267. Reducing numbers of higher denominations to a number of a lower is called Reduction Descending.

268. To reduce numbers of higher denominations of a compound number to a number of a lower denomination,

Rule.

Multiply the number of the highest denomination given by the number of units it takes of the next lower to make one of this higher, and to the product add the given number (if any) of the lower denomination; multiply this result by the number of units it takes of the next lower denomination to make one of this; add as before, and so proceed till the number is brought to the denomination required.

76. Reduce 222 gills to gallons, quarts, etc.

OPERATION.

4) 222 gi.

2) 55 pt. 2 gi.
4) 27 qt.+ 1 pt.

6 gal. + 3 qt.

Ans. 6 gal. 3 qt. 1 pt. 2 gi.

1 pt., we divide the 222 gi. by 4 and obtain 55 pt. and 2 gi. remainder. As 2 pt. 1 qt., we divide the 55 pt. by 2 and obtain 27 qt. and 1 pt. remainder. As 4 qt.

=

=

1 gal., we divide the 27 qt. by 4 and obtain 6 gal. and 3 qt. remainder. Therefore,

222 gi. 6 gal. 3 qt. 1 pt. 2 gi. Hence,

=

269. Reducing a number of a lower denomination to numbers of higher is called Reduction Ascending.

270. To reduce a number of a lower denomination to numbers of higher denominations,

Rule.

Divide the given number by the number of units it takes of that denomination to make one of the next higher; divide the quotient in the same way, and so proceed till the number is brought to the denomination required. The last quotient and the remainders (if any) will be the result required.

271. Reduction Ascending and Reduction Descending prove each other.

77. Reduce 18 bush. 3 pk. 7 qt. 1 pt. to pints.

78. Reduce 1577048 seconds to minutes, hours, etc.

79. Reduce 20° 25′ 30′′ to seconds.

80. Reduce 4 t. 1473 lb. 7 oz. to ounces.

81. Reduce 2548 square inches to higher denominations. 82. Reduce 4 sq. m. 25 a. 154 sq. rd. to square rods.

83. Reduce 8 cu. yd. 1727 cu. in. to cubic inches.

84. Reduce 4 sq. yd. to square inches.

85. Reduce 4 gal. 1 pt. to gills..

86. Reduce 2 wk. 6 d. 8 h. 16 sec. to seconds.

87. Reduce 4 m. 319 rd. to rods.

88. Reduce 3795 rods to miles, etc.

89. Reduce 6598 pints to quarts, pecks, etc.

90. Reduce 368294" to higher denominations.
91. Reduce 4680 gills to higher denominations.
92. Reduce 195261 cubic inches to feet and yards.

93. Reduce 310556 square rods to higher denominations.

NOTE 1. The rules given in Arts. 268 and 270 are equally applicable when the number given is a fraction.

NOTE 2. In the first statement of Ex. 8, the 16, in the numerator, is equal to 32, and, in the second statement, the 33 is retained in the numerator as a factor in the dividend, and the 2 is put in the denominator as a factor in the divisor. In practice the first statement should not be written at all.

95. Reduce of a gallon to integers of lower denominations.

NOTE 3. When the given fraction is a decimal the operation is much more simple.

96. Reduce 0.713 m. to integers of lower denominations.

NOTE 4. The principle is the same as in the Rule in Art. 269, the lowest denomination being reduced to the next higher.

98. Reduce 2 qt. 1 pt. 3 gi. to the decimal of a gallon.

The 1.75 pt. are reduced to quarts by dividing by 2 (2 pints=1 quart), giving 0.875 qt., which is written at the right of the 2 qt., making 2.875 qt.; and in like manner we continue dividing on the same principle as in Arts. 268-9.

99. Reduce

100. Reduce

of a gallon to lower denominations.

of an acre to lower denominations.

101. Reduce of a ton to lower denominations.

102. Reduce 0.375° to lower denominations.

103. Reduce 7654.825 rods to miles, rods, feet, and inches. 104. Reduce 0.984375 of a bushel to lower denominations. 105. Reduce 0.40625 of a gallon to lower denominations. 106. Reduce 6 rd. 5 ft. 9 in. to the fraction of a mile.

107. Reduce 42 sq. rd. 181 sq. ft. 72 sq. in. to the fraction of

an acre.

108. Reduce 875 lb. 10 oz. to the fraction of a ton.

109. Reduce 175 ft. 6 in. to the decimal of a mile.

110. Reduce 145 sq. rd. 68 sq. ft. 9 sq. in. to the decimal of

an acre.

111. Reduce 3 pk. 4 qt. 1 pt. to the decimal of a bushel.

272.

ADDITION OF COMPOUND NUMBERS.

The difference between operations with the compound numbers of these tables and with simple numbers is, that in these compound numbers the scale is not decimal, nor regular, but varies; the principles, however, are the same.

Therefore no special rules are needed for addition, subtraction, multiplication, or division of compound numbers.

112. Find the sum of 3 gal. 3 qt. 1 pt. 3 gi., 4 gal. 2 qt. 1 pt. 2 gi., 7 gal. 2 qt. 3 gi., 5 gal. 3 qt. 1 pt., 9 gal. 2 gi.

the 2 qt. to the sum of the quarts in the next column. Proceed in this manner until all the columns are added.

NOTE. A fraction occurring in the amount should be reduced to integers of lower denominations; or, if possible, enough taken from the lower to cause the fraction to disappear. Thus, in Ex. 114, yd., or 1 ft. 6 in., taken from the 2 ft. 4 in. increases the 1 yd. to 2 yd. and leaves 0 ft. 10 in. in the columns of feet and inches.

115. A farmer raised in one field 21 bush. 3 pk. 7 qt. 1 pt. of wheat; in another, 48 bush. 2 pk. 1 pt.; in another, 28 bush. 6 qt.; and in another, 75 bush. 1 pk. 5 qt. 1 pt.: how much wheat did he raise in the four fields?

116. Add 29° 59′ 59′′, 25° 20′ 30′′, 8° 25′ 55′′, and 10° 10′ 10" together.