ANALYSIS.*Since the least common multiple of the denominators of the fractions (c) and is 12, then twelfths will be the least number of equal parts into which and can be divided, and have all the

pieces of equal size.

ANAL. STEPS.-1. Change to twelfths.

Since in one unit there are 12 twelfths, in § of a unit there are 1 third of 12 twelfths, or 4 twelfths; and in of a unit there are 2 times 4 twelfths, or 8 twelfths.

2. Change to twelfths.

Since in 1 unit there are 12 twelfths, in of a unit there are of 12 twelfths, or 3 twelfths; and in & of a unit there are 3 times 3 twelfths, or 9 twelfths: hence 1o2, and 3=12.

From the above analyses is deduced the following: RULE I. - Find the least common multiple of the denominators for a common denominator. Divide the common denominator by the denominator of each fraction, multiply the quotient by the numerator, and write the result over the common denominator, or,

RULE II. Multiply the terms of each fraction by any number that will make the denominators common.

ORAL, SLATE, AND BLACKBOARD EXERCISES.

Change the fractions in each series of the following numbers to equivalent fractions having a common

NOTE.

When one denominator can be divided exactly by the

other, it is necessary to multiply only one of the fractions; thus,

=

The teacher should refer to the illustration at every step of the analysis.

4

WRITTEN EXERCISES.

PROBLEM.-Change 38, 21, 3, 42, to equivalent fractions having a common denominator.

ANALYSIS.-Require the pupil to write an analysis of this problem in full, and present it to the teacher as a class-exercise.

21'

64. §, §, 12, 14, 1. 65. §, 14, 27, 11, 11. 3 $ 66. 8, 27, 4, 25, 21. 67. 11, 121, 14, 28. 68. 43, §, 14, 115. 69. 35, 34, 53, 914.

3

70. 48, 52, 12, 15.

ADDITION OF FRACTIONS.

93. Since things can be added only to like things, it is evident, that, before fractions can be added or subtracted, they must be reduced to equivalent fractions having a common denominator.

PROBLEM. — What is the sum of 53, 5, and 43?

FORMULA. 5+ 5 +48 = 5 1 8 +239 +424=1033.

PROCESS.

41=4

10号

RULE.

ANALYSIS.(1.) The least common denominator of ,, and, is twenty-fourths.

(2.) 5=511; §=24; and 48=424·

(3.) The sum of 18 twenty-fourths, 20 twenty-fourths, and 9 twenty-fourths, is 47 twenty-fourths, equal to 1 unit and 24; which, added to the sum of 4 and 5, equals 103.

Change the fractions to equal fractions having a common denominator, and add the numerators.

REVIEW.-What is multiplication? (40.) Define the terms. (41, 42, 43, 44.) Which are factors? (44.) Describe the sign of multiplication. (45) Show its use. (45.) Repeat the axiom. (46.) Give the principles. (47.) Analyze a short example in multiplication. (48.)

NOTE.-1. When adding mixed numbers, find the sum of the fractions, and add it to the sum of the integers.

2. Answers should always be given in the lowest terms, or reduced to integral or mixed numbers when possible.

ORAL, SLATE, AND BLACKBOARD EXERCISES.

Prepare the following like the model, and present on plates as a class-exercise.

N.B. For brief business methods of adding fractions, see Key to Drill Cards.

EXERCISE X.-Find the sum of V and W; as, A. }+}, &c. Ex. XI. — Find the sum of W and X; as, A. §+4; B. f +4, &c. Ex. XII. - Find the sum of V and X; as, A. †††; B. } +‡, &c. Ex. XIII. - Find the sum of V, W, and X; as, A. 1+1+†, &c. Ex. XIV.-Prefix 1 to V, and find the sum of two numbers; thus, A+B, 1+6; B+C, 63 +8; C+D, 8}+9}{ ; D+E, 9+4},

&c.

Ex. XV.-Prefix 1 to W, and find the sum of two numbers; thus A+ B, 13+63; B+C, 61+81, &c.

Ex. XVI.—Prefix 1 to X, and find the sum of two numbers as before.

Ex. XVII. — Prefix 2 to V, and find the sum of three numbers; thus, A+B+C, 381+591 + 161; B+C+D, 59 +16}+2011, &c. Ex. XVIII. - Prefix 2 to W, and find the sum of three numbers as before.

Ex. XIX. - Prefix 3 to V, 2 to W, and 1 to X, then find the sum of the resulting mixed numbers; thus, A. 472+381 +1; B. 604 +591 +67, &c.

1. If a boy earns

ORAL EXERCISES.

of a dollar at one time, g of a dollar at another time, and $ at another, how many

dollars does he earn in all?

FORMULA.

ANALYSIS.

equal $14.

$3+$3+$4=$14.

$3 and $3 equal $or, 1 dollar; and 1 dollar and $4

SUGGESTION. In adding more than two fractions, it is often more convenient to add two of them first, reduce, and add each of the others to the sum of the preceding addition.

2. If a boy spends of a dollar at one time, $18 at another time, and $1 at another, how much does he spend in all?

3. There are 7 quarters of an apple in one pile, and 3 halves in another pile: how many apples in both?

4. A girl has 2 apples in one hand, and 3 quarterapples in the other; her brother has 1 apple in one hand, and 1 half of an apple in the other: how many apples have both of the children?

5. Josie gave her mother 2 2 melons: to which did she much did she give to both?

melons, and her father give the most? How

oranges, to Charlie 2

6. Lillie gave to Laura 2 oranges, and to Delia 23 oranges: to which did she give the least? To which the most? How much did she give away in all?

7. A boy bought a pair of skates for $23, a knife for

and a book for $13: how much did he give for all?

REVIEW.-Give method of multiplying by 10, 100, &c. (49.) Give the table of United States currency. (50.) What is understood by division? (51.) How many terms in division? (52, 53, 54, 55) Detine each. Give an ample illustrating each. Describe the sign of division. (56.)

WRITTEN EXERCISES.

78. A man bought 4 pieces of calico. The first contained 375 yards; the second, 41 yards; the third, 278 yards; and the fourth, 38: how many yards did he buy?

79. If a pound of sugar costs 125 cents, a pound of tea $1.214, and a pound of coffee 224 cents, what will all cost?

80. A rubber cost 12 cents, a slate 15 cents, a pencil of a cent, and a sponge 5 cents: how much

did they all cost?

81. A farmer raised 326 bushels of potatoes, 126} bushels of turnips, and 1281 bushels of beets: how many bushels did he raise in all?

82. Henry has 24 dollars, Edward has 131 more than Henry: how many has Edward? How many

have both?

83. One-third of a mill is valued at $116: what is the value of the whole mill?

84. A farm was divided into 5 parts; four of the parts contained 237 acres, one part contained 753 acres: how many acres in the farm?

85. $347 is $127 less than the value of my horses and carriage: what are they worth?

86. I sold a house for $13413, which was $137 less than it cost me: did I make, or lose? and how much? 87. A boy sold a sled for $1.863, which was 59-54 cents less than it cost him: did he make, or lose? and how much?

division?

REVIEW.-Describe the fractional sign. (57.) How many signs are there in What is their difference? What is the difference between short division and long division? (58, 59.)