7. .00004, .00137, .02376, .01007. 8. .001768, .040035, 70.360004. 9. .1010101, .000133, 60.04, 45.1003. 13. 1007.3154, 7496.35491768. Change the following decimals to common fractions: 16. .91, .347, .513, .691, .851, .917. 19. .001, .5302, 8.01, .000053. VII. The operations with decimals are Reduction, Addition, Subtraction, Multiplication, and Division. REDUCTION OF DECIMALS. VIII. Reduction of decimals is changing their form without alter ing their value. There are four cases. CASE I. IX. 1. Annexing decimal ciphers to an integer does not change its value. Thus: 7.00 is the same as 7.; for 7.00 is 7 and no hundredths. 2. Conversely: Omitting decimal ciphers to the right of an integer does not change its value. CASE II. X. 1. Annexing ciphers to a decimal does not change its value. Thus: .70 is the same as .7, for is the same as 10. 70 2. Conversely: Omitting ciphers from the right of a decimal does not change its value. CASE III. XI. To reduce a decimal to a common fraction. 1. Reduce .75 to a common fraction. 75 100. 100 Solution-75 hundredths written as a common fraction is 7050 reduced to its lowest terms is 4. RULE: Write the decimal as a common fraction and reduce to its low est terms. 7 Answer, . 2. Reduce .6 to a common fraction. 3. Reduce .25 to a common fraction. 4. Reduce .375 to a common fraction. 5. Reduce .035 to a common fraction. 6. Reduce .5625 to a common fraction. 7. Reduce .34375 to a common fraction. 8. Reduce .1484375 to a common fraction. 9. Express 4.02 as an integer and a common fraction. Answer, 45. 10. Express 8.415 as an integer and a common fraction. Answer, 19 128. 1 CASE IV. XII. To reduce a common fraction to a decimal. 1. Reduce to a decimal. 4) 3.00 Solution.-Annexing a decimal cipher to 3 it is 3.0; 30 tenths divided by 4 is 7 tenths and 2 tenths remaining. Annexing a cipher to .2 it is .20. 20 hundredths divided by 4 is 5 hundredths. The result is .75. Explanation: is 3 divided by 4. Annexing a decimal cipher to 3 does not change its value. Annexing a decimal cipher to .2 does not change its value. RULE: Annex decimal ciphers to the numerator and divide by the denominator. Point off as many decimal places in the quotient as there are decimal ciphers annexed to the numerator. XIII. Addition of decimals- is the process of finding the sum of two or more decimal numbers. 1. Add 375.83, 49.627, 5842.1963, 813.9762. 375.83 49.627 5,842. 1963 813.9762 7.081.6295 Solution. Write the numbers so that the four decimal points are in a column, the units, 5, 9, 2, and 3, in the first column to the left; the tenths, 8, 6, 1, and 9, in the first column to the right, etc., then adding as in simple numbers; place the decimal point in between the 1 and 6 under the column of decimal points. RULE: Write the numbers so that the decimal points and the figures of the same order may stand in the same column. Add as in simple numbers. Place the decimal point in the sum under the column of decimal points. 2. Add 37.105, 432.07, 4.20733, 11.706. Answer, 485.08983. 3. Add 4 and 4 ten-thousandths, 28 and 35 thousandths, 8 and 7 hundredths, and 9404 hundred-thousandths. Answer, 40.1944. 4. Find the sum of 3 units and 25 hundredths, 6 units and 4 tenths, and 35 hundredths. Answer, 10. 83143-13--5 5. Add 21.611, 6,888.32, 3.4167. 6. Add 6.61, 636.1, 6,516.14, 7,231.0967. Answer, 6,913.3477. 67.1234, and 5.1233. Answer, 7. Add 4 and 8 tenths, 43 and 31 hundredths, 74 and 19 thousandths, 11 and 204 thousandths. Answer, 133.333. 8. Add 45 and 19 thousandths, 7 and 71 hundred-thousandths, 93 and 4327 ten-thousandths, 6 and 401 ten-thousandths. Answer, 151.49251. 9. Add 432 and 432 thousandths, 61 and 793 ten thousandths, 100 and 7794 hundred-thousandths, 6.009, 1,000 and 1001 ten-thousandths. Answer, 1,599.69834. 10. Add 16 and 41 thousandths, 9 and 94 millionths, 33 and 27 hundredths, 8 and 969 thousandths, 32 and 719906 millionths. Answer, 100. 11. Add 204 and 9 ten-thousandths, 103 and 9 hundred-millionths 42 and 9099 millionths, 430 and 99 hundredths, 220.0000009. Answer, 999.99999999. 12. Add 35 ten-thousandths, .00035, 35 millionths, and 35 ten millionths. Answer, .0038885. 13. The estimated amount of coal in tons in a ship's bunkers was as follows: 36.54, 80.2, 79.70, 83.265, 21.232, 54.46, 21.324, 56.41, 80.7, 56.42. How much coal was there in all? Answer, 570.251 tons. 14. The following stores were received on board a vessel: 20 gallons cylinder oil, $35; 50 gallons machine oil, $74.54; 200 pounds waste, $14.66; 8 monkey wrenches, $7.76. What was the total value of the stores received? Answer, $131.96. 15. The evaporators of a vessel during one week made the following amounts of water: First day, 70.26 tons; second day, 68.52 tons; third day, 65.726 tons; fourth day, 74.526 tons; fifth day, 54.2 tons; sixth day, 32.926 tons. How many tons of water were made? Answer, 366.158 tons. 16. After taking coal aboard from 4 lighters there remained on the lighters 5.316 tons on one, & tons on another, 12.25 tons on another, and 13.569 tons on the last. How much coal was left on the lighters? Answer, 39.385 tons. 17. A man saved $6.75 one month, $7.25 another month, $7.37 another month, and $8.13 another month. How much did he save in all? Answer, $29.50. 