VULGAR FRACTIONS. A FRACTION, or broken number, is one or more parts of a UNIT. EXAMPLE.-12 inches are 1 foot. Here, 1 foot is the unit, and 12 inches its parts; 3 inches, therefore, are the one fourth of a foot, for 3 is the quarter or fourth of 12. A Vulgar Fraction is a fraction expressed by two numbers placed one above the other, with a line between them; as, 50 cents is the of a dollar. The upper number is called the Numerator, because it shows the number of parts used. The lower number is called the Denominator, because it denominates, or gives name to the fraction. The Terms of a fraction express both numerator and denominator; as, 6 and 9 are the terms of . A Proper fraction has the numerator equal to, or less than the denominator; as, ,&c. An Improper fraction is the reverse of a proper one; as, 2, &c. A Mixed fraction is a compound of a whole number and a fraction; as, 5}, &c. A Compound fraction is the fraction of a fraction; as, of, &c. A Complex fraction is one that has a fraction for its numerator or denominator, or A Fraction denotes division, and its value is equal to the quotient obtained by dividing the numerator by the denominator; thus, 12 is equal to 3, and 21 is equal to 41. REDUCTION OF VULGAR FRACTIONS. To find the greatest Number that will divide Two or more Numbers without a Remainder. RULE. Divide the greater number by the less; then divide the divisor by the remainder; and so on, dividing always the last divisor by the last remainder, until nothing remains. EXAMPLE.-What is the greatest common measure of 1908 and 936 ? 936) 1908 (2 36) 936 (26 72 216 216 So 36 is the greatest common measure. To find the least Common Multiple of Two or more Numbers. RULE.-Divide by any number that will divide two or more of the given numbers without a remainder, and set the quotients with the undivided numbers in a line beneath. Divide the second line as before, and so on, until there are no two numbers that can be divided; then the continued product of the divisors and quotients will give the multiple required. EXAMPLE. What is the least common multiple of 40, 50, and 251 5) 40.50. 25 5) 8.10. 5 2) 8. 2. 1 4. 1. 1 Then 5×5×2×4=200 Ans To reduce Fractions to their lowest Terms. RULE.-Divide the terms by any number that will divide them without a remainder, or by their greatest common measure at once. EXAMPLE.-Reduce 120 of a foot to its lowest terms. 960 To reduce a Mixed Fraction to its equivalent, an Improper Fraction. NOTE.-Mixed and improper fractions are the same; thus, 5=11. For illus tration, see following examples: RULE.-Multiply the whole number by the denominator of the fraction, and to the product add the numerator; then set that sum above the denominator. EXAMPLE.-Reduce 23 to a fraction. 23×6+2=140 EXAMPLE.-Reduce 123 inches to its value in feet. 123÷6=203; that is, 20 feet and 2 or of a foot. To reduce a Whole Number to an equivalent Fraction having a given Denominator. RULE.-Multiply the whole number by the given denominator, and set the prod uct over the said denominator. EXAMPLE.-Reduce 8 to a fraction whose denominator shall be 9. 8X9=72; then 12 the answer. To reduce a Compound Fraction to an equivalent Simple one. RULE.-Multiply all the numerators together for a numerator, and all the denominators together for a denominator. NOTE. When there are terms that are common, they may be omitted. EXAMPLE. Reduce of of a pound to a simple fraction. X= Ans. To reduce Fractions of different Denominations to equivalent ones having a common Denominator. RULE.-Multiply each numerator by all the denominators except its own for the new numerators; and multiply all the denominators together for a common denominator. NOTE. In this, as in all other operations, whole numbers, mixed, or compound fractions, must first be reduced to the form of simple fractions. EXAMPLE.-Reduce,, and to a common denominator. 1X3X4=12 2X2X4=16=18=18 Ans. 3X2X3=18 2×3×4=24 To reduce Complex Fractions to Simple ones. RULE.-Reduce the two parts both to simple fractions; then multiply the numerator of each by the denominator of the other. EXAMPLE.--Simplify the complex fraction 2 41 RULE.-If the fractions have a common denominator, add all the numerators together, and then place the sum over the denominators. NOTE.-If the prepared fractions have not a common denominator, they must be reduced to one. Also, compound and complex must be reduced to simple fractions. EXAMPLE.-Add 4 and together. = 2 of x=1. Then, 18+5=1131. Ans. SUBTRACTION OF VULGAR FRACTIONS. RULE.-Prepare the fractions the same as for other operations, when necessary; then subtract the one numerator from the other, and set the remainder over the common denominator. EXAMPLE.-What is the difference between and †? RULE.-Prepare the fractions as previously required; multiply all the numerators together for a new numerator, and all the denominators together for a new denominator. EXAMPLE.-What is the product of 2 and 3 ? EXAMPLE.-What is the product of 6 and 3 of 5? ex of 5=x10 == 20 Ans. C 26 APPLICATION OF REDUCTION OF VULGAR FRACTIONS. DIVISION OF VULGAR FRACTIONS. RULE.-Prepare the fractions as before; then divide the numerator by the numerator, and the denominator by the denominator, if they will exactly