DIVISION OF DECIMAL FRACTIONS. Art. 109. When a whole number or a fraction is divided by a whole number, the quotient will be less than the dividend in the same proportion that the divisor is greater than a unit; thus, 25÷5=5; here the divisor is 5 times as great as a unit, and the quotient is only 1 fifth as great as the dividend; and .8÷4.2; here the divisor is 4 times as great as a unit, and the quotient is only 1 fourth as great as the dividend. When a whole number is divided by a fraction, the quotient will be greater than the dividend in the same proportion that the divisor is less than a unit; thus, 5.510; here the divisor is 5 tenths or 1 half of a unit, and the quotient is 2 times as great as the dividend. When a greater fraction is divided by a less one, the quotient will be greater than the dividend in the same proportion that the divisor is less than a unit; thus, .4.2=2; here the divisor is 2 tenths or 1 fifth of a unit, and the quotient is 5 times as great as the dividend. When a less fraction is divided by a greater one, the quotient will be less than a unit in the same proportion that the divisor is greater than the dividend; thus, .2.4=.5; here the divisor is 2 times as great as the dividend, and the quotient is only 5 tenths or 1 half of a unit. From the above remarks and illustrations it is plain, that when the divisor is a whole number, there must be as many decimal figures in the quotient as there are in the dividend. It is obvious, that when the divisor only contains decimal figures, or when there are more decimal figures in the divisor than there are in the dividend, if we make the number of decimal figures in the dividend equal to the number of decimal figures in the divisor, by annexing decimal ciphers to the dividend, the quotient will be a whole number, except when the divisor is greater than the dividend. It is also obvious, that when there are more decimal figures in the dividend than there are in the divisor, there will be as many decimal figures in the quotient as the number of decimal figures in the dividend exceeds the number of decimal figures in the divisor. Hence the following general rule for the division of decimal fractions. RULE. Divide as in whole numbers, then point off so many of the quotient figures for decimals, counting from the right, as the number of decimal figures in the dividend exceeds the number of decimal figures in the divisor. If there are not as many figures in the quotient, prefix so many ciphers as are necessary to make up the required number. When there are more decimal figures in the divisor than there are in the dividend, or when the divisor exceeds the dividend, annex so many decimal ciphers to the dividend as are necessary to obtain any required number of decimal figures in the quotient. When there is a remainder, ciphers may be annexed to it, and the division continued until the required number of decimal figures is obtained. 16. What will be the quotient of 150 divided by .75? Ans. 200. 17. What will be the quotient of .75 divided by 200? Ans. .00375. 18. What will be the quotient of 500 divided by .05? Ans. 10000. 19. What will be the quotient of .05 divided by 500? Ans. .0001. 20. When wood is worth $6.75 a cord, how many cords can you buy with 81 dollars? 21. If 1.875 yard of cloth is sufficient to make a coat, how many coats can be made of 37.5 yards? When the divisor is 10, 100, 1000, &c., division is performed by removing the decimal point in the dividend so many figures towards the left as there are ciphers in the divisor. Thus, 4.75 + 10.475. 100.475 REDUCTION OF DECIMAL FRACTIONS. TO CHANGE A VULGAR FRACTION TO AN EQUIVALENT DECIMAL FRACTION. Art. 110. Suppose we wish to change the vulgar fraction to an equivalent decimal fraction. OPERATION. 8) 7.0 (.975 Ans. 64 60 56 We write the numerator 7, and place a point at the right, for a dividend, and the denominator 8 for a divisor. We annex a cipher to the dividend, which reduces it to tenths; we then divide the 70 tenths by 8, the quotient is 8 tenths, and 6 tenths remain. We annex a cipher to the remainder, which reduces it to hundredths; we then divide the 60 hundredths by 8, the quotient is 7 hundredths, and 4 hundredths remain. We annex a cipher to the remainder, which reduces it to thousandths; we then divide the 40 thousandths by 8, the quotient is 5 thousandths. 