Examples. 1. Reduce 14 drams to the decimal of a lb. Avoirdupois. 2. Reduce 78d. to the decimal of £. 3. Reduce 63 pints to the decimal of a peck. 4. Reduce 9 hours to the decimal of a day. 5. Reduce 375678 feet to the decimal of a mile. 6. Reduce 7 oz. 19 pwt. of silver to the decimal of a pound 7. Reduce 3 cwt. 7 lb. 8 oz. to the decimal of a ton. 8. Reduce 2.45 shillings to the decimal of a £. 9. Reduce 1.047 roods to the decimal of an acre. 10. Reduce 176.9 yards to the decimal of a mile. 11. Reduce 2 qr. 14 lb. to the decimal of a cwt. 12. Reduce 10 oz. 18 pwt. 16 gr. to the decimal of a lb. 13. Reduce 3 qr. 2 na. to the decimal of a yard. 14. Reduce 1 gal. to the decimal of a hogshead. 15. Reduce 17 h. 6 m. 43 sec. to the decimal of a day. 16. Reduce 4 cwt. 23 qr. to the decimal of a ton. 17. Reduce 19s. 18. Reduce 1 R. 19. Reduce 2 qr. 20. Reduce 2 yd. 5d. 2far. to the decimal of a pound. 3 na. to the decimal of an English ell. 2 ft. 6 in. to the decimal of a mile. 21. Reduce 15' 221" to the decimal of a degree. 22. Reduce 1 cwt. 1 qr. 1 lb. to the decimal of a ton. 23. Reduce 3 bush. 3 pk. to the decimal of a chaldron. 24. Reduce 17 yd. 1 ft. 6 in. to the decimal of a mile. 25. What decimal part of a year is 9 months? 26. What decimal part of an acre is 1 R. 14 P.? 27. What decimal part of a chaldron is 45 pk. ? 28. What decimal part of a mile is 72 yards? 29. What part of a ream of paper is 9 sheets? 30. What part of a rod in length is 4.0125 inches? 31. Reduce 10 wk. 2 da. to the decimal of a leap year. 32. Reduce 43 13 10 10 gr. to the decimal of a lb. 33. Reduce 3 qt. 1.75 pt. to the decimal of a hhd. 34. Reduce 24 sq. yd. 1.8 sq. ft. to the decimal of an acre. CASE IV. 215. To find the value of a decimal in integers of lower denominations. 1. What is the value of .832296 of a £? ANALYSIS.-First multiply the decimal by 20, which brings it to the denomination of shillings, and after cutting off from the right as many places for decimals as there are in the given number, we have 16s. and the decimal .645920 over. This is reduced to pence by multiplying by 12, and then to farthings by multiplying by 4. OPERATION. .832296 20 16.645920 12 7.751040 4 3.004160 Ans. 16s. 7d. 3far. Rule I. Multiply the decimal by the units of the descending scale, and point off as in the multiplication of decimals: II. Multiply the decimal part of the product as before, and continue the operations to the lowest denomination. The integers cut off at the left, form the answer. Examples. 1. What is the value of .6725 of a hundredweight? 2. What is the value of .61 of a pipe of wine? 3. What is the value of .83229 of a £? 4. Required the value of .0625 of a barrel of beer. 5. Required the value of .42857 of a month. 6. Required the value of .05 of an acre. 7. Required the value of .3375 of a ton. 8. Required the value of .875 of a pipe of wine. 9. What is the value of .375 of a hogshead of beer? 215. How do you find the value of a decimal in integers of lower denominations? 10. What is the value of .911111 of a pound troy? 12. What is the value of .001136 of a mile in length? 15. Required the value of .875 of a yard. 16. Required the value of .3489 of a pound, apothecaries. 17. Required the value of .759 of an acre. 18. Required the value of .01875 of a ream of paper. 21. Required the value of 3375 of an acre. 22. Required the value of .785 of a year of 365 days. REPEATING DECIMALS. 216. In changing a common to a decimal fraction, there are two general cases: 1st. When the division terminates; and 2d. When it does not terminate. In the first case, the quotient will contain a limited number of decimal places, and the exact value of the common fraction will be expressed decimally. In the second case, the quotient will contain an infinite number of decimal places, and the exact value of the common fraction cannot be expressed decimally. CASE I. 217. When the division terminates. When a common fraction is reduced to its lowest terms (which we suppose to be done in all cases that follow), there will be no factor common to its numerator and denominator. 216. How many cases are there in changing a common to a decimal fraction? What are they? What distinguishes one of these cases from the other? 1. Reduce to its equivalent decimal. ANALYSIS.-Annexing one 0 to the numerator multiplies it by 10, or by 2 and 5; hence, 2 and 5 become prime factors of the numerator every time that a 0 is annexed. But if the division is exact, these prime factors, and none others, must also be found in the denominator. 2. Reduce to its equivalent decimal. ANALYSIS.-3618 x 2 1 9 × 2 × 2 = 8 × 3 × 2 × 2; in which we see that the denominator contains other factors than 2 and 5; hence, the fraction cannot be exactly expressed decimally. Rule.-I. Reduce the fraction to its lowest terms, then decompose the denominator into it prime factors; and if there are no factors other than 2 and 5, the exact division can be made: OPERATION. 50)17.00(.34 15 0 200 OPERATION. 36)5.0(.1388+ 36 140 108 320 288 320 288 II. If there are other prime factors, the exact division cannot be made. NOTE.-Every 0 annexed to the numerator, introduces the two factors 2 and 5; and these factors must be introduced until we have as many of each as there are in the denominator after it shall have been decomposed into its prime factors 2 and 5. But the quotient will contain as many decimal places as there are decimal O's in the dividend. Hence, The number of decimal places in the quotient will be equal to the greatest number of factors, 2 or 5, in the divisor. 3. Can be exactly expressed decimally? How many places? ANALYSIS.-25=5 × 5; hence, the fraction can be exactly expressed decimally, and by two decimals, because 5 is taken twice as a factor in the divisor. OPERATION. 25)7.0(.28 50 200 200 Examples. Find the decimals and number of places in the following: 218. When the division does not terminate. 1. Let it be required to reduce to its equivalent decimal. ANALYSIS.-By annexing decimal ciphers to the OPERATION. numerator 1, and making the division, we find the equivalent decimal to be .3333+, &c., giving 3's as far we choose to continue the division. 3)1.0000 .3333+ The further the division is continued, the nearer the value of the decimal will approach to, the exact value of the common fraction. We express this approach to equality of value, by saying, that if the division be continued without limit, that is, to infinity, the value of the decimal will then become equal to that of the common fraction; thus, .3333..., continued to infinity = }; for, each succeeding 3 brings the value nearer to . Also, .9999..., continued to infinity = 1; for, each succeeding 9 brings the value nearer to 1. 2. Find the decimal corresponding to the common fraction 3. ANALYSIS.-Annexing decimal ciphers and dividing, we find the decimal to be .2222+, in which we see that the figure 2 is continually repeated. OPERATION. 9)2.0000 .2222 |