TO CHANGE A DECIMAL FRACTION OF ANY HIGHER DENOMINATION TO ITS EQUIVALENT VALUE EXPRESSED IN WHOLE NUMBERS OF LOWER DENOMINATIONS. Art. 113. Suppose we wish to change .790625£. to its equivalent value in shillings, pence, and farthings. OPERATION. 20 s. 15.812500 12 d. 9.750000 4 Multiplying the given decimal of a pound by 20, the product must be the numerator of a decimal fraction 20 times as great, or the fraction of a shilling equal in value to the given fraction of a pound; this product being an improper fraction, we divide it by the denominator 1000000, by pointing off six figures, counting from the right; the figures at the left of the point express the quotient, or number of shillings, and the figures at the right of the point express the remainder, or decimal of a shilling. Multiplying this decimal of a shilling by 12, the product is the decimal of a penny of equal value; pointing off six figures as before, the figure at the left of the point expresses the quotient, or number of pence, and the figures at the right express the remainder, or decimal of a penny. Multiplying this decimal of a penny by 4, the product is the decimal of a farthing of equal value; pointing off six figures, the figure at the left of the point expresses the quotient, or number of farthings. Therefore, .790625 £.= 15s. 9d. 3qrs. qrs. 3.000000 From the above example and its illustration we obtain the following RULE. Multiply the given decimal by that number of the next lower denomination which is equal to a unit of the higher, and, from the right of the product, point off so many figures for decimals as there are figures in the given decimal. Multiply the decimal of each lower denomination in the same manner, and the several numbers at the left of the decimal points will express the equivalent value of the given decimal in the proper denominations. 1. Change .125 of a shilling to its value in pence and farthings. 3. Change .75 of a dollar to its value in shillings and pence. 2. Change .125 of a pound to its value in shillings and pence. 4. Change .625 of a yard to its value in quarters and nails. 5. Change .628125 of a pound to its equivalent value in shillings, pence, and farthings. Ans. 12s. 6d. 3qrs. 6. Change .8890625 of a troy pound to its equivalent value in ounces, pennyweights, and grains. Ans. 10 oz. 13 pwts. 9 grs. 7. Change .796875 of a bushel to its equivalent value in pecks, quarts, and pints. Ans. 3 pks. 1 qt. 1 pt. 8. Change .6625 of a mile to its equivalent value in furlongs and rods. Ans. 5 fur. 12 rods. 9. Change .6 of an acre to its equivalent value in roods and rods. Ans. 2 roods 16 rods. 10. Change .5625 of a day to its equivalent value in hours and minutes. Ans. 13 hours 30 minutes. TO CHANGE ANY NUMBER OF SHILLINGS, PENCE, AND FARTHINGS, TO THE DECIMAL OF A POUND, THE DECIMAL TERMINATING AT THOUSANDTHS. Art. 114. Suppose we wish to change 17s. 10d. 2qrs. to the decimal of a pound. OPERATION. = .8 = half of 16s. As shillings are twentieths of a pound, half their number must be tenths of a pound; consequently, we write a figure expressing half the even num.894 in 101d. increased by 2. ber in the place of tenths. One shilling is £..05£.; therefore, when there is an odd shilling, we write 5 in the place of hundredths. Again, farthings are 960ths of a pound, and if there were 1000 instead of 960 farthings in a pound, it is plain that any number of farthings would be so many thousandths of a pound. But 960 plus 2 of 960 or 40= 1000; consequently, any number of farthings, plus of the number, will be so many thousandths of a pound. Whenever the number of farthings is more than 12, of the number is more than of a farthing; therefore 1 must be added to the number; and when the number of farthings is more than 36, 4 of the number is more than 1 farthings; therefore, 2 must be added to the number; and the figures expressing the number thus increased we write in the second and third places. Hence we have the following RULE. Write half the greatest even number of shillings in the place of tenths, and 5 in the place of hundredths, when the number of shillings is odd. Write the number of farthings in the given pence and farthings in the second and third places; increasing their number by 1 when it exceeds 12, and by 2 when it exceeds 36. When there are no shillings, or only one shilling, in the given sum, a cipher must be written in the place of tenths. When the number of farthings in the given pence and farthings does not exceed 9, a cipher must be written in the second place, or place of hundredths. 1. Change 4s. 6d. 1qr. to the decimal of a pound. 3. Change 1s. 1d. 3qrs. to the decimal of a pound. 2. Change 7s. 7d. 2qrs. to the decimal of a pound. 4. Change 9s. 9d. 2qrs. to the decimal of a pound. 5. Change 5s. 5d. 1qr.; 6s. 6d. 2qrs.; 7s. 7d. 3qrs.; Ss. 8d. 1qr.; 9s. 9d. 2qrs., and 10s. 10d. 3qrs., each to the decimal of a pound. Ans. .272; .327; .382; .434; .490, and .545. 6. Change 1d. 1qr.; .1d. 3qrs.; 2d. 1qr.; .2d. 3qrs.; 5d. 2qrs.; 1s. Id.; 1s. 6d., and 1s. 11d., each to the decimal of a pound. Ans. .005; .007; .009; .011; .023; .054; .075; .096. ΤΟ CHANGE ANY DECIMAL OF A POUND, TERMINATING AT THOUSANDTHS, TO ITS VALUE IN SHILLINGS, PENCE AND FAR THINGS. Art. 115. Suppose we wish to change .894 of a pound to its value in shillings, pence and farthings. OPERATION. 