OBS.-Let question 2d be written upon the blackboard, in the following manner, and illustrated. Teacher. Express by writing upon the blackboard, what part of a pound is 138. 131 20 Scholar. 138. is of a pound, which equals 40 3 20 T. What kind of fractions are those you have written? 7. Remove the denominator of the numerator, and illustrate. S. 40 40 3 60 20 To multiply the denominator is the same as to divide the numerator, for, to multiply the divisor is the same as to divide the dividend. 7. Remove the denominator of the fraction, and illustrate. 3. To divide the numerator divides the fraction, for, to divide the dividend divides the quotient. Let other questions be written and illustrated in a similar manner. OBS.-It will be recollected that it was said (Art. 60) that a mixed number is the quotient of a division, whose divisor was the denominator of the fraction. Consequently, example 10 is the quotient of a division whose divisor was 11. We have therefore only to multiply quotient and divisor together, or to reduce a mixed number to an improper fraction, and we have the fraction of a cwt., which we divide by 20, to reduce it to the fraction of a ton. Thus : QUESTIONS.-1. Rule for reducing fractions to integers of lower denominations? 2. Rule for reducing integers of lower denominations to a fraction of a higher ? If there should be no fraction in the question, the lowest denomination may be reduced to a fraction of the higher. In example 12, the 36 minutes may be reduced to the fraction of an hour; thus, 38=3. We then have 93 hours, a mixed number. of 11. What is the value of of a month? 13. What is the value of of a pound Troy? 15. What is the value of of an acre? 17. What is the value of of a yard of cloth? 19. What is the value of of a dollar in shillings? 21. What is the value of 5 of a ton? 23. What is the value of of a hogshead? 6. Reduce 6 furlongs, 26 po. 11 ft. to the fraction of a mile. 8. Reduce 8 mi. 5 fur. 20 po. to the fraction of a degree. 10. Reduce 16 cwt. 1 qr. 12 lbs. 11 oz. 10 dr. to the fraction of a ton. 12. Reduce 3 w. 1 da. 9 hr. 36 m. to the fraction of a month. 14. Reduce 8 oz. 11 pwt. 102 grs. to the fraction of a pound. 16. Reduce 3 roods, 13 rods, 90 feet, 108 in. to the fraction of an acre. 18. Reduce 3 qrs. 2 na. to the fraction of a yard. 20. Reduce 4s. 6d. to the fraction of a dollar. 22. Reduce 11 cwt. 0 qr. 12 lbs. 7 oz. 17 drs. to the fraction of a ton. 24. Reduce 49 gals. to the fraction of a hogshead. Reduction of Vulgar Fractions to Decimal. Art. 130.-1. Reduce to a decimal fraction. In this example, being a proper fraction, the numerator will not contain the denominator; but, by annexing a cipher, which reduces it to tenths, we can divide by the denominator. In 1 unit there are 10 tenths, but the example is one half of a unit; therefore, one half of 10 tenths, which is 5 tenths, will be the answer. Hence the RULE. I. Annex a cipher, or ciphers, to the numerator, and divide by the denominator. II. If there be a remainder, a cipher, or ciphers, may be annexed, and the process of division carried on until there be no remainder, or the quotient is sufficiently exact. The decimal places in the quotient must be equal to the number of ciphers annexed to the numerator. If, after division, the quotient does not contain so many, supply the deficiency by prefixing ciphers. 2. Reduce to a decimal fraction. 3. Reduce,, and 3, to decimal fractions. Ans. .75. Ans. .625, .125, .6. 4. Reduce,, , and, to decimal fractions. 3 42 Ans. .875, .375, .75, .25, .5. 5. Reduce of of 2 to a decimal fraction. Art. 131. To reduce a decimal fraction to a vulgar. RULE. Write down the given decimal, as a numerator, and for a denominator, write 1, with as many ciphers annexed as there are figures in the numerator, and then reduce the fraction to its lowest terms. (See Art. 61.) 1. Reduce .25 to a vulgar fraction. QUESTIONS.-1. Rule for reducing a vulgar fraction to a decimal? 2. How many decimal places must there be in the quotient? 3. If the quotient does not contain a sufficient number of figures, what is to be done? To reduce Integers of different denominations to a Decimal Fraction of a higher denomination, and the reverse. Art. 132.-1. Reduce 4 Art. 133.-2. Reduce .375 pence 2 farthings to the decimal of a shilling to integers of lower denominations. of a shilling. Operation. 4/2.0 12 4.500 2 farthings is of a penny; then, by the rule for reducing vulgar frac.375 tions to decimal, we have .5, or of a penny. This, placed at the right of 4 pence, 4.5, and divided by 12, the number of pence in a shilling, or because 4 pence is of a shilling, gives .375 of a shilling. Hence the As this question is the reverse of the former, and as the decimal, .375, was obtained by dividing the integers, it is plain, that the integers may be obtained by multiplying the decimal by the same numbers. Operation. .375 12 4.500 4 2.000 RULE. Place the numbers one above another, the highest denomination at the bottom. Divide the lowest denomination by that number which expresses how many of that it takes to make 1 of the next higher denomination, writing the quotient at the right of the next higher denomination; and so proceed until the whole shall be reduced to the required decimal. OBS.-Integers of different denominations may be reduced to a decimal of a higher, by reducing the given numbers to the lowest denomination mentioned for a numerator, and the integer, to which the given numbers are to be reduced, to the same denomination for a denominator, and dividing the numerator by the denominator, Hence the RULE. Multiply the given decimal by that number which expresses how many of the next lower denomination it takes to make one of that in which the decimal is given; observing to point off as many places in the product, for decimals, as there are figures in the given decimal; and so proceed through all the denominations; and the several numbers at the left of the decimal points will be the answer required. OBS.-Pointing off the product is the same as dividing by the denominator of the decimal. Art. 134.-To reduce shillings, pence, and farthings to the decimal of a pound, by inspection. 1. Reduce 7s. 8d. 2qrs. to the decimal of a pound? One shilling is of a pound: therefore, two shillings is , or. Having, therefore, any number of shillings given, if we take one half the even number, they will be reduced at once to the decimal of a pound. If there is an odd shilling, it is the same as 10 of a pound: 20.05. The farthing, which 960 of a pound, is made to occupy the 1000ths place. But 9 is greater than 1000 by 24000; there will, therefore, be a loss of 4000 on every farthing; but if we add one to the number, when they exceed 12 and do not exceed 36, and two when they exceed 36, the expression will be nearly so many 1000ths of a pound. is 60 2. Reduce 4s. 6d. to the decimal of a pound. Operation. .2 half of the even shillings. .001 for excess of 12. .225 Ans. |