From the table it appears, that the first figure on the right hand of the decimal point signifies so many tenth parts of a unit; the second figure, so many hundredth parts of a unit; the third figure, so many thousandth parts of a unit, &c. It takes 10 thousandths to make 1 hundredth, 10 hundredths to make 1 tenth, and 10 tenths to make 1 unit, in the same manner as it takes 10 units to make 1 ten, 10 tens to make 1 hundred, &c. Consequently, we may regard unity as a starting point, from whence whole numbers proceed, continually increasing in a tenfold proportion towards the left hand, and decimals continually decreasing, in the same proportion, towards the right hand. But as decimals decrease towards the right hand, it follows of course, that they increase towards the left hand, in the same manner as whole numbers. T68. The value of every figure is determined by its place from units. Consequently, ciphers placed at the right hand of decimals do not alter their value, since every significant figure continues to possess the same place from unity. Thus, '5, 50, 500 are all of the same value, each being equal to, or But every cipher, placed at the left hand of decimal fractions, diminishes them tenfold, by removing the significant figures further from unity, and consequently making each part ten times as small. Thus, '5, '05, '005, are of different value, '5 being equal to, or ; '05 being equal to ʊ, or ; and '005 being equal to Too, or zoo. Decimal fractions, having different denominators, are readily reduced to a common denominator, by annexing ciphers until they are equal in number of places. Thus, '5, '06, 234 may be reduced to '500, '060, 234, each of which has 1000 for a common denominator, T 69. Decimals are read in the same manner as whole numbers, giving the name of the lowest denomination, or right hand figure, to the whole. Thus, '6853 (the lowest denomination, or right hand figure, being ten-thousandths) is read, 6853 ten-thousandths. Any whole number may evidently be reduced to decimal parts, that is, to tenths, hundredths, thousandths, &c. by annexing ciphers. Thus, 25 is 250 tenths, 2500 hundredths, 25000 thousandths, &c. Consequently, any mixed number 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 = 46 4667. Write the following numbers in the same manner : Eighteen and thirty-four hundredths. 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. 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. TT 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 $350, 7 pounds of sugar for 83 cents, an ounce of pepper for 6 cents; what did he give for the whole? $8 = 835 = '06 = OPERATION. 8000 mills, or 1000ths of a dollar. 835 mills, or 1000ths. 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 T 28.) 2. A man, owing $375, paid $175 75; how much did he then owe? $375 OPERATION. =37500 cents, or 100ths of a dollar. 175675 = 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 Too, 110 and 17%? Ans. 808, or 808'143. 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 25. Remainder, 188, or 1'98. 13. What is the amount of 29%, 3741008000, 97100%, 315Too, 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 of 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 7%, and ‘25 is : now, x 100 = 100% = '1875, Ans. same as be fore. 3. Multiply '125 by '03. OPERATION. 125 '03 '00375 Prod. 1875 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 them at the left hand. The correctness of the rule may appear from the following process: 125 is 12%, and '03 is 18: now, 12% × 180 = 100%‰000375, the same as 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. 5. 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 cents. 15. Multiply forty-seven tenths by one thousand eighty six hundredths. |