18. A man bought tools at the following prices: 1 scale, $0.45; 1 screw pitch gauge, $1.25; 1 wire gauge, $2.45; 1 set feelers, $1.65; 1 universal bevel level protractor, $7.95; and 1 pair of micrometer calipers, $5.50. How much did he spend for tools? Answer, $19.25. 19. A ship, in making a journey, steamed the following distances each day: 249.56 miles, 253.461 miles, 196.275 miles, 210.375 miles, 230.5 miles, 225.36 miles. How many miles did she travel? Answer, 1,365.531 miles. 20. During a four-hour watch the following coal was used by hours: .85 tons, .75 tons, .675 tons, and .5 tons. How much coal was used? Answer, 2.775 tons. SUBTRACTION OF DECIMALS. XIV. Subtraction of decimals is the process of finding the difference between two decimal numbers. 1. From 729.835 subtract 461.5738. 729.835 461.5738 268.2612 Solution.-Write the two numbers so that the decimal points may be in a column; the units 9 and 1 in the first column to the left; the tenths, 8 and 5, in the first column to the right, etc. Then, subtracting as in simple numbers, place the decimal point in the remainder between 8 and 2, under the column of decimal points. NOTE. The ten-thousandth place in the minuend may be regarded as occupied by a cipher. RULE: Write the numbers so that the decimal points and figures of the same order may stand in a column. Subtract as in simple numbers. Place the decimal point in the remainder under the column of decimal points. 2. From 97.5168 subtract 38.25942. Answer, 59.25738. 3. From 20.014 subtract 7.0021. Answer, 13.0119. 4. From 5.03 subtract 2.115. Answer, 2.915. 5. From 24.0042 subtract 13.7013. Answer, 10.3029. 6. From 170.0035 subtract 68.00181. Answer, 102.00169. 7. From .0142 subtract .005. 8. From .05 subtract .0024. 9. From 13.5 subtract 8.037. Answer, .0092. Answer, .0476. Answer, 5.463. 10. From 3 subtract .00003. Answer, 2.99997. 11. From 29.0029 subtract 19.003. Answer, 9.9999. 12. From 5 subtract .125. Answer, 4.875. 13. From 1 thousand subtract 1 ten-thousandth. Answer, 999.9999. 14. From 1 subtract 1 millionth. Answer, .999999. 15. From 25 thousandths subtract 25 millionths. Answer, .024975. 16. If, at the beginning of a w tch there were 36.875 cubic feet of oil in a tank and at the end of the watch there were 33.195 cubic feet in it, how many cubic feet of oil were used for the watch? Answer, 3.68 cubic feet. 17. A tank holds 1,920.2564 cubicfeet of water when full. There are 1,765.31 cubic feet in it now. How many cubic feet does it lack of being full? Answer, 154.9464 cubic feet. At 18. A ship's position at noon is 426.526 miles from Boston. noon the next day she is 782.321 miles from the same port. How many miles has she traveled? Answer, 355.795 miles. 19. The altitude of a V-thread is .866 inches. The altitude of a 1-inch pitch United States standard thread is .6495 inch. How much more altitude has the V-thread than the United States standard? Answer, .2165 inch. 20. The heating surface of a boiler is 4,681.1 square feet. The grate surface is 96.6 square feet. How much more heating surface than grate surface is there? Answer, 4,584.5 square feet. 21. A ship on her standardization run made 19.2 knots per hour. On her speed trials she only made 18.87 knots. How much more was made on her standardization runs than on her speed trials. Answer, .33 knot. MULTIPLICATION OF DECIMALS. XV. Multiplication of decimals is the process of finding the product of numbers involving decimals. Placing the decimal point in the product depends on the following principle: The number of decimal orders in the product is equal to the number of decimal orders in both the factors. Thus: Let the factors be .2 and .03; then the number of decimal orders in the product will be three. For .2 equals and .03 equals 18. The product of .2 by .03 will be the same as the product of fo by 180. fo X180=1000 or .096. Therefore .2X.03.006, in which there are three decimal orders. 6 1. Multiply 2.149 by 6.34. EXAMPLES. 2. 149 6. 34 8596 6447 12894 13. 62466 Solution. Multiply as in simple numbers, 2,149 by 634. There are three decimal orders in 2.149 and two decimal orders in 6.34; hence there must be five decimal orders in the product. Therefore, the product must be 13.62466. 2. Multiply .0276 by .035. .0276 .035 1380 828 .0009660 Solution. Multiply the numerator 276 by the denominator 35. The result is 9,660. There are four decimal orders in .0276 and three decimal orders in .035; hence there must be seven decimal orders in the product. Three ciphers therefore must be prefixed to the 9,660. Therefore, omitting the cipher on the right, the product is .000966. 3. Multiply 2.075 by 100. 207.5 Solution.-Write 2,075 and place the decimal point between the 7 and the 5, two places farther to the right than it is in 2.075. NOTE. To multiply 207.5 by 100, annex a cipher and move the decimal point two places to the right. RULE: Multiply together the numerators of the decimals as in simple numbers. Point off as many decimal places in the product as there are decimal places in both factors. 4. Multiply 33.21 by 4.41. Answer, 146.4561. |