2. Multiply by 2; by 4; by 8. Multiply by 3; by 5; by 15.

3. Multiply by 2; by 3; 6; 9; 18. Multiply by 3; by 6; 8; 12; 24.

8. Divide a pound of raisins equally among 3 boys; what part of a pound will each boy have?

NOTE. If a pound be divided into 3 equal parts, each of the parts will be of a pound. Sayofis. If the half pound were divided equally among 4 boys, each boy would have § of, or § of a pound.

80. As the numerator shows the number of parts expressed by the fraction, multiplying the numerator multiplies the fraction, and dividing the numerator divides the fraction.

1. Multiply by 2; by 5; by 9; by 12. Say, 2 times are 4. 5 times are 5:

5; 8. 9; 12.

24. 2. Multiply by 3; by 4; by 6; 8. 3. Multiply by 2; by 3; 4. Multiply 4 by 2; by 5; Say 2 times 4 are 8; 2 times 5. Multiply 5 by 4; by 5;

9; 12.

by 4; by 7; 9; 11. 17 by 5; by 6; 7 ; 8.

are, which added to 8 make 89. ; 7; 12. 15g by 3; by 5; 7;

6. Multiply 25 by 2; by 4; 7; 8; 12.
32 by 4; by ; 16.

7. Divide 25 by 5.

by 7; 21. 8 by 3; 4; 6; 8; 12.

68 by 9. 7 by 12.

8. Divide 21 by 3;

9. Divide 34 by 5. 10. Divide 4 11. Divide 8 to be divided;

Note. 3425.

by 7.

of 8 is 24.

=

by 3. One third of 8 is 2, and 22 remain of 12, which added to 2 is 2; therefore,

NOTE. After dividing by 5, (see margin,) 23 remain, which is equal to 17, of which is 1.

20. Divide 318478 by 3; by 4; by 5; by 7.

21. Divide 5763 by 3; by 6; by 9.

22. Divide 5763 by 12; by 16; by 25.

81. From the last two articles we may derive the following rules:

RULE 1. To multiply a fraction, Divide the denominator, if it can be done; if not, multiply the numerator.

RULE 2. To divide a fraction, Divide the numerator, if it can be done; if not, multiply the denominator.

1. Multiply 2. Multiply

by 25; by 125; by 375.

by 8; by 128; by 160; by 320. 3. Multiply 37812 by 6; by 8; by 24; by 12. 4. Multiply by 8; by 15; by 27; by 80. 5. Multiply 18471 by 3; by 15; by 19; by 30. 6. Divide 240 by 3; by 12; by 16; by 48. 7. Divide 31 by 8; by 12; by 27; by 67. 8. Divide by 5; by 25; by 75; by 225. 9. Divide 75 by 12; by 36; by 15.

67

10. Divide 31578 by 15; by 37; by 6700.

11. Divide 1408071

by 3500; by 87100.

82.

COMPOUND FRACTIONS.

A Compound Fraction is a fraction of a fraction; as, 3 of t

of; of; of.

Reduce to a simple fraction of; } of 1; } of 1; of; of; 8 of 1; of;

of are

RULE. Reduce all the numbers to a fractional form; and, after cancelling the factors which are common to the numerator and denominator, multiply all the numerators together, for a new numerator, and all the denominators, for a new denominator.

1. Reduce the following compound fractions to simple ones. ofofofof; of of 9 of 75.

7

3 4 6 The last example, after cancelling, is XXX = ठ 7

2. Reduce to simple fractions of 4 of 43 of 16;

of 154.

1

15 15

of 21

3. Reduce of of 10 of 100; 1 of 11 of 15 of 50. 4. Reduce to simple fractions of .15 of 3 of 40; # of .7 of .08 of 5; of .046 of 31.07.

CIRCULATING DECIMALS.

83. In reducing common fractions to decimals, if the denominator contains any prime factors other than the factors of 10, viz., 2 and 5, the division can never be completed, the same quotient figure or figures being continually repeated. Such decimals are called repeating, or circulating decimals; and the figures repeated are called repetends. If only one figure is repeated, as in .333, &c., it may be expressed by writing the figure repeated once only, with a dot over; thus, }=.3; }=.6.

