may be read together, giving it the name of the lowest denomination or right hand figure. Thus, 25'63 may be read 2563 hundredths, and the whole may be expressed in the form of a common fraction, thus, 2563. The denominations in federal money are made to correspond to the decimal divisions of a unit now described, dollars being units or whole numbers, dimes tenths, cents hundredths, and mills thousandths of a dollar; consequently the expression of any sum in dollars, cents, and mills, is simply the expression of a mixed number in decimal fractions. Forty-six and seven tenths 467 46'7. = Write the following numbers in the same manner : Fifty-two and six hundredths. Nineteen and four hundred eighty-seven thousandths. One and five thousandths. 135 and 3784 ten-thousandths. 9000 and 342 ten-thousandths. 10000 and 15 ten-thousandths. 974 and 102 millionths. UNIV. 320 and 3 tenths, 4 hundredths and 2 thousandths. 500 and 5 hundred-thousandths. 47 millionths. Four hundred and twenty-three thousandths. ADDITION AND SUBTRACTION OF DECIMAL FRACTIONS. ¶ 70. As the value of the parts in decimal fractions increases in the same proportion as units, tens, hundreds, &c., and may be read together, in the same manner as whole numbers, so, it is evident that all the operations on decimal fractions may be performed in the same manner as on whole numbers. The only difficulty, if any, that can arise, must be in finding where to place the decimal point in the result. This, in addition and subtraction, is determined by the same rule; consequently, they may be exhibited together. 1. A man bought a barrel of flour for $8, a firkin of but ter for $3'50, 7 pounds of sugar for 834 cents, an ounce of pepper for 6 cents; what did he give for the whole? Ans. $12'395 = 12395 mills, or 1000ths. As the denominations of federal money correspond with the parts of decimal fractions, so the rules for adding and subtracting decimals are exactly the same as for the same operations in federal money. (See π 28.) 2. A man, owing he then owe? $375 $375, paid $175 75; how much did OPERATION. 37500 cents, or 100ths of a dollar. 175'75 = 17575 cents, or 100ths of a dollar. © 199′25 — 19925 cents, or 100ths. The operation is evidently the same as in subtraction of federal money. Wherefore,-In the addition and subtraction of decimal fractions,-RULE: Write the numbers under each other, tenths under tenths, hundredths under hundredths, according to the value of their places, and point off in the results as many places for decimals as are equal to the greatest number of decimal places in any of the given numbers. EXAMPLES FOR PRACTICE. 3. A man sold wheat at several times as follows, viz. 13'25 bushels; 8'4 bushels; 23'051 bushels, 6 bushels, and "75 of a bushel; how much did he sell in the whole ? Ans. 51'451 bushels. 4. What is the amount of 429, 21, 355, 100 and 176? Ans. 80814, or 808'143. A 5. What is the amount of 2 tenths, 80 hundredths, 89 thousandths, 6 thousandths, 9 tenths, and 5 thousandths? Ans. 2. 6. What is the amount of three hundred twenty-nine, and seven tenths; thirty-seven and one hundred sixty-two thousandths, and sixteen hundredths ? 7. A man, owing $4316, paid $376'865; how much did he then owe? Ans. $3939'135. 8. From thirty-five thousand take thirty-five thousandths. 9. From 5'83 take 4'2793. 10. From 480 take 245'0075. Ans. 34999'965. Ans. 1'5507. Ans. 234'9925. 11. What is the difference between 1793'13 and 817' 05693 ? 98 Ans. 976'07307. 12. From 48 take 2. Remainder, 18, or 1'98. 13. What is the amount of 29, 3741008000, 97700, 315, 27, and 100? Ans. 942'957009. MULTIPLICATION OF DECIMAL FRACTIONS. ¶ 71. 1. How much hay in 7 loads, each containing 23'571 cwt.? OPERATION. 