Find the true value, in the form of an improper fraction, of 10. 5.328. 12. 2.43. 14. 12.227. 11. 9.752. 13. 7.86. 15. 5.39. ADDITION. PREPARATORY PROPOSITION. 350. Any two or more decimals can be reduced to a common denominator by annexing ciphers. Thus, .r = 7, and, according to (235—II), %= %6= 17% = 100, and so on; therefore, .7 =.70=.700 = .7000, Hence any two or more decimals can be changed at once to the same decimal denominator by annexing ciphers. ILLUSTRATION OF PROCESS 351. Find the sum of 34.8, 6.037, and 27.62. (1.) (2.) EXPLANATION.-1. We arrange 34.800 34.8 the numbers so that units of the same order stand in the same 6.039 6.03% column. 27.620 27.62 2. We reduce the decimals to a 68.45 y 68.45 common denominator, as shown in (1), by annexing ciphers. 3. We add as in integers, placing the decimal point before the tenths in the sum. In practice, the ciphers are omitted, as shown in (2), but the decimals are regarded as reduced to a common denominator. Thus the 3 hundredths in the second number and the 2 hundredths in the third, when added are written, as shown in (2), as 50 thousandths ; in the same manner, the 8 tenths and 6 tenths make 1400 thousandths, or 1 unit and 400 thousandths. The 1 unit is added to the units and the 4 written in the tenths' place as 400 thousandths. From this it will be seen that the addition of decimals is subject to the same laws (261—I and II) and rule (263) as other fractions. EXAMPLES FOR PRACTICE. 352. Find the sum of the following, and explain as above: 1. 38.9, 7.05, 59.82, 365.007, 93.096, and 8.504. 2. 9.07, 36.009, 84.9, 5.0036, 23.608, and .375. 3. $42.08, $9.70, $89.57, $396.02, and $.89. 4. .039, 73.5, .0407, 2.602, and 29.8. 5. 395.3, 4.0701, 9.96, and 83.0897. 6. 8.0093, .805, .03409, 7.69, and .0839. 17. $.87, $32.05, $9, $75.09, $.67, and $3.43. 8. .80003, 3.09, 13.36, 97.005, and .9999. SUBTRACTION. 353. Find the difference between 83.7 and 45.392. 83.100 EXPLANATION.—1. We arrange the numbers so 45.392 that units of the same order stand in the same column. 2. We reduce the decimals, or regard them as 38.308 reduced to a common denominator, and then subtract as in whole numbers. The reason of this course is the same as given in addition. The ciphers are also usually omitted. EXAMPLES FOR PRACTICE. 354. Subtract and explain the following: 1. 834.9 — 52.47. 6. 379.000001 – 4.0396. 2. 39.073 – 7.0285. 17. 54.5 – 37,00397. 3. $67.09 – $29.83. 8. 96.03 – 89.09005. 4. $95.02 — $78.37. 9. .09 –.0005903. 5. 83.003 — 45.879. 10. . - .099909. TICE 11. A man paid out of $3432.95 the following sums; $342.06, $593.738, $729.039, $1362.43, $296.085, $37.507. How much has he left ? Ans. $72.091. 12. In a mass of metal there are 183.741 pounds ; } of it is iron, 25.305 pounds are copper, and 3.0009 pounds are silver, and the balance lead. How much lead is there in the mass ? 13. A druggist sold 74.52 pounds of a costly drug. He sold in March 10 pounds, in April 25.125, in May 213, and the balance in June. How many pounds did he sell in June? Find the decimal value of MULTIPLICATION. _327 83 3 27 83 27141 -= 27.141 EXPLANATION.-1. Observe that 3.27 and 8.3 are mixed numbers: hence, according to (282), they are reduced before being multiplied to improper fractions, as shown in (1). 2. According to (276), 167 is, as shown in (2), equals 27.141. Hence 27.141 is the product of 3.27 and 8.3. The work is abbreviated thus: (3.) We observe, as shown in (2), that the product of 3.29 3.27 and 8.3 must contain as many decimal places as there are decimal places in both numbers. Hence 8.3 we multiply the numbers as if integers, as shown in 981 (3), and point off in the product as many decimal 2616 places as there are decimal places in both numbers. Hence the following 27.141 - addition - 40396 356. RULE.—Multiply as in integers, and from the right of the product point off as many figures for decimals as there are decimal places in the multiplicand and multiplier. pound ? EXAMPLES FOR PRACTICE. 357. Multiply and explain the following: 1. 7.3 x 4.9. 6. 34.0007 X 8.43. 11. .009 x .008. 2. 13.4 x.37. 17. 73.406 x.903. 12. .0007 X.036. 3. 35.08 x 6.2. 8. .4903 X.06. 13. .0405 X.09. 4. $97.03 x 42. 9. .935 x.008. 14. .00101 x .001. 5. $83.65 x.7. 10. 5.04 x.072. 15. .307 x .005. Multiply and express the product decimally: 16. $353 by 9%. 19. 73 thousandths by . 17. 37 by 6%. 20. 121 by 34 hundredths. 18. $.054 by 181 21. 97 tenths by .0003. 22. What is the value of 325.17 pounds of iron at $.023 per Ans. 7.47891 dollars. 23. What would 12.34 acres of land cost at $43.21 per acre ? 24. A merchant sold 86.43 tons of coal at $9.23 a ton, thereby gaining $112.12 ; what was the cost of the coal ? 25. A French gramme is equal to 15.432 English grains; how many grains are 144 grammes equal to ? 26. A metre is equal to 39.3708 inches; how many inches are there in 1.325 metres ? 27. A merchant uses a yardstick which is .00538 of a yard too short; how many yards will he thus gain in selling 438 yards measured by this yardstick ? Find the value of the following: 31. A dealer in wood and hay bought 2395 tons of hay at $14.75 a ton, and 23874 cords of wood at $4.50 a cord; how much did he pay for all ? Ans. $46070. 32. Bought 18 books at $1.371 each, and sold them at a gain of .50 cents each; what did I receive for the whole ? 33. A boy went to a grocery with a $10 bill, and bought 34 pounds of tea at $.90 a pound, 7 pounds of flour at $.07 a pound, and 4 pounds of butter at $.35 a pound; how much change did he return to his father? Ans. $4.96. 34. What would 15280 feet of lumber cost, at $2.371 for each 100 feet? Ans. $362.90. DIVISION. PREPARATORY PROPOSIT10ns. 358. PROP. I.- When the divisor is greater than the dividend, the quotient expresses the part the dividend is of the divisor. Thus, 4:6 =$=. The quotient f expresses the part the 4 is of 6. 1. Observe that the process in examples of this kind consists in reducing the fraction formed by placing the divisor over the dividend to its lowest terms. Thus, 32 = 56 = 34, which reduced to its lowest terms gives 4. 2. In case the result is to be expressed decimally, the process then consists in reducing to a decimal, according to (338), the fraction formed by placing the divisor over the dividend. Thus, 5 :8= f, reduced to a decimal equals .625. Divide the following, and express the quotient decimally. Explain the process in each case as above. 1. 3:4. 4. 13:40. 9. 5% 10. 320. 2. 7:20. 5. 15:32. 8. 8:11. 11. 7:-88. 3. 5:8. 6. 9:80. 9. 5:6. 12. 4; 13. |