1s. £.8 X 2=16s. £.05= £.044.002=42 qrs.: = 10дd. It is evident that the operation in this Art. must be the reverse of the operation in Art. 114. As 1 tenth of a pound is equal to 2s., there must be twice as many shillings as there are tenths of a pound, and .8 X 2 = 16s. hundredths of a pound being equal to 1s. -we deduct 5 hundredths from the 9 hundredths, and add its equal, 1s., to 16s., and there remain 44 thousandths of a pound; and 44 thousandths, less 2 thousandths, equal 42 farthings, or 10 pence. Hence we obtain the following Ans. 17s. 10.d. : 5 RULE. Write twice the number of tenths of a pound for shillings; and if the number of hundredths is 5, or more than 5, add another shilling; then deduct the 5 hundredths, and call the remaining number of thousandths so many farthings; deducting 1 when the number exceeds 12, and 2 when it exceeds 36. 1. Change .226 of a pound to its value in shillings, pence and farthings. 3. Change .057 of a pound to its equivalent value. 5. Change .542 of a pound to its equivalent value. 2. Change .381 of a pound to its value in shillings, pence and farthings. 4. Change .490 of a pound to its equivalent value. 6. Change .999 of a pound to its equivalent value. 7. Change .272; .327; .382; .434; .490, and .545, of a pound to their equivalent value in shillings, pence and farthings. Ans. 5s. 5d. 1qr.; 6s. 6d. 2qrs.; 7s. 7d. 3qrs.; 8s. 8d. 1qr.; 9s. 9d. 2qrs., and 10s. 10d. 3qrs. REPEATING OR CIRCULATING DECIMALS. Art. 116. We have seen, (Art. 110,) that a vulgar fraction whose denominator is any prime number other than 2 or 5, or when it contains any prime factor except 2 and 5, cannot be exactly expressed in decimals. Thus, if we reduce to a decimal we shall obtain the repeating figure .1, which we distinguish by placing a point. over it. Since the repeating decimal figure .1 is equal to the vulgar fraction §, .2 must equal, .3 must equal §, &c.; therefore, every single repeating decimal figure must be equal to a vulgar fraction whose numerator is the repeating figure, and whose denominator is 9. If we reduce to a decimal, we shall obtain the repeating figures .01, which we distinguish by placing a point over each figure. Since the repeating decimal figures .01 are equal to the vulgar fraction, .02 must be equal to, .03 must be equal to, &c. Again, if we reduce repeating figures .001, which we point over the first and last figure. decimal figures .001 are equal to to a decimal, we shall obtain the distinguish by placing a Since the repeating .002 must be equal to , .003 must be equal to, &c. This correspondence exists universally; therefore, any repeating or circulating decimal is equal to a vulgar fraction whose numerator is the repeating figure or figures, and whose denominator is a number of 9s equal to the number of repeating figures. TO CHANGE ANY REPEATING OR CIRCULATING DECIMAL TO ITS EQUIVALENT VULGAR FRACTION. RULE. Make the repeating figure or figures the numerator, and a number of 9s equal to the number of repeating figures the denominator; this fraction, reduced to its lowest terms, will be the equivalent vulgar fraction required. 1. Change .4 to its equivalent vulgar fraction. 3. Change to its equivalent vulgar fraction. 2. Change .6 to its equivalent vulgar fraction. 4. Change .8 to its equivalent vulgar fraction. 5. Change .36 to an equivalent vulgar fraction. Ans. 6. What is the equivalent vulgar fraction of .81? 12 Ans.. TO CHANGE A REPEATING OR CIRCULATING DECIMAL TO AN EQUIVALENT VULGAR FRACTION WHICH DOES NOT BEGIN TO REPEAT AT THE FIRST PLACE. Art. 117. If we change to a decimal, we shall obtain .53. This decimal begins to repeat at the second place, or hundredths. The first figure, 5, is , and the repeating figure, 3, is of or 36. 18=18+36= 16, or, the original vulgar fraction. Again, if we change 261 to a decimal, we shall obtain .4745. This decimal begins to repeat at the third place or thousandths. The first two figures are, and the two repeating figures of do, or 80-110 =11%, and +100=110%, or 38, the original vulgar fraction. From the above illustrations we obtain the following RULE. Change the finite part, or figures which do not repeat, to a vulgar fraction by writing its decimal denominator; then change the repeating figure or figures to a vulgar fraction, (Art. 116,) which will be a fraction of the finite part; reduce this compound fraction to a simple one; then reduce the two fractions to their least common denominator and find their sum; this fraction, reduced to its lowest terms, will be the equivalent vulgar fraction required. 1. Change .46 to its equivalent vulgar fraction. Ans. 75. 2. Change .38 to an equivalent vulgar fraction, and prove the result to be correct. 3. Change .374 to an equivalent vulgar fraction. Ans. 88. 4. Change .4635 to an equivalent vulgar fraction, and prove the answer to be right. 5. Change .47647 to an equivalent vulgar fraction, and then change the result to a circulating decimal. 6. Change .5925 to an equivalent vulgar fraction, and then change the vulgar fraction to a decimal. 7. If 1 pound sterling of English currency be equal to $4.4, what is the value of 500 pounds sterling? 8. If 1 yard of linen is worth $.6, what is the value of 12 yards? 9. If 1 yard of silk is worth $.83, what is the value of 20 yards? |