4. Make 85.8, 6.84, .117, and .7587 similar and conterminous. 5. Make .5046633, .42, .42, .048, and .0075 similar and conterminous. 6. Reduce, and § to decimals, and make their repetends similar and conterminous. ADDITION OF REPEATING DECIMALS. Art. 198. To find the sum of two or more repeating decimals. Ex. 1. What is the sum of 1.7, 4.787, 2.03631, and 6.1728? WRITTEN PROCESS. SIMILAR AND CONTERMINOUS. DISSIMILAR. 1.7 4.787 2.03631 6.1728 =1.77777777 Ans. 14.77484575. EXPLANATION. Since fractions can be added only when they express like parts of a unit, the repetends must first be made similar, because they then have a common denominator, which, if expressed, would be 99999900. They can then be 14.77 48 4575 added as common deci mals. To the sum of the right-hand column must be carried as much as would be carried if the decimals were extended to form more columns. In this case 2 must be carried, making 25, of which we write the 5, and proceed as usual. The result can be seen to be correct by finding the sum of the actual repetends, which is 2484573, and dividing by their common denominator, 99999900, giving .02181535, which, added to 14.75, gives 14.77181535, = 14.77484575. Rule.—Make the repetends similar and conterminous, and add as in finite decimals, observing to increase the sum of the right-hand column by as many units as are carried from the lefthand column of the repetends. Make a repetend in the result similar and conterminous with those added. EXAMPLES For Practice. 2. What is the sum of .3, .87, .5643, and .12? Ans. 1.8976. 3. What is the sum of .16, .792, .21431, and .56? 4. What is the sum of .09, .2045, and .25? 5. Add 7.124943, 5.0770, and .24. 6. Add .61,.25, .5635, and .104. Ans. .54. Ans. 1.536. SUBTRACTION OF REPEATING DECIMALS. Art. 199. To find the difference of two repeating decimals. common denominator, which, if expressed, would be 99999900. Then we can subtract as in common decimals, observing to carry to the first right-hand figure, 7, the 1 which would come from the extended decimals. The result is seen to be correct by taking 5.41337337 from 16.72777373, giving 11.30733881. Rule.-Make the repetends similar and conterminous, and subtract as in finite decimals, observing to increase the righthand figure of the subtrahend by 1, if the lower repetend is greater than the upper. Make a repetend in the result similar and conterminous with those above. EXAMPLES FOR PRACTICE, 2. From 25.127 take 14.65. 3. From .107 take 043. Ans. 10.5112. Ans. .063672763. MULTIPLICATION OF REPEATING DECIMALS. Art. 200. To multiply a repeating decimal. Ex. 1. Multiply 9.256 by 5.7. FIRST METHOD. 9.256256256 5.7 SECOND METHOD. 64793793793 462812812813 52.7606606606 = 52.7606 Rule. If the multiplier contains no repetend, first extend the repetend of the multiplicand enough to secure every figure of the repetend of the product, then multiply as if the multiplicand were a finite decimal. If the multiplier contains a repetend, reduce it to a common fraction, multiply, and reduce the result to the form of a repetend, as in addition of repetends; or reduce both factors to common fractions, and multiply, and reduce the result to a decimal. 7. Multiply .23 by 3.215: 5.128 by .46. DIVISION OF REPEATING DECIMALS. Art. 201. To divide a repeating decimal. Rule.—If the divisor contains no repetend, first extend the repetend of the dividend enough to secure every figure of the repetend of the quotient, and then divide as if the dividend were a finite decimal. If the divisor contains a repetend, reduce both repetends to common fractions, divide, and reduce the quotient to a decimal. 7. Divide 5.324 by 2.5: .45 by .3. MISCELLANEOUS EXAMPLES IN DECIMALS. 1. How much land in 4 fields containing 7.35 acres, 4.875 acres, 6.465 acres, and 10.7 acres, respectively? 2. A man owning a farm of 500 acres, sold to A, 80.5 acres, to B, 100.45 acres, and to C, 90.75 acres. How many acres had he left? 3. There are 16.5 feet in a rod; how many feet are there in 60.48 rods? 4. At $.62 per bushel, how many barrels of apples, each containing 23 bushels, can be bought for $41.25? Ans. 24. 5. If 12 boxes of soap, each weighing 60 pounds, can be bought for $46.80, what is the cost per pound? Ans. $.065. 6. Divide 21.6 by 21 tenths. Ans. 96. 7. If .75 tons of hay cost $20.25, what is a ton worth? 8. If 1 ton of hay is worth $20.25, what is .6 ton worth? 9. A stock-dealer bought 80 head of sheep at $4.75, and 64 head at $5.5 a head; at what price per head must he sell them to gain $116 on the whole? Ans. $5.75. 11. Reduce .0875 to a common fraction. 12. A miller wished to buy an equal number of bushels of corn, rye, and wheat; the corn cost $.684, the rye $.933, and the wheat $1.87 per bushel. How many bushels of each could he buy for $2264.76? Ans. 648. 13. If 25 barrels of flour cost $256.25, what will 33 barrels cost? 14. A grocer bought 124.5 pounds of butter at $.15 per pound, of which he sold at $.1875 a pound, and the rest at cost; how much did he gain? 15. From an oil tank containing 4325 gallons, 81.5 barrels, of 42.25 gallons each, were drawn off. How many gallons remained? 16. There are 5.5 yards in one rod, and 320 rods in a mile; how many yards in a mile? 17. A man sold 10 bushels of onions at $2.75 a bushel, and received in payment 30 pounds of coffee, at $.28 a pound, 50 pounds of sugar, at $.14 a pound, 20 pounds of cheese, at $.165 a pound, and the balance in molasses, at $.80 a gallon; how many gallons did he receive? |