by of ? 42 758 multiplied by ? of of divided 19. A man owns of a ship; he sells of his share: what part is that of the whole, expressed in decimals? 20. Bought of 87 bushels of wheat for of 7 dol lars a bushel: how much did it come to, expressed in decimals? 21. If a man receives of a dollar at one time, $7 at another, and $8 at a third: how much in all, expressed in decimals? 22. What mixed decimal is equal to the sum of of 18, of 11, and 7? 23 What decimal is equal to of 3 taken from of 83 24. What decimal is equal to the sum of 1, 4, and ? 8 CASE II. 213. To change a decimal to the form of a common fraction. ANALYSIS.-A decimal fraction may be changed to the form of a common fraction by simply writing its denominator (Art. 190). Examples. Express the following decimals in common fractions. 1. Reduce 25 aud .75. 6. Reduce .01875. 7. Reduce .22575. 8. Reduce .265625. 9. Reduce .333}. 10. Reduce 57147. CASE III. 214. To reduce a compound number to a decimal of a given denomination. 1. Reduce £1 4s. 93d. to the decimal of a £. ANALYSIS.-We first reduce 3 farthings to the decimal of a penny, by dividing by 4. We then annex the quotient .75d. to the 9 pence. We next divide by 12, giving $125, which is the decimal of a shilling. This we annex to the shillings, and then divide by twenty OPERATION. 3d. = .75d.; hence, .8125s., and £240625; therefore, £1 4s. 93d. = £1.240625 Rule.-I. If the lowest denomination contains a fraction, reduce it to a decimal and prefix the integral part: II. Then divide by the units of the ascending scale, annet the quotient to the next higher denomination, and proceed in the same manner through all the denominations, to the quired unit. NOTE.-When any denomination, between the lowest and the highest is wanting, the number to be prefixed to the corresponding quotient, is 0. 212. How do you change a common to a decimal fraction? 213. How do you change a decimal to the form of a common fraction! 214. How do you reduce a compound number to a decimal of a given denomination? 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 ua. 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. 19. Reduce 2 qr. 3 na. to the decimal of an English ell. CASE IV. 215. To find the value of a decimal in integers of lower de nominations. 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? 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? 315. 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. 19. Required the value of .0055 of a ton. 20. Required the value of .625 of a shilling. 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 deci mal fraction? What are they? What distinguishes one of these cases from the other? |