33. To reduce a fraction of any given denomination to whole denominate numbers. Suppose we wish to know the value of of a yard; we know that is of a yard equals of 4 of a quarter= of a quarter=1 quarter+} of a quarter. Again, 1 of a quarter equals 3 of 4 of a nail=2 nails. Therefore of a yard equals 1 quarter 2 nails. Ilence, we deduce this RULE. Multiply the numerator by the units in the next inferior denominate value, and divide the product by the denominator ; mul. tiply the remainder, if any, by the next lower denominate value, and again divide the product by the denominator; continue this process until there is no remainder, or until we reach the lowest denominate value. The successive quotients will form the successive denominate values. Examples 1. What is the value of of an hour ? In this example, is of an hour= is of of a minute=12 minutes. 2. What is the value of of 1 yard ? . Ans. 1 quarter 29. nails. 3. What is the value of } of of one mile ? Ans. 1 furlong 20 rods. 4. What is the value of of of 1 cwt. Ans. 1 quarter 12 pounds. 5. What is the value of £ of 14 miles 6 furlongs? .. Ang. 2 miles 3 furlongs 26 rods 11 feet. 6. What is the value of $ of of 2 days of 24 hours each? Ans. 9 hours 36 minutes.. 7. What is the value of 1 of 1 of is of an hour ? Ank. 5 minutes 374 seconds. CHAPTER III. DECIMAL FRACTIONS. 34. A decimal fraction is that particular form of a vulgar fraction whose denominator consists of a unit followed by one or more ciphers. Thus, to, or Toooand 3.0.326, are decimal fractions. In practice, the denominators of decimal fractions are not written; but they are always understood. Thus, instead of io; Tos idly, and today, we write 0.3; 0.07; 0.037; and 0.0001, The first figure on the right of the period, or decimal point, is said to be in the place of tenths, the second figure is said to be in the place of hundredths, the third in the place of thou. sandths, and so on, decreasing from the left towards the right, in a tenfold ratio, the same as in whole numbers. The fol. lowing table will exhibit this more clearly. NUMERATION TABLE OF WHOLE NUMBERS AND DECIMALS.* &c. &c. &c. &c. 33. 3 3 3 3 3 3 3 3 3 3 Ascending. me O Descending. * This table is in accordance with the French method of numbering, where each period of three figures changes its denominate value. Examples. 1. Write 7 tenths; 365 thousandths ; 75 millionths. . Ans. 0.7; 0.365; 0.000075. 2. Write 37 hundredths; 5 tenths ; 3781 ten millionths. Ans. 0.37; 0.5; 0.0003781. 3. Write 43 hundredths ; 3456 ten thousanths. Ans. 0.43; 0.3456. 4. Write 13 billionths; 3 ten billionths. Ans. 0.000000013; 0.0000000003. 35. Since decimals, like whole numbers, decrease from the left towards the right in a tenfold ratio, they may be connect. ed together by means of the decimal point, and then operated upon by precisely the same rules as for whole numbers, provi. ded we are careful to keep the decimal point always in the right place. Annexing a cipher to a decimal does not change its value. Thus, 0.3=0.30=0.300=&c. But prefixing a cipher, is the same as removing the decimal figures one place further to the right, and therefore, each cipher thus prefixed reduces the value in a tenfold ratio. Thus, 0.3 is ten times 0.03, or a hundred times 0.003. 2. ADDITION OF DECIMALS. 36. From what has been said under Art. 35, we deduce the following RULE. Place the numbers so that the decimal points shall be directly over each other, and then add as in whole numbers. Examples. . 1. Find the sum of 47.3; 37.672; 1.789101 ; 89.9134 ; and 0.0037. OPERATION. 1.789101 0.0037 Ans. 175.678201 2. What is the sum of 0.67; 0.0371; 47.5; 1100.0001; 29.0037; 1.000005; and 33.033 ? Ans. 1211.243905. 3. What is the sum of 1.8; 40.06; 120.365; 1100.0001; 47.003; 31.11101; and 3.0001 ? Ans. 1343.33921, • 4. What is the sum of 13.29; 14.2835; 111.117; 4.006; 67.88864; and 496.446 ? Ans. 707.03114: 5. What is the sum of 37.345 ; 8.26 ; 7.534 ; 19.0005. 10.94 ; and 103.729 ? Ans. 186.8085 6. What is the sum of 0.90058; 7.634; 3.007956 ; 1.1? Ans. 12.642536. 7. What is the sum of 47.635; 3.13; 4.5787; 0.003001; 0.40005; and 4112.3789 ? , Ans. 4168.125651. 8. What is the sum of 17.154; 32.004501 ; 6.4; 49.345; and 1.0005 ? - Ans. 105.904001. 9. What is the sum of 4.996; 38.37; 421.633 ; 5.65; and 4.29 ? Ans. 474.939. 10. What is the sum of 57.41; 365.0001; and 1.101 ? 11. What is the sum of 2.4999; 47.121212; 0.1; and 411.001 ? 12. What is the sum of 433.9; 777,5; 67.06; and 35.88 ? SUBTRACTION OF DECIMALS. 37. From what has been said under Art. 35, we infer the following RULE. .. Place the smaller number under the larger, so that the decimal point of the one is directly under that of the other. Then pro. ceed as in subtraction of whole numbers.. Examples. 1. From 213.5734 subtract 87.657237. 213.5734 87.657237 Ans. 125.916163 2. From 385.76943 subtract 72.57. Ans. 313.19943. 3. From 0.975 subtract 0.483764. . Ans. 0.491236. 4. From 0.5 subtract 0.0003. • Ans. 0.4997. 5. From 96.5 subtract 0.000783. Ans. 96.499217, 6. From 23.005 subtract 13.000378. Ans. 10.004622. 7. From 110.001001 subtract 11.010002. Ans. 98.990999. MULTIPLICATION OF DECIMALS. 38. Let us multiply 0.47 by 0.6. If we put these decimals in the form of vulgar fractions they become to and %; these multiplied, by rule under Art. 25, give 10x%= 180o. Now it is obvious that there will be, in all cases, as many ciphers in the denominator of the product as there are in both of the factors added together. Hence, the following i |