40 40 From the above illustration, we obtain the following general rule for changing a vulgar fraction to an equivalent decimal fraction. RULE. Annex a cipher to the numerator which reduces it to tenths, then divide the number of tenths by the denominator; if there is a remainder, annex a cipher which reduces it to hundredths, then divide the number of hundredths; and thus continue until there is no remainder, or until a sufficient number of quotient figures is obtained. There are some vulgar fractions which cannot be expressed exactly in decimals. Since we multiply each successive remainder by 10, and as 10 is composed of the prime factors 2 and 5, it follows, that whenever the denominator of a vulgar fraction is a prime number other than 2 or 5, or when it contains any prime factor except 2 and 5, the vulgar fraction cannot be accurately expressed in decimals. 9. Change to an equivalent decimal. Ans. .9375. 10. Change T20 to an equivalent decimal. Ans. .0008. 11. What decimal is equivalent to? Ans. .34375. 12. What decimal is equivalent to? Ans. .1484375. 13. Change to its equivalent decimal. Ans. .16015625. 14. Change to its approximate decimal. Ans. .66666+ 15. Change to its approximate decimal. Ans. .54545+. TO CHANGE A DECIMAL FRACTION TO AN EQUIVALENT VULGAR FRACTION. Art. 111. Since the denominator of a decimal fraction is a unit with so many ciphers annexed as there are figures in the given decimal, we have the following RULE. Change the decimal fraction to a vulgar fraction by writing its denominator under it, then reduce this vulgar fraction to its lowest terms. 1. What vulgar fraction is equal to .5? 3. What vulgar fraction is equal to .75? 5. What vulgar fraction is equal to .375? 2. What vulgar fraction is equal to .25? 4. What vulgar fraction is equal to .125? 6. What vulgar fraction is equal to .625? 7. Change .9375 to an equivalent vulgar fraction. Ans. 18. NUMBER TO AN EQUIVALENT DECIMAL FRACTION OF ANY Art. 112. Suppose we wish to change 7s. 6d. to the decimal of a pound. Sixpence iss., and s., changed to a decimal, is equal to .5s. If we divide .5s. by 20, we shall change it to the decimal of a pound, because any fraction of a shilling is only 1 twentieth as great a fraction of a pound; thus, .5s. 20 = .025 €. Seven shillings is £., and £., changed to a decimal, is equal to .35£.; and .35£.+.025£. .375., the decimal required. Instead of changing .5s. and £. separately to decimals, we may annex the .5s. to the 7s. ; thus, 7.5s. ; and 7.5s.20 = .375 £., as above. Hence the following 20 RULE. Write the given numbers under each other for dividends, proceeding in order from the lowest denomination to the highest. Then begin with the lowest denomination, and divide the number of each denomination, in succession, (annexing as many decimal ciphers as may be necessary,) by that number which is equal to a unit of the next higher, and write each quotient at the right of the next lower dividend; the last quotient will be the decimal required. Change 15s. 9d. 3qrs. to an equivalent decimal of a pound. OPERATION. 4) 3.00 12) 9.7500 Three farthings are d.; dividing 3.00qrs. by 4, changes d. to the decimal .75d., which we place at the right of the 9d.; dividing 9.75d. by 12, changes 9.75d. to the decimal of a shilling, which we place at the right of the 15s.; dividing 15.8125s. by 20, changes 15.8125s. to Ans. .790625£. the decimal of a pound. 20) 15.812500 A compound number may be changed to the decimal of a higher denomination, by reducing the given compound number to a vulgar fraction, (Art. 91,) and then reducing the vulgar fraction to a decimal. (Art. 110.) In the above question, 15s. 9d. 3qrs. 759qrs. 1£.=960qrs.; then 15s. 9d. 3qrs. is equal to 75%£., and 788. =.790625 £. 1. Change 1d. 2qrs. to the decimal of a shilling. 3. Change 4s. 6d. to the decimal of a dollar. ,י 960 2. Change 2s. 6d. to the decimal of a pound. 4. Change 2qrs. 2n. to the decimal of a yard. 5. Change 12s. 6d. 3qrs. to the decimal of a pound. Ans. .628125£. 6. Change 6d. 1qr. to the decimal of a shilling. Ans. .52083+s. 7. Change 10 oz. 13 pwts. 9 grs. to the decimal of a pound. Ans. .8890625 lb. 8. Change 7 cwt. 3 qrs. 17 lbs. 10 oz. 12 drs. to the decimal of a ton. Ans. .39538+ ton. the decimal of a bushel. Ans. .796875 bushel. decimal of a mile. 9. Change 3 pks. 1 qt. 1 pt. to 10. Change 5 fur. 12 rods to the Ans. .6625 mile. 11. Change 2 roods 16 rods to the decimal of an acre. Ans. .6 acre. 12. Change 13 hours 30 minutes to the decimal of a day. Ans. .5625 day. 13. Change 3 quarters 2 nails to the decimal of a yard. Ans. .875 yard. 14. Change 15 gallons 3 quarts to the decimal of a barrel. Ans. .5 barrel. |