E 11). Multiply 25.238 by 13. Seren+117 X 1.024= 12.17. Prou. 307.14:40. how many? 11. Multiply 5 thousand by it. 128.73 1111.25 X 0.58 5 thousandths. Prod. 25. 16.3 Ans. 12. “Twenty-five x 1.25 are 15.01.004+1.0004x0.oouse how many? 1.000000038 Ans.. SUBTRACTION OF DECIMALS. ANALYSIS. 123. 1. What is the difference between 43.25 rods and 22.5 roos We write down the numbers as fur Addition, with the 43.25 22.5 largest uppermost. As there are no hundredths in the subtrahend, we bring down the 5 hundredths. Proceed. Ans. 20.75 rods. ing to the 10ths, we are unable to take 0.5 from 0.2, we therefore borrow a unit from the 3 units, which being 10 tenths, we join 10 to the 2, making 12 tenths; from which we take 5 tenths, and write the remainder, 7 tenths, in the place of tenths beluw the line. The rest of the operation must be obvious. 2. From 24 hours take 18.75 hours, what remains? 24. Here, as we cannot take 5 from nothing, we borrow 0.10 from the 4 units, or 400 hundredths; then taking 5 (=0.05) 18.75 froin 0.10, the remainder is 0.05. The 400 hun:lredths has Aps. 5,25 h. now become 300 hundredths, or 39 tenths, or 3.9; then 0.7 from 0.9 leaves 0.2, and so on. RULE. 124. Write down the numbers as in Addition of Decimals, observing to place the largest number uppermost. Beginning at the right, subtract as in Simple Subtraction, (99) and place the decimal point in the remainder directly under those in the given numbers. NOTE 1.- When the numbers are all properly written, and the results correctly pointed, the decimal points will all fall in one vertical column, mr directly under one another, both in Subtractior und Addition. Note 2.-In numbers given for Addition or Subtraction, the decimal places may all be made equal by annexing ciphers to a part of them,(116) without altering their value, and then all the decimals will express similar parts of a unit, or be of the same denomination. 'QUESTIONS FOR PRACTICE. 3. A person bought 27.631b. 4. From 468.742 rods, take of cinnamon, and sold 19.814 76.4815 rods. Ib. how much had he left? Rem. 392.2605. 27.63 5. From 9 ft. take 0.9 ft. 19.814 what remains? Ans. 8.1 ft. 6. From 2.73 take 1.9185. Ans. 7.816 lb. Rem. 0.8115. 7. What is the difference il. From 1 take 1 hunbetween 999 and nincty-nine dredth. Hem. 0.99. hundredths? Rem. 998.01. 12. From 1000 take 1 tbou 8. From *0.9173 subtract sandth. 0.2134. 13. How many are 71.019. From 742 take 195.127. 19,71 ? Rem. 546.873. 14. How many are 100 10. From 9.005 take 8.728. | 0.01? DIVISION OF DECIMALS. Since any ANALYSIS. 125. 1. If 14.25 lb. of' butter be divided into 3 equal shares, bow wany pounds will there be in each? 8)14.25(4.75 Here we wish to divide 14.25 into two factors, one of 12 which shall be 3, and the other such a number as, mul:iplied by 3, (101) will produce 14.25. We first seek how many 22 times 3 in 14, and find it 4 times, and 2 units over. The Ź 21 units being 20 tenths, we join them to the 2 tenths, making 22 terths, and, dividing these by 3, the quotient is 0.7, and 15 0.1 over:' but 0.1 being 0.10,(116) we join the 1 to the 5 hundredths, making 0.15, and dividing by 3, the quotient is 5 hundredths. The whole quotient then is 4.75 lb. To prove that this is the true quotient, we multiply it by the divisor, 3, (4.75X3=14.25) and reproduce the dividend. dividend may be regarded as the product of the divisor and quotient taken as factors, (101) and since the product must have as many decimal places as are contained in both the factors,(121) it follows that the number of decimal places in the divisor and quotient, counted together, must be jast equal to the number of decimal places in the dividend. 126. 2. If 18 bushels of wheat be divided equally among 4 men, how "much will each receive? Here we find that 18 bushels will give each man 4 bushele, 4)18(4.5.bu. and that there will be 2 bushels left. We now add a cipher 16 to the 2, which multiplying it by 10,(91) reduces it to tenths, and dividing 20 tenths by 4, the quotient is 0.5; -20 cach inan will, therefore, receive 4.5 bushels. Hence by annexing ciphers to the remainder of a division, the opera tion may be continued, and in pointing the result, the ciphers amnexed are to be regarded as decimals belonging to the dividend. 127. 8. What is the quotient of 0.0084 by 0.42? Omitting the ciphers, we find 42 in 84 just 2 times; 0.42)0.0084 0.02 but since there are 4 places of decimals in the divi. 84 Ang. dend, and only 2 in the divisor, there must be 2 places also in the quotient; we therefore place a cipber as die left of the 2 in the quotient, between it and the separatrix, to make up the deficiency. We see by this example, that if a quantity be divided by a deciinal, the quotient will be larger than the dividond. RULE. 128. Write down the divisor and dividend, and divide as in whole numbers. Point off as many places for decimals from the right hand of the quotient, as the decimal places in the dividend exceed those in the divisor. Note 1.- If there are not so many figures in the quotient as the number of decimal places required, supply the deficiency by prefixing ciphers. 2.-Should the decimal places in the divisor exceed those in the diviSlend, make them equal by annexiug ciphers to the latter. 8.