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. § 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 O to the 1 and divide by the 3, we find 3.3}}; 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 ; in the second, it is of 1 hundredth, equal to ; and in the third, it is of 1 thousandth, equal to go. By omitting these small values, 3%, 300, 3000, we have .3 for a near or approximate value of 1, .33 for a nearer value, and .333 for a still nearer value of 1. 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. Ans. .333'. Ans. .285'. 21. Reduce $ to an approximate decimal. 22. Reduce $ to an approximate decimal. 23. Reduce $3 to an approximate decimal. Ans. .444'. ADDITION, SUBTRACTION, MULTIPLICATION AND DIVISION OF DECIMALS, AND FEDERAL MONEY. $148. 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. § 149. 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 3 6 1.2 63 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. 1005.41. Ans. 3.7722. 387+534.6+83.81. 2.25+.7373+.7849. Ans. 8418.3651. 3686.373. Ans. 5. Find the Sum, 6. Find the Sum, 7. Find the Sum, 48.36+ 8370+.0051. 8. Find the Sum, 8.773+974.6+ 2703. 9. Find the Sum, 74.03+ 3737+4301. 10. Find the Sum, .9346+203.7+.7376. 11. Find the sum of 100 dollars 72 cents, 34 dollars 5 cents, and 119 dollars 482 cents. $100.72 Ans. 205.3722. cents, 25 dollars 6 $279.32279 dollars 32 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, 4, 4 and 1, and find the sum to be 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 871⁄2 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 184 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 18 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 683 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. 17. Sold a barrel of sugar, for $15; a sack of coffee, for 13 dollars 5 cents; a keg of rice, for 5 dollars 43 cents; and a box of candles, for 9 dollars 8 cents; required the sum received. Ans. $42.56. 18. A merchant's bill was as follows: for 3 yards of cloth, $21; for 3 pair of stockings, 1 dollar 87 cents; for a dozen skeins of silk, 75 cents; required the amount of the bill. Ans. $23.62. 19. A farmer sold produce as follows, namely: wheat, for $300; corn, for 97 dollars 93 cents; hay, for 56 dollars 12 cents; and oats, for 18 dollars 64 cents; required his amount of sales. Ans. $472.12. 20. Bought a quantity of flour, for 75 dollars; a quantity of bacon for 57 dollars 183 cents; and a quantity of corn, for 42 dollars 64 cents; for what sum must the whole be sold to make a profit of 25 dollars? Ans. $199.25. SUBTRACTION OF DECIMALS. RULE XXX. $150. For the subtraction of decimals. 1. Set the less value under the greater, with tenths under tenths, hundredths under hundredths, &c., and subtract as in integers; observing to make in the right of the remainder as many decimal figures as will be equal to the greatest number of decimal figures in either of the given numbers. 2. When the minuend has no decimal figures, or not so many as the subtrahend, conceive the deficient places to be occupied by decimal Os. EXAMPLES. 1. To find the difference between 23.0623 and 380.75. 38 0.75 Having set the less quantity under the greater, with tenths under tenths, &c., we suppose the two vacant places over 23 to be occupied by 00, (§ 135;) and subtract as in integers; thus, 3 from 10 leaves 7, &c. Four decimal figures are made in the right of the remainder. 2. To find the difference between 525 and 9.87534. 525 9.875 34 5 15.12466 The places for decimals over .87534 must be regarded as occupied by Os; thus, 4 from 10 leaves 6, &c. 11. Find the Difference between $325 and 93 dollars 64 cents. Expressing the 64 cents in a decimal of a dollar, supplying the vacant decimal places in the upper number with 00,-and annexing, mentally,, equal to a unit,-we say from leaves ; then 1 to 6 makes 7, and 7 from 10 leaves 3, &c. (§ 34). Two decimal figures are made in the remainder, for cents or hundredths of a $. 12. A person having 95 dollars 64 cents, pays 43 dollars 182 cents for fuel, what sum will he have remaining? Ans. $51.87. |
