129. A merchant purchased 4 pieces of silk; the first contained 37 yds., the second 434 yds., the third 27

yds.,

and the fourth 38 yds.; how many yards did all the pieces contain?

130. If a book cost 123 cts., a slate 103 cts., a pencil of a cent, and a sponge 5 cts.; how much did they all cost? 131. A butcher bought, at one time, 1293 lbs. of beef; at another, 439 lbs.; at another, 372 lbs.; and, at another, 473 lbs.; how many pounds did he purchase in all?

132. A boy spent of a dollar for a geography, $ for a spelling-book, and $19 for a slate; how much did he spend for all?

133. If I have of a pound of sugar in one paper, fr lbs. in another, 3 lbs. in another, and 2 lbs. in another; how many 385ths of a pound have I in all?

134. I have three strings; the first is 23 inches long, the second is 2 inches long, and the third is of an inch long; how many twelfths of an inch are there in the three strings?

135. I have three apples; the 1st weighs 53 oz., the 2nd 117 oz., and the 3rd 122 oz.; how many 72nds of an ounce do the three apples weigh?

136. What is the size of the largest equal pieces into which,, and of a pear can be cut? How many pieces in each part? How many pieces in all the parts?

137. What is the size of the largest equal pieces into which 3, 7, and of an orange, can be divided? How many pieces in all the parts ?

138. Reduce, 4, 5, 371, to equivalent fractions having a common denominator.

139. What is the size of the largest pieces that can be made from 24 of an apple, each piece to be of equal size? How many pieces will there be?

6

Which term of a Which to the divisor? ANS.-13÷4. What is

QUESTIONS.-What do fractions indicate? (135.) fraction answers to the dividend? (136.) (136.) What, then, is the meaning of 13? the meaning of 5? Ans.—5÷11. In what other way can 15÷8 be expressed? ANS.-15. What is the value of a fractional expression? ANs.—The number of times the denominator is contained in the numerator. What is the value of an expression of unperformed division; as, 13÷6? ANS.—The number of times the divisor is contained in the dividend. Give the fractional forms of the following expressions:-3÷4; 4÷8; 9÷8; 13÷11; 6÷27; 9÷16; 93; 23-37.

ANALYSIS.-1. (a.) For convenience find the least common multiple of the denominators, 16 and 12, which is 48.

2. (b.) Change 13 and to equivalent fractions having the denomination of 48ths, and subtract.

Since twelfths can not be subtracted from sixteenths, change both 16ths and 12ths to 48ths; 13=39 48ths; 7=28 48ths; and 3-28 leaves a remainder of 11.

Therefore, the difference between 13 and is

QUESTIONS.-If the divisor is less than the dividend, is the value of the expression greater, or less, than a unit? ANS.-Greater. If the divisor is greater than the dividend, is the value greater, or less, than a unit? ANS.-Less. If the numerator is greater than the denominator, is the value of the expression greater, or less, than a unit? (130.) If the denominator is greater than the numerator, is the value of the expression greater, or less, than a unit? (129.) If both numerator and denominator be increased in an equal ratio, how is the value of the fraction affected? (137., a.)

ANALYSIS.-(a.) As can not be taken from no sevenths, take

1 unit from 13 units, which is equal to 7 sevenths; 4 sevenths from 7 sevenths leave 3 sevenths, which write in the remainder; 3 units from 12 units leave 9 units.

Therefore, 3 taken from 13 leave 92.

(d.) 13 equals 1333, and 123-1225

Since 25 30ths can not be taken from 24 30ths, take one unit from the 13 units; one unit is equal to 30 3uths; 30 30ths and 24 30ths equal 54 30ths; and 25 30ths from 54 30ths leave 29 30ths, which write in the remainder; 12 units from 12 units leave no units.

Therefore, 12 taken from 133 leave 3.

*NOTE.-Reduce mixed numbers to improper fractions.

EXAMPLES FOR PRACTICE.

152. 463-23=how many? | 157. 25—143=how many? 153. 37-63=how many? 158. 374-16=how many? 154. 141-53=how many? 159. 36-how many? 155. 11-34-how many? 160. 4937=how many?

156. 37—163=how many? 161. 5572-331=how many?

162. 57-3+=how many?

163. 143-63+4=how many?
164. 54-38+44-how many?
165. 374—54+8=how many?
166. 4247+3=how many?
167. 67-153+14=how many?
168. 147-863+34=how many?
169. 387-13+4173-how many?
170. 467-34313+2318-how many?
171. 8347-41381+63-how many y?

LESSON XVII.

152. To add any two fractions, or find their difference, when each has a unit for its numerator.

RULE I.*-To add two fractions haviug units for numerators :- Write the sum of the denominators over their product.

RULE II.-To find the difference between two fractions having units for numerators:- Write the difference of the denominators over their product.

*NOTE.-Require the pupil to write the analysis from which these rules

are deduced.

MISCELLANEOUS EXERCISES FOR PRACTICE

What is the value of each of the following expres

sions?

172. ++-4; 2+1 +1-7.
173. +++§; 3 + 4 +8 +8+4.
174. 4+4-3+4; 4+8+4−4+
175. 4+-+63; 3+64-3+1 +1.
176. 44+4-(+4); 4+7+63-(2-4)..
177. 24-+-(+9); +14-(123-1).
178. 4+113+−(6+9); (‡+3)+&+(4+9).
179. +12-11+3−2; 53+4~}+(4−4)· ·
180. 4+13-(6-3)+1; 4+16~+(#−√).
181. 8+11-(3-2)+4; (+12) 4+.
182. (14+3+4-1)−44; 4+4−4-4+4.
183. +(163-4)−4+4; (13-17)+4+4.
184. +(§−4)+(3+4); (11−1)+5−3.
185.53+4+(7+13); 5+7+1-7.

186. (6÷3)+++(6×8)+(3÷9); 6×5+3-44.

187. (4÷16)+(9×3)+3⁄4−(§+4); (8÷11)+(3÷9).