominator. Ans. 15.

Ex. 1. Add

and

together.

2. Add 1, 5, and 17, together.

3. Add 17, 17, 1, and together. 4. Add and together.

5. Add, 15, 5, and

together.

Ans. 1.

Ans. 392127.

Ans. 121=11.
Ans. 13.

6. Add 24, 24, 24, 24, and 4 together.

87

48
969

67

11. Add together 15, 15, 15, 135, and 1.

12

12. Add together 7, 8, 8, and .

13. Add together 34, 18, 16, and 14.

14. Add together 3, 1, and f.

Ans. 23.

Ans. 43.

3 +12+18=18+38+48 (Art. 147, Rule 2) 1, Ans.

15. Add together and .

3+8=18+38 (Art. 147, Rule 1) = 1}=133, Ans.

16. Add together, 3, and 4.

17. Add together 2, 1, and g.

6

307

P2++=1+1+1==14, Ans.

=

152. Rule for adding fractions? Can unlike numbers be added? Of what kind is the sum of two or more numbers?

23. Add s. to 3d.

s.3d.d.+3d.=73d.+1gd.=f3d.=57d., Ans.

or, fs.d.3s.+1s. =30s.+s. 11s., 2d Ans.

1st Ans.

24. Add gal. to jqt.

25. Add together bush. pk. and qt.

26. Add together ton gewt. and qr.

Ans. 11qt. or gal

(c) To add two fractions that have a common numer

ator:

Multiply the sum of the denominators by either numerator, and place the product over the product of the denominators.

27. What is the sum of and } ?

1 8+7 15 3

- ; .. +

7

56

and §?

and?

45 = Ans. 56'

3

=

3 X 15
56

14=17, Ans.

28. What is the sum of

29. What is the sum of

(d) To add mixed numbers:

Add the sum of the fractions to the sum of the integers.

30. What is the sum of 3 and 43?

7

1+3=38+18=43=1%; 3+4=7;
3+43=7+1847, Ans.

31. What is the sum of 54, 3, and 123?
32. What is the sum of 18, 5, and 24g?
33. What is the sum of 153, 24, 74, and ?
34. What is the sum of 3, 6, 41, and 243?

37. What is the sum of 3, 425, 835, and 25?

38. How many are 83 +33 +83 +14?

152. Mode of adding two fractions that have like numerators? Mode of adding mixed numbers?