=

If the same figures are repeated alternately, as in=.5454, &c., or in .621621, &c., the repetend is expressed by writing the figures once, with a dot over the first and last figures; thus, &=.54; 7=.621.

Repeating decimals are changed to common fractions by writing the repetend for the numerator, and as many 9's as

there are figures in the repetend for the denominator; thus, 3=};.54—§ƒ; 621=fff; .45=1+§ of ‚&; 38714 =b+745 of Tho; 07 = 7 of 1b.

1. Change to decimals the following common fractions: }; 3$; 1; 111; 8.

2. Change to common fractions .4; 45; ‚414; 5.432.

3. Change to common fractions .45; .38717; 07; .005. 4. Change to common fractions .318; 5.451071; 30.41001; 2.005109; .401064.

84. TO REDUCE A GIVEN INTEGER OR FRACTION TO A FRACTION HAVING A GIVEN DENOMINATOR.

1. One unit is how many 4ths? Ans. 4 4ths. 3 units are how many 4ths? Since there are 4 4ths in 1 unit, there will be 4 times as many 4ths as units. Ans. 12 4ths.

2. Reduce 5 to halves. Since there are 2 halves in 1 unit, there will be 2 times as many halves as units. Reduce 5 to 3ds; to 4ths; 5ths; 6ths; 7ths; 8ths; 10ths; 12ths.

3. In 1 unit how many 3ds? In of a unit how many 3ds? There are 3 times as many 3ds as units.

of a unit = 1. Ans. 1 3ds. 4. In how many 3ds? Ans.

Ans. 21 3ds.

5. Change to 4ths. Ans. 2 4ths.

3 times of a unit are

of a 3d.

In how many 3ds?

There 4 times as many 4ths as units.

6. Reduce to halves. 3ds; 4ths; 5ths; 6ths; 9ths;

X 2 = of a half. Reduce to 12ths.

RULE. Multiply the given integer or fraction by the required de

nominator.

7. Reduce & to 8ths; to 12ths; 16ths; 20ths; 24ths; 36ths. 8. Reduce & to 12ths; 18ths; 24ths; 30ths; 42ds; 54ths. 9. § of a bushel are how many 4ths of a bushel?

§×4==21. Ans. 21 4ths of a bushel.

10. of a pk. are how many 8ths of a pk.? 11. Reduce of a lb. to 16ths of a lb.

20ths of an oz.

of an oz. Troy to

12. Reduce & of a day to 24ths of a day. of an hour to 60ths of an hour. of a minute to 60ths of a minute.

$5. TO REDUCE FRACTIONS TO THEIR LEAST COMMON DENOM

INATOR.

Fractions are said to have a common denominator when they have the same denominator. Thus, the fractions 4, 4, 4, have a common denominator.

1. Reduce to 10ths; to 15ths; 20ths; 25ths; 30ths; 45ths. by 10, the common factor 5 should be

NOTE. In multiplying cancelled. Thus

Xixi or 9.

2. Reduce to 18ths; to 27ths. to 24ths; to 36ths; to 60ths. 3. Reduce and to 6ths.and to 10ths. and to 24ths.

NOTE. The least common denominator is the least common multiple of all the denominators.

RULE. Reduce the fractions to their lowest terms; and multiply each fraction by the least common denominator; the several products will be ine numerators to the common denominator. (84.)

Or, which is the same, Divide the common denominator by the denom inator of each of the fractions; the several quotients, multiplied by the respective numerators, will be the numerators to the common denomi

nator.

4. Reduce, and, to their least common denominator. The least common multiple of 8, 12 and 15, is 120.

5. Reduce to their least common denominators and ; and ; and ; and ; &, and ; &, and ; 3, and 7.

6. Reduce to their least common denominators & and ;

75 and 2.

7. Reduce, & and ; §, † and 1.

8. Reduce, 12, 1, and 2; 15,

and

9. Reduce, 7, 24 and fo; &,' and

10. Reduce, 18, and 1; 2, 10 and 1.