23'571 cwt. = 23571 1000ths of a cwt. Ans. 164'997 cwt. 164997 1000ths of a cwt. We may here ( 69) consider the multiplicand so many thousandths of a cwt., and then the product will evidently be thousandths, and will be reduced to a mixed or whole number by pointing off 3 figures, that is, the same number as are in the multiplicand; and as either factor may be made the multiplier, so, if the decimals had been in the multiplier, the same number of places must have been pointed off for decimals. Hence it follows, we must always point off in the product as many places for decimals as there are decimal places in both factors. 2. Multiply 75 by $25. OPERATION. 675 '25 375 150 '1875 Product. M* In this example, we have 4 decimal places in both factors; we must therefore point off 4 places for decimals in the product. The reason of pointing off this number may appear still more plain, if we consider the two factors as common or vulgar fractions. Thus, "75 is 75%, and ‘25 is : now, × 10% = 76875 1875, Ans. same as be fore. 3. Multiply '125 by '03. OPERATION. 125 '03 '00375 Prod. Here, as the number of significant. figures in the product is not equal to the number of decimals in both factors, the deficiency must be supplied by prefixing ciphers, that is, placing The correctness of the rule may appear from the following process: 125 is, and '03 is To: now, 25 X 130 = 100%‰000375, the same as them at the left hand. before. These examples will be sufficient to establish the following RULE. In the multiplication of decimal fractions, multiply as in whole numbers, and from the product point off so many figures for decimals as there are decimal places in the multiplicand and multiplier counted together, and, if there are not so many figures in the product, supply the deficiency by prefixing ciphers. EXAMPLES FOR PRACTICE. 4. At $5'47 per yard, what cost 8'3 yards of cloth? Ans. $45'401. 15. At $'07 per pound, what cost 26'5 pounds of rice? Ans. 1'855. 6. If a barrel contain 175 cwt. of flour, what will be the weight of '63 of a barrel? Ans. 1'1025 cwt. 7. If a melon be worth $'09, what is 7 of a melon worth? Ans. 6 8. Multiply five hundredths by seven thousandths. 9. What is '3 of 116? 10. What is '85 of 3672? cents. Product, '00035. Ans. 31212. 11. What is '37 of '0563 ? 12. Multiply 572 by '58. Ans. '020831. Product, 331'76. 13. Multiply eighty-six by four hundredths. 14. Multiply '0062 by '0008. Product, 3'44. 15. Multiply forty-seven tenths by one thousand eightysix hundredths. 16. Multiply two hundredths by eleven thousandths. 17. What will be the cost of thirteen hundredths of a ton of hay, at $11 a ton? 18. What will be the cost of three hundred seventy-five thousandths of a cord of wood, at $2 a cord? 19. If a man's wages be seventy-five hundredths of a dollar a day, how much will he earn in 4 weeks, Sundays excepted? DIVISION OF DECIMAL FRACTIONS. 72. Multiplication is proved by division. We have seen, in multiplication, that the decimal places in the product must always be equal to the number of decimal places in the multiplicand and multiplier counted together. The multiplicand and multiplier, in proving multiplication, become the divisor and quotient in division. It follows of course, in division, that the number of decimal places in the divisor and quotient, counted together, must always be equal to the number of decimal places in the dividend. This will still further appear from the examples and illustrations which follow: 1. If 6 barrels of flour cost $44718, what is that a barrel? By taking away the decimal point, $44'718 44718 mills, or 1000ths, which, divided by 6, the quotient is 7453 mills, $7'453, the Answer. Or, retaining the decimal point, divide as in whole numbers. OPERATION. 6)44'718 Ans. 7'453 As the decimal places in the divisor and quotient, counted together, must be equal to the number of decimal places in the dividend, there being no decimals in the divisor,-therefore point off three figures for decimals in the quotient, equal to the number of decimals in the dividend, which brings us to the same result as before. 2. At $475 a barrel for cider, how many barrels may be bought for $31 ? In this example, there are decimals in the divisor, and none in the dividend. $475 475 cents, and $31, by annexing two ciphers, 3100 cents; that is, reduce the di |