RULE. I. Expand the repetends, and place the first point in each over the same order in the decimal. II. Place the second point so that each new repetend shall contain as many places as there are units in the least common mul tiple of the number of places in the several given repetends. NOTE. Since none of the points can be carried to the left, some of them must be carried to the right, so that each repetend shall have at least as many finite places as the greatest number in any of the given repetends. EXAMPLES FOR PRACTICE. 1. Make .43, .57, .4567, and .5037 similar and conterminous. 2. Make .578, 37, .2485, and 04 similar and conterminous. 3. Make 1.34, 4.56, and .341 similar and conterminous. · 4. Make .5674, .34, .247, and .67 similar and conterminous. 5. Make 1.24, .0578, 4, and .4732147 similar and conterminous, 6. Make .7, .4567, .24, and .346789 similar and conterminous. 7. Make .8, .36, .4857, .34567, and .2784678943 similar and conterminous. ADDITION AND SUBTRACTION. 241. The processes of adding and subtracting circulating deci mals depend upon the following properties of repetends: I. If two or more repetends are similar and conterminous, their denominators will consist of the same number of 9's, with the same number of ciphers annexed. Hence, II. Similar and conterminous repetends have the same denominators and consequently the same fractional unit. 1. Add .54, 3.24 and, 2.785. OPERATION. .54 54444 = = 3.24 3.24242 ANALYSIS. Since fractions can be added only when they have the same fractional unit, we first make the repetends of the given decimals similar and conterminous. We then add as in finite decimals, observing, however, that the 1 which we carry from the left hand column of the repetends, must also be added to the right hand column; for this would be required if the repetends were further expanded before adding. 6.57214 2. From 7.4 take 2. 7852. OPERATION. 7.4444 2.7852 4.6581 ANALYSIS. Since one fraction can be subtracted from another only when they have the same fractional unit, we first make the repetends of the given decimals similar and conterminous. We then subtract as in finite decimals; observing that if both repetends were expanded, the next figure in the subtrahend would be 8, and the next in the minuend 4; and the subtraction in this form would require 1 to be carried to the 2, giving 1 for the right hand figure in the remainder. 242. From these principles and illustrations we derive the following RULE I. When necessary, make the repetends similar and conterminous. II. To add ;-Proceed as in finite decimals, observing to increase the sum of the right hand column by as many units as are carried from the left hand column of the repetends. III. To subtract; - Proceed as in finite decimals, observing to diminish the right hand figure of the remainder by 1, when the repetend in the subtrahend is greater than the repetend of the minuend. IV. Place the points in the result directly under the points above. NOTE. When the sum or difference is required in the form of a common fraction, proceed according to the rule, and reduce the result. EXAMPLES FOR PRACTICE. 1. What is the sum of 2.4, .32, .567, 7.056, and 4.37 ? Ans. 14.7695877. 2. What is the sum of .478, .321, .78564, .32, .5, and .4326 ? Ans. 2.8961788070698. 3. From 7854 subtract .59. Ans. .1895258. 5. What is the sum of .5, .32, and .12 ? Ans. 29.7455. Ans. 1. 6. What is the sum of .4387, .863, .21, and .3554? 8. From .432 subtract .25. 9. From 7.24574 subtract 2.634. Ans. .18243. Ans. 4.61. 10. From .99 subtract .433. Ans. .5 11. What is the sum of 4.638, 8.318, .016, .54, and .4 12. From .4 subtract .23. Ans. Ans MULTIPLICATION AND DIVISION. 243. 1. Multiply 2.428571 by .063. ANALYSIS. We 244. From these illustrations we have the following RULE. Reduce the given numbers to common fraction multiply or divide, and reduce the result to a decimal. UNITED STATES MONEY. 245. By Act of Congress of August 8, 1786, the dollar was declared to be the unit of Federal or United States Money; and the subdivisions and multiples of this unit and their denominations, as then established, are as shown in the 246. By examining this table we find 1st. That the denominations increase and decrease in a tenfold ratio. 2d. That the dollar being the unit, dimes, cents and mills are respectively tenths, hundredths and thousandths of a doilar. 3d. That the denominations of United States money increase and decrease the same as simple numbers and decimals. Hence we conclude that I. United States money may be expressed according to the decimal system of notation. II. United States money may be added, subtracted, multiplied and divided in the same manner as decimals. NOTATION AND NUMERATION. 247. The character $ before any number indicates that it expresses United States money. Thus $75 expresses 75 dollars. 248. Since the dollar is the unit, and dimes, cents and mills are tenths, hundredths and thousandths of a dollar, the decimal point or separatrix must always be placed before dimes. Hence, in any number expressing United States money, the first figure at the right of the decimal point is dimes, the second figure is cents, the third figure is mills, and if there are others, they are tenthousandths, hundred-thousandths, etc., of a dollar. Thus, $8.3125 expresses 8 dollars 3 dimes 1 cent 2 mills and 5 tenths of a mill or 5 ten-thousandths of a dollar. 249. The denominations, eagles and dimes, are not regarded in business operations, eagles being called tens of dollars and dimes tens of cents. Thus $24.19 instead of being read 2 eagles cents, is read 24 dollars 19 cents. Hence, of United States money is as follows: 4 dollars 1 dime 9 practically, the table 250. Since the cents in an expression of United States money may be any number from 1 to 99, the first two places at the right of the decimal point are always assigned to cents. Hence, when the number of cents to be expressed is less than 10, a cipher must be written in the place of tenths or dimes. Thus, 7 cents is expressed $.07. NOTES.1. The half cent is frequently written as 5 mills and vice versa. Thus, $.37 $.375. 2. Business men frequently write cents as common fractions of a dollar. Thus, $5.19 is also written $5, read 5 and dollars. 100 3. In business transactions, when the final result of a computation contains 5 mills or more, they are called one cent, and when less than 5 they are rejected. Thus, $2.198 would be called $2.20, and $1.623 would be called $1.62. EXAMPLES FOR PRACTICE. 1. Write twenty-eight dollars thirty-six cents. 2. Write four dollars seven cents. 3. Write ten dollars four cents. 4. Write sixteen dollars four mills. Ans. $28.36. 5. Write thirty-one and one-half cents. Ans. $48 01. 6. Write 48 dollars 14 cents. 7. Write 1000 dollars 1 cent 1 mill. 8. Write 3 eagles 2 dollars 5 dimes 8 cents 4 mills. |