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RULE. -Convert the fractions, when of different denominators, into others having a common denominator ; then subtract the numerator of the one from that of the other, and under the difference write the common denominator : the resulting fraction may then be reduced to its lowest terms.
NOTE.—When mixed numbers are given, before subtracting, convert them into improper fractions : also, convert compound into simple fractions, and if of different denominations, convert them into the same denomination. Example.-From take = ?
These are first converted to fractions having a common denomina
tor; the numerator of the one is 30 – 28 =
then subtracted from that of the 2
other, and the common denominaCommon denominator 35
tor written below the difference. Exercises. What is the difference between1. and
Ans. ਤ 4. and f Ans, ਲੰਨ 2.
| 5. of r " of 7 3. u
12 | 6. 49 238 " 253
MULTIPLICATION OF VULGAR FRACTIONS. RULE.-Multiply all the numerators together for a new numerator, and all the denominators together for a new denominator : the resulting fraction may then be reduced to its lowest terms.
NOTE.- Before multiplying, convert any mixed number into improper fractions, and compound into simple fractions; also, in order to shorten the multiplication, the fractions may be reduced to their lowest terms.
A fraction is multiplied by an integer by multiplying its upper figure only: thus multiplied by 6 are = 4.3.
The result of multiplying by a fraction is to lessen the multiplicand. This will be obvious if we consider that to multiply by 1 does not increase a number, but leaves it the same as before, and consequently to multiply by less than 1-that is, by a fraction-must diminish it. Example.—. Multiply } by .
7 y 4 _ 28
ở x j = 2 = 1 Answer. Exercises.—Multiply the following1. by Ang. I 4. 58 by Ans. 475 2. 16 " 12 15 | 7. of } , 127
5 3. 4 3 1 1
223 | 6. 79 , 9.1 739
DIVISION OF VULGAR FRACTIONS. RULE.-Invert. the given divisor—thus, if it is , write it as }; then multiply the two fractions together, and the resulting fraction is the quotient : it may then be reduced to its lowest terms.
NotE.—Before proceeding, convert any mixed numbers into improper fractions, and compound into simple fractions.
A fraction is divided by an integer by dividing its upper figure only; thus divided by 4 are &
The result of dividing by a fraction is to increase a number, on the same principle that multiplying lessens it. See Multiplication, page 72. Example.- Divide by 4.
Here the divisor is written as ; the two fractions are then multiplied together, and the resulting fraction reduced to its lowest terms.
Miscellaneous Exercises in Vulgar Fractions. 1. What is the sum of ý and ? .
• Ans. 15 2. If }, , and be added together, what is the amount ? , 111
3. If we take from £, what remains ? . . . . ', 4. What is the difference between 4 and ? . 1 17 5. Whether is or the greater number, and what is the excess ? . . .
. Ans. is greater by a 6. How many sevenths are there in three and two-sevenths ?
Ans. 23 sevenths. 7. Multiply by }, . . . . .. Ans. 8. I J " , . . . . . . " 9. 1
. . . . . . " ii 10. Divide er · · · · · · " 11. " I " TE .
. . " : 12. 11
. . . 13. three-elevenths by eleven-twelfths, . . 14. What is the half of a half ? . . 15. How much is two-thirds of two-thirds ? . . ! 16. Of what number is the double ? . . .
17. What number multiplied by 4 gives & ? . . Ans. 18. What number is that, of which is five-elevenths ? 14 19. What is the seventh part of the half of three-eighths ? ,
20. If two-thirds of 1 lb. cost 52d., what will three pounds and a quarter cost ?.
. . Ans. 2s. 27d. 1 21. What is the half of a third part of £1 sterling? Ans. 35. 4d. 22. What part of £l is 17s. 6d. ?
. . Ans. ? 23. If three-eighths of a ship be worth £15,000, what is the value of the whole vessel ? .
. Ans. £40,000 24. If a whole ship is valued at £50,000, what are three sixtyfourths of it worth ?
. , Ans. £2343, 15s. 25. If one man can do a piece of work in three days, how much of it can he do in a day ? ..
Ans. } 26. If five men can execute a job in one day, how much can each of them perform in the same time? . . . Ans.
27. If one man require three days to complete an undertaking, and another five, how much will they do in one day, if they work together at the same rate ? . . .
. . Ans. 28. If a man can perform three-eighths of a piece of work in one day, in what time will he finish it ? . . Ans. 2 days.
29. A person pays away two-thirds of his money, and gets back two shillings; what had he at first, if he has now the half of his original sum ? .
. . . . . Ans. 12s. 30. A man spends half of his money at one shop, and the third of the remainder at another; he finds that he has six shillings left; what was the sum the man originally had ? . Ans. 18s.
31. If we cut an apple into five parts, and one of these parts into seven others, what part of the whole apple is one of the smaller parts ? . . . . . . Ans. The thirty-fifth part.
32. Whether is three-fourths of 3 or four-thirds of 4 the greater number?
. Ans. Ads of 4, by 34 33. If a man give away three-sevenths of £14, how much money has he left ? .
. Ans. £8 34. If a man who has twenty-five guineas, give away two-fifths of it to one man, two-fifths of the remainder to a second, and the rest to a third, how much money does each of them receive ?
Ans. The first, £10, 10s. ; Second, £6, 6s.; Third, £9, 9s. 35. If one cock empty a cistern in five, and another in four hours, in what time will they empty it if used together? Ans. 2 hours.
36. If a greyhound's leap is one-fourth longer than a hare's, in what time will it overtake the hare that is fifty leaps before it, and that they both take one leap in a second ? Ans. 3 min. 20 sec.
37. Two men set out at the same time, the one from London to York, and the other from York to London; the one travels at the rate of 51 miles an hour, and the other 63; in how many hours will they meet, the distance between the two places being 197 miles? .
. . . . . . Ans. 1639 hours. DECIMAL FRACTIONS.
DECIMAL FRACTIONS are those which express tenths, or combinations of tenths, and are usually indicated by figures in the same way as integers, except that a point is placed before them for distinction: thus—3, three-tenths; .47, forty-seven hundredths. The term is derived from the Latin word decem, signifying ten.
In decimal fractions, instead of the numerator and denominator both being expressed as in Vulgar Fractions, the numerator only is written down (hence the fraction is written in the same way as integers), and the denominator is always understood to be either 10, 100, 1000, or some other combination of tenths, according to the number of figures in the numerator. If the numerator consists of 1 figure, the understood denominator is 10; if of 2 figures, it is 100, and so on; the denominator being always 1, with as many nothings annexed as there are figures in the numerator: for example-4 are written decimally.9; or 19as 99. An integer with a fraction, as 33%, is written thus-3:9; the point [:] marking the separation between them.
As in whole numbers, a figure increases in value ten times by every removal to the left of the units' place, so in decimals, a figure decreases in value ten times by every removal to the right of the units' place: the first figure to the right of the point, or, as it is termed, the first place of decimals, expresses so many tenths; the second figure to the right, so many hundredths ; the third figure, so many thousandths; and so on, according to the following table :'1...= io or 1-tenth.
•1 - 10th. •02 . . = 10 » 2-hundredths.
•12 - 100ths. •003. . = 1000 - 3-thousandths.
•123 - 1000ths. *0004. = 10,000 " 4-ten-thousandths.
•1234 -10,000ths. •00005. = 100.000 - 5-hundred-thousandths. •12345 - 100,000ths. •000006 = 1,000,000 " 6-millionths. 1 .123456 - 1,000,000ths.
Decimals are read from left to right, as in whole numbers, beginning at the first figure after the point, and naming them according to the number of parts expressed by the figures ; thus-5•375, is read five, and three hundred and seventy-five thousandths.
The annexing of nothings to decimals does not alter or increase their value: thus •5 and 500 express the same value, because •5 bears the same proportion to 10, that •500 does to 1000, each being equal to one-half ().
The prefixing of nothings to decimals decreases their value tenfold for every nothing prefixed : thus •1 (+) becomes •01 Géo) by prefixing one nothing.
The calculations in Decimals are performed in the same way as in integers, and hence their great convenience and utility.
Decimals are useful in astronomical, chemical, and other scientific calculations, which require much nicety and minuteness.
Decimals may be reduced, added, subtracted, multiplied, or divided.
REDUCTION OF DECIMALS.
RULE.-Annex as many nothings to the numerator as will make it greater than the denominator, and then divide it by the latter ; continuing to annex nothings and to divide till there is no remainder, or till the same figures come to be repeated in the quotient.
The answer must contain as many decimals as there have been nothings annexed to the numerator ; if there are not as many after dividing, prefix the requisite number to the quotient.
NOTE.-It is only some Vulgar Fractions that admit of being exactly expressed by decimals, as in Example 1, where, on dividing, the fraction terminates exactly in the decimal ; which is hence called a terminate decimal. There are other fractions, as in Examples 2 and 3, which cannot be exactly expressed by decimals, as they do not terminate exactly; however, by carrying the division to several places of decimals, as 8333 &c., the difference between the decimal and the exact fraction becomes too trifling to be appreciable: such decimals are called interminate. Example 1.-Convert the Vulgar Fraction into a decimal. 8) 3000
Here 3 nothings require to be annexed before the division comes to a close; the answer, therefore, con-.
sists of 3 decimals. Example 2.-Convert into a decimal. 6) 500
This is an example of an interminate decimal: after
annexing two nothings, and dividing by 6, the answer .83 &c., Ans. is 83 and 2 over: by annexing more nothings, the
division might be carried to an unlimited extent, as the decimals would always be 3 repeated continually, with a remainder
of 2 over, however many nothings might be annexed. Example 3.-Convert za to a decimal. 22 ) 700000
Here the division may also be carried to an un. limited extent, by annexing more nothings: the rest of the decimals would always be 18 repeated.
THE TERM, RECURRING DECIMAL, is applied to those decimals in which the same figure or figures are continually repeated: they are called Repeating, or Circulating, according to the number of figures repeated.
A repeating decimal is where the same figure is repeated, and is indicated by a dot placed over the recurring figure; thus, •833 &c., is written as •83.
A circulating decimal is where two or more figures are repeated, and is indicated by a dot over the first and last recurring figures; thus 31818 &c., is written as •318, and 73925925 &c., as •73925.
When the decimal consists entirely of recurring figures, as :3, it is tormed a pure recurring decimal: when it consists partly of recurring, and partly of non•recurring figures, as •83, it is termed a mixed recurring decimal.