VULGAR FRACTIONS. A FRACTION means a part of a whole : the term is derived from a Latin word signifying broken. Whole or unbroken numbers, as 1, 2, 3, &c., are termed integers; broken numbers as, i, a half; }, a third, &c., are termed fractions. Fractions are of two kinds : vulgar fractions, from a Latin word signifying common; and decimals, from a word signifying ten. VULGAR FRACTIONS are the common fractions of halves, thirds, fourths, and so on; the term is applied to all fractions when expressed by figures in this form—, two-thirds ; , five-sixths, &c. They are called vulgar fractions, in distinction from decimals, a term applied to fractions of tenths, hundredths, &c. (see page 75), when expressed by figures in the same way as integers, except that a dot is placed before them, thus—2, two-tenths. If we suppose a loaf to be divided into two equal parts, each of the parts is a half, and forms a fraction of the whole; in figures, it is written as a Vulgar Fraction, thus-2. Again, if the loaf is divided into four equal parts ; each of these is called a fourth, or a quarter, and is written thus- ; two of them may be expressed as X, but as two-fourths are the same as one-half, they are writt three of them are written, 3, expressing three-fourths. If the whole be divided into three equal parts, each part is called a third ; if into five, each is called a fifth; if into six, a sixth ; and so on, according to the number of parts into which the whole is divided; thus-means two-thirds of a whole ; , three-fifths; , five-sixths. To represent a Vulgar Fraction, therefore, two numbers are required, written the one above the other, with a short line between. The number under the line shews into how many parts the whole of the article, whatever it may be, is divided ; and the number above the line, shews how many of these parts we mean to express. The upper number is called the numerator, because it shews the number of the parts-as three-fourths, six-sevenths; the lower number is called the denominator, because it denominates the nature of the fraction-such as thirds, eighths, &c. All the parts are together equal to the whole. Thus--two-halves, or three-thirds, or four-quarters, make each a whole. When the upper figure of the fraction is less than the under, the fraction expresses less than a whole; when both numbers are the same, the fraction is equal to one whole; and when the upper number is greater than the under, it expresses more than one whole. Thus-pa is less than a whole, 4 is equal to a whole, and is more than a whole. It will now be understood that such quantities as 35 yards, or 23 pounds, mean 3 whole yards and five-eighth parts of a yard, and 2 whole pounds and three-fourth parts or 3 quarters of a pound. An improper fraction is that of which the numerator is equal to or greater than the denominator-as , &. The true way of expressing 4 would be 1, and instead of 5 it would be 11. A compound fraction is a fraction of a fraction; as-1 of 1 REDUCTION OF VULGAR FRACTIONS. RULE.—Divide the numerator and denominator by any number that will divide both without a remainder; then divide the new fraction in a similar way; and continue the process till the fraction cannot be reduced any lower : this last fraction is the answer. Or, when a divisor cannot be readily got, find the greatest common divisor (see page 66) of the numerator and denominator; then divide both by it, and the result will be the fraction in its lowest terms. NOTE.—A fraction is not altered in value when both of its terms are multiplied or divided by the same number. Example.Reduce the fraction 44 to its lowest terms. 4) 44( 17 Ans. . Here we divide 44 and 76 by 4. Exercises.—Reduce the following to their lowest terms :- , 1. Ans. s 1 4. 4 Ans. 1 7. 875 Ans ? 1 .5. do 1 } 8. 288 minima II. TO REDUCE A MIXED NUMBER, AS 2*, TO A FRACTIONAL FORM. RULE.—Multiply the integer by the denominator of the fraction, and include the numerator in the product; then write the denominator below it. An integer is reduced to a fractional form by writing 1 below it; thus-4 is written as 4. Example.-Reduce 24 to a fractional form. 23 12 = Answer. Exercises.-Reduce the following to a fractional form :1. 7 Ans. | 3. 6 Ans. f 1 5. 1217 Ans. 446 2. 2 1 18 | 4. 9 | 6. 147 1391 III. TO REDUCE AN IMPROPER FRACTION, AS 4, TO A WHOLE OR MIXED NUMBER. RULE.-Divide the numerator by the denominator, and the quotient is either a whole or a mixed number, as the case may be. Example-Reduce to a mixed number. 6 )17 nswer. Exercises.--Reduce the following to whole or mixed numbers :1. Ans. 2 | 3. Ans. 11 5. 4, Ans. 3 2. 10 31 1 4. 3: 2 161 16. , 98 IV. TO REDUCE A COMPOUND TO A SIMPLE FRACTION. RULE.—Multiply all the numerators together for the numerator, and all the denominators together for the denominator ; the resulting fraction may then be reduced to its lowest terms. Example. Reduces of it to a simple fraction. x = * = Answer. . Exercises.—Reduce the following to simple fractions :1, í of Ans. } | 3. of Ans. zo | 5. of Ans. 2. " " 1 4. & " t " za 1 6. " " V. TO CONVERT FRACTIONS HAVING DIFFERENT DENOMINATORS, TO OTHER EQUIVALENT FRACTIONS HAVING A COMMON DENOMINATOR. Rule 1. Find the least common multiple (see page 66) of all the denominators, and place it as the common denominator; then multiply this common denominator by the upper figure of each fraction, and divide the product by the under figure of each, for the respective numerators: the resulting fractions will be in the lowest common terms, if the given fractions are stated in their lowest terms. Rule 2. Another method is to multiply all the denominators together for a common denominator ; then, as before, multiply this common denominator by the upper figure, and divide it by the under figure of each fraction, for the respective numerators: the resulting fractions may then be reduced to their lowest common terms. Example.-Convert , , , to other fractions having a common denominator. First Method. Here the least com30 mon multiple of the de nominators is found to 3) 60 590 6150 be 30, and is multiplied, divided, &c., according 20 to the rule. Ans, jo Here all the denomiSecond Method. nators are multiplied together for a common denominator ; 3 X5 X6 = 90: and this is multi3 ) 180 plied, divided, &c., as be fore: the resulting frac 5 tions are reduced to their Ans. = 18 = 18 8 = lowest common terms. Exercises.-Convert to fractions having a common denominator : 1. Ans. | 4. i sis Ans. It no 2. f f goo 36 5. f f 1 " To Me 1918 3. Ã Ã Ã " Menu 4 41 | 6. 36 13 11 9 16 30 5) 270 VI. TO CONVERT A FRACTION OF ONE DENOMINATION INTO THE FRACTION OF ANOTHER, WITHOUT ALTERING ITS VALUE. RULE.-Ascertain how many of the smaller denomination make one of the greater : if the conversion is from a higher to a lower denomination, multiply the numerator of the fraction by that number; if from a lower to a higher, multiply the denominator. Example 1.-Convert tio of a pound to the fraction of a penny. Here, as the change is from a 2 X 240 = 480 = 16 of 1d. Ans. higher to a lower, we multiply the numerator by 240, the num ber of pence in a pound. Example 2.- Convert of a penny to the fraction of a shilling, Here, as the change is from a 21 lower to a higher, we multiply the denominator by 12, the number of pence in a shilling. number ominator bve multiply Exercises. 1. Convert t of a farthing to the fraction of 1d., Ans. penny " 1 shilling £1, 1 shilling in pound o 1 pound 7. o , shilling d . 8. " 3o 1 ton 1qr. " VII. TO FIND THE VALUE OF A FRACTION OF A GIVEN DENOMINATION. RULE.—Reckon the upper figure of the fraction as so many of the given denomination, and then divide it by the under figure, as in Compound Division. Example.—What is the value of } of a pound ? £2 Here the 2 is reckoned as £2, and divided 3)40 by 3, as in Compound Division. 13s. 4d. Ans. Exercises.—What is the value of 1. of a shilling, Ans. 8d. 4. of a shilling, Ans. 630. 2. " pound, 12s.6d. 5. pound, I 5s.54d. 3. " pound, 4s. 2d. 6. . ton, 15 cwts. VIII. TO CONVERT A GIVEN SUM, AS 14s. 6d., TO A FRACTION OF ANOTHER DENOMINATION. RULE.—Convert the given sum into its lowest denomination, and place it as the numerator; then convert one of the proposed denomination into the same denomination as the other, and place it as the denominator: the resulting fraction may then be reduced to its lowest terms. Example.—Convert 3s. 3d. to the fraction of a pound. 3s. 3d., reduced to pence, its lowest denomination = 39 _ 13 of 4 £1, reduced to pence. . . . . = 24080 Exercises. 1. Convert 6s. 9d. to the fraction of a pound, , . Ans. £ 1 14s. 74d. " . i £11; 3. 1 16s. 8. 1 qr. 12 lbs. to the fraction of a cwt., Ans. cwt. ADDITION OF VULGAR FRACTIONS. RULE.-Convert the fractions, if they have different denominators, into others having a common denominator; then add together all the numerators, and under the sum write the common denominator : the resulting fraction may then be reduced to its lowest terms. If the answer is an improper fraction, reduce it to a whole or mixed number. NOTE.—When mixed numbers are given, add the fractions first, and then the integers: when fractions are compound, before adding, convert them into simple fractions, and if of different denominations, convert them into the same denomination. Example.--Add together 4, Ķ, io. These are first con$ = 1 verted to fractions having s = 138 a common denominator, Io = 16 by Reduction, Rule V., * page 69; the numera120 + 100 + 45 = 265 = 63 = 123 Ans. tors are then added toCommon denominator 150 gether, and the common denominator written below their amount: the answer being an improper fraction, is converted into the mixed number 138. • The second method under Rule V., is used in this example. Exercises.—Add together1. Ans. lady | 4. $ 1 Ans. 2146 2. $ 1 14 5. of — of of 4 1 151 3. 1 6. 121 24 17 1 54% |