It may be useful to observe here, conversely, that, § 141. In a decimal fraction of a dollar, the first figure after the point denotes tenths of a dollar, or tens of cents; the second denotes cents; the first two together denote cents; the third, mills; the fourth. tenths of a mill, &c. Thus $.5 is 5 tenths of a dollar, equal to 50 cents; $.0625 is 6 cents, 2 mills, and 5 tenths of a mill. DECIMALS REDUCED TO VULGAR FRACTIONS. RULE XXVII. § 142. To reduce a decimal to a vulgar fraction. 1. Remove the decimal point from the numerator, and underneath set the proper denominator. (§ 128). 2. The fraction thus formed may often be reduced to lower Reduce each of the following decimals of a dollar to a vulgar fraction of a dollar. 13. Express the value of $13.75 by an integer and a vulgar fraction. Ans. $13 14. Express the value of $20.125 by an integer and a vulgar fraction. Ans. $20 15. Express the value of $35.625 by an integer and a vulgar fraction. Ans. $355 16. Express the value of $49.0625 by an integer and a vulgar fraction. Ans. $497 17. Express the value of $57.9375 by an integer and a vulgar fraction. Ans. $571 VULGAR FRACTIONS REDUCED TO DECIMALS. RULE XXVIII. § 143. To reduce a vulgar fraction to a decimal. 1. Divide the denominator into the numerator, with as many Os annexed to the latter as may be necessary to find an exact quotient, or the number of decimal figures required. 2. Point off in the right of the quotient as many decimal figures as there were Os annexed to the numerator; observing to prefix Os to the quotient, when necessary to make up the number. 125) 3000 (24. Then 13.024. Annexing three Os to the numerator, and dividing 3000 by the denominator, we find the quotient 24. Prefixing a 0 to the quotient, to make up three decimal figures, for the three Os annexed to the numerator, we find 1. -.024. The fraction is equal to its numerator 3 divided by its denominator 125 (§ 92.) Each 0 annexed to the numerator, multiplies the fraction by 10 (§ 119-2); but each decimal figure made in the quotient, divides the quotient by 10, since each quotient figure thus becomes of its former value. Thus multiplying and dividing by the same number, we preserve a constant value to the fraction. (§ 93). EXERCISES. Reduce each of the following fractions of a dollar to a decimal of a dollar. Ans. $.625. 5. $5. Ans. $31.8125. 10. $16 Ans. $.5625. Ans. $17.125. Ans. $13.4375 Ans. $25.6875 A Complex Reduced to a Simple Decimal. $144. In a complex decimal, instead of the vulgar fraction annexed, we may put its equivalent decimal, without the point prefixed to it. Thus in .6, the 16. Reduce .25 17. Reduce .314 18. Reduce .18 19. Reduce .23 .04; then .6.604. to a simple decimal. to a simple decimal. 20. Reduce .90' to a simple decimal. Ans. .255. Ans. .3125. Ans. .1875. Ans. .23125. Approximate Decimals. Ans. .900625. 145. A decimal which expresses a near, but not the exact, value of a vulgar fraction, or other quantity, is an approximate decimal. In reducing, for example, to a decimal,-if we annex one 0 to the 1 and divide by the 3, we find = 3; by annexing two Os to the 1, we find .33}; by annexing three Os to the 1, we find }=.333}; and so on. In the first of these mixed decimals, the annexed is of 1 tenth, equal to 3; in the second, it is of 1 hundredth, equal to ; and in the third, it is of 1 thousandth, equal to 300• By omitting these small values, 3'%, 380, 30'00, we have .3 for a near or approximate value of 3,.33 for a nearer value, and .333 for a still nearer value of . The sign thus is commonly affixed to an approximate decimal; .33, 33 hundredths, nearly. Instead of the sign+, we shall employ a comma ', after the manner of an apostrophe, to denote an approximate decimal. Thus .33', 33 hundredths, nearly. The number of figures to which an approximate decimal need be carried, in any particular case, will depend on the value of the whole quantity of which the decimal expresses a part. In a decimal of a dollar, for example, two figures will give the number of cents, which is near enough for ordinary purposes. When greater accuracy is required, the third figure may be found, which will give the number of mills. 21. Reduce $3 to an approximate decimal. 22. Reduce $ to an approximate decimal. 23. Reduce $ to an approximate decimal. Ans. .333'. Ans. .285'. ADDITION, SUBTRACTION, MULTIPLICATION AND DIVISION OF DECIMALS, AND FEDERAL MONEY. $146. The objects of Addition, Subtraction, Multiplication and Division, are the same for every kind of quantity; and, having been defined for integers and vulgar fractions, the definitions need not be repeated. § 147. Addition, Subtraction, &c., of Federal Money, are brought under the Rules to be given for the same operations, respectively, on decimal fractions, by regarding dollars as integers, and making cents and mills decimals of a dollar. $148. The principles of notation being the same for decimals and integers, (§ 133 and 137) the methods of adding, subtracting, &c., will be the same for decimals and integers. ADDITION OF DECIMALS. RULE XXIX. § 119. For the addition of decimals. Set tenths under tenths, hundredths under hundredths, &c., and add as in integers; observing to make in the right of the sum as many decimal figures as will be equal to the greatest number of decimal figures in any one of the given numbers. EXAMPLE. To find the sum of .25+84.346+.73+275.937. .25 .73 36 1.263 The sum is 361 and 263 thousandths. Having set tenths under tenths, hundredths under hundredths, &c.,-this order also causing units to fall under units, tens under tens, &c., when mixed numbers are to be added-we add up the several columns of figures as in integers; and make three decimal figures .263 in the sum, this being the greatest number of decimal figures in any one of the given numbers. Ans. 205.3722 10. Find the Sum, .9346+203.7+.7376. 11. Find the sum of 100 dollars 72 cents, 25 dollars 61. cents, 34 dollars 5 cents, and 119 dollars 48 cents. Having made each number of cents a decimal of a dollar (§ 140) and placed tenths under tenths, &c., we first add up the fractions of a cent, namely, 3, 4 and 1, and find the sum to bo 1. We set down the, and carry the 1 to 8. In the sum we point off two decimal figures for cents, or hundredths of a $; (§ 132.) 12. What sum should be paid for a hat, at 5 dollars 871 cents; a vest, at 3 dollars 184 cents; and a pair of shoes, at 2 dollars 62 cents? Ans. $11.683. 13. What should be paid for a quarter of beef, at $7; a barrel of flour, at 4 dollars 564 cents; a lot of groceries, at 13 dollars 37 cents; and a lot of butter, at 2 dollars 64 cents? Ans. $27.00. 14. Find the sum that must be paid for a quire of paper, at 25 cents; a bottle of ink, at 12 cents; a dozen books, at 1 dollar 183 cents; and a bunch of quills, at 37 cents. Ans. $1.933. 15. Find the sum that should be paid for a set of chairs, at $18; a pair of tables, at 35 dollars 50 cents; a looking-glass, at 5 dollars 183 cents; and a bedstead, at 9 dollars 314 cents. Ans. $68.00. 16. Bought a cord of wood, for 2 dollars 50 cents; a ton of hay, for 12 dollars 682 cents; a barrel of apples, for 2 dollars 564 cents; and quarter of beef, for 5 dollars 75 cents; required the sum paid. Ans. $23.50. |