-Whenever there is a remainder after division, by annexing ciphers to it, one or more additional figures may be obtained in the quotient.(126) QUESTIONS FOR PRACTICE. 4. In 68.43 hours, how ma- Let the pupil point the folny times 1.5 hours? lowing answers according to 1.5)68.43(45.62 Ans. the rule. 60 9. What is the quotient of 4263 by 2.5? Ans. 17052. 84 10. What is the quotient of 75 4.2 by 36? Ans. 116t 11. What is the quotient of 93 3298 by 7.54? Ans, 437+ 90 12. What is the quotient of 243by 5,47 Ans, 45, 30 NOTE.- When the quotient is 30 not complete, it is denoted by pla cing the sign + after it, in which 5. Divide 1 by 0.5. case more quotient figures may be Quot. 2:* obtained by annexiog ciphers to the remainder. 13. 6;=how many? 7. Divide 7.02 by 0.18. Quot. 39. 14. =8.3. Ans. 8. Divide 0.0081892 'by 21.75-16.75 0.347. Quot. 0.0236, 115 9.3175.09–1.75–8.46.0-3.58 . These are called Reciprocals. =1 17 + VULGAR FRACTIONS CHANGED TO DECIMALS. ANALYSIS. 129. If we divide an apple equally between 2 boys, the part which each will receive will be. $ an applc, or the quotient of 1 divided by 2; if we divide 1 apple between 3 boys, each will receive , or the quotient of a divided by 3. In like manner, if 3 apples be divided between 4 boysa, each boy will received of an apple, or the quotient of 3 divided by 4, and gencrally a Vulgar, or Common Fraction denotes the division of the numerator by the denominator (22, 103) The fraction $, for example, denotes that 1 is divided by 2, but since 1 does not contain 2, the quotient is less than 1, and must therefore be expressed in parts of unity. Now if we add a cipher to the dividend, 1, it becomes 10 tenths ;(126) and 10 tenthe divided by 2, the quotient is 0.5.(125) Hence the decimal 0.5 is equivalent to . Again, in the fraction }, if we add a cipher to the 1, it becomes 10 tenths as before, and 10 tenths divided by 3, the quotient is 0.3, and 0.1 remains. Joining a cipher to 0.1, it becomes 0.10, and dividing agaiu by 3, the quotient is 0.03, and thus may we go on as far as we please, getting by each additional cipher a 3 in the quotient, which is 10 times Jess than the preceding, as 0.333+, which is the decimal expression for f. And again in the fraction 4, adding a cipher to 3, and dividing by 4, the quotient is 0.7, and 0.2 remain; adding a cipher to 0.2, and dividing again by 4, the quotient is 0.05;- 0.75 then is the decimal expression for * And generally, 130. To change Vulgar Fractions to Decimals. Role.-Annex ciphers continually to the numerator, and divide by the denominator, so long as there shall be a remainder, or until the decimal be obtained to a sufficieat degree of exactness. The quotient will be the decimal required; and it must consist of as many decimal places, as the number of ciphers annexed. If the quotient does not contain so many figures, make up the deficiency lig prefixing ciphers. (127) QUESTIONS FOR PRACTICE. 1. What is the decimal ex- 5. What are g of a month pression for ? in decimals? Ans. 0.375 mo. 25)1.0020.04 Ans. 6. Change 44 to a decimal. 1.00 Ans. 0.70454 2. Change , d, and I to 7. Change to a decimal. equivalent decimals. Ans. 0.173+ Ans. J=0.5, 1=0.25, and 1=0.75. 8. Change go to a decimal. Ans. 0.002. 3. What is the decimal expression for of a day? 9. Change 71 to a mixed Ans. 0.2 day. number, 4. Change ds of a rod to a 10. Change to a decimal decimal. 11. Having become familiar with the method of changing Vulgar Fractions to Decimals, whenever fractions occur, the pupil has only to substitute for them their equivalent decimal values, and proceed as if they had been given in decima's. To illustrate this remark, take the following 152. FEDERAL MONEY. 33 QUESTIONS FOR PRACTICE. 1. There are 3 pieces of 4. What is the product of cloth, one contains 44 yards, | 24 by }? 24 X0.5=12, Ans. one 34, and the other 51 yds. 5. In 28 rods how many how many yards in the yards, 55 yards being equal whole? to one rod? 4=4.5 5.=5.5, and 28X5,5=154 3=3.75 rods, Answer. =-5.25 6. In 154 yards how many rods? Ans. 13.50=13} 16=154 rds. :-5.5=28 2. There are 4 boxes, each rods, Ans. of which contains 53 lb. of 7. What is the quotient of sugar; how many pounds in the whole? 12 by f? 53=5.375. -Ans. 21.51h. 12:=12+=24, Ans. 3. A person having 174 By these examples it appears tons of hay, sold 6 tons; that a number is diminished by mulhow much had lie lelt? tiplication and increased by divis. ion, when the multiplier and divisor Ans, 10.925 tons. are fractions or decimals. FEDERAL MONEY. 132. Federal Money is the established currency of the United States. Its denominations are all in a decimal or ten-fold proportion, as exhibited in table 1, page 38. The dollar is considered the unit money, and all the lower denominations are regarded as decimal parts of a dollar. Thus the dime is 1 tenth, or 0.1 of a dollar, the tent 1 hundredth, or 0.01 of a dollar, and the mill 1 thousandth, or 0.001 of a dollar; and placing these together, dol, d, c, m. 1 1. 11, They might be read, one dollar, one dime, one cent and one mill, or, one dollar, eleven cents and one mill, or, one dollar, 'one hundred and eleven mills or thousandths. The place next to dollars, on the left, is eagles, and 11. may be read, 1 cagle and I dollar, or elevén dollars. Twenty-five eagles, 8 dollars, 4 dimes, 6 cents and 3 mills, may be written and read, decim. |