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 of a yard equals of of a quarter= of a quarter=1 quarter of a quarter. Again, of a quarter equals of † of a nail=2 nails. Therefore, of a yard equals 1 quarter 2 nails. Hence, we deduce this RULE. Multiply the numerator by the units in the next inferior denominate value, and divide the product by the denominator; multiply 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. Ans. 1 quarter, 24 nails. Ans. 1 furlong, 20 rods. 2. What is the value of of 1 yard? 3. What is the value of of of one mile? 4. What is the value of of 5 of 1 cwt.? 5. What is the value of Ans. 1 quarter, 12 pounds. of 14 miles, 6 furlongs ? Ans. 2 miles, 3 furlongs, 26 rods, 11 feet. 6. What is the value of of each? of 2 days of 24 hours Ans. 9 hours, 36 minutes. 7. What is the value of of of of an hour? Ans. 5 minutes, 37 seconds. 8. What is the value of of of 9 hours and 18 minAns. 1 hour, 33 minutes. utes? 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. 37 Thus, 10, 100, 10, and 363, are decimal fractions. In practice, the denominators of decimal fractions are not written, but they are always understood. 37 Thus, instead of 3, 1, 3, and T, 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 thousandths, and so on, decreasing from the left towards the right, in a ten-fold ratio, the same as in whole numbers. The following table will exhibit this more clearly. *This table is in accordance with the French method of numbering, where each period of three figures changes its denominate value. Decimal Point. cths. Ten Hundred Thousandths. &c. &c. EXAMPLES. 1. Write 7 tenths; 365 thousandths; 75 millionths. Ans. 07; 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 thousandths. Ans. 043; 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 ten-fold ratio, they may, when connected together by means of the decimal point, be operated upon by precisely the same rules as for whole numbers, provided 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 ten-fold ratio. Thus, 0.3 is ten times 0'03, or a hundred times 0·003. 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; 88-9134, and 0.0037. Operation. 47.3 37.672 1.789101 88.9134 0.0037 Ans. 175 678201. 2. What is the sum of 0.67; 0·0371; 1100·0001; 47.5; 29-0037; 1.000005, and 33.033? Ans. 1211.243905. 3. What is the sum of 18; 40'06; 120-365; 47'003; 1100-0001; 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; 19-0005; 7.534; 10.94, and 103.729 ? Ans. 186 8085. 6. What is the sum of 0.90058; 7634; 3'007956, and 1.1? Ans. 12.642536. 7. What is the sum of 47-635; 3.13; 0.003001; 4.5787; 0.40005, and 4112.3789 ? Ans. 4168 125651. Ans. 105.904001. 8. What is the sum of 17154; 32 004501; 49-345; 6'4, and 1.0005 ? 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? Ans. 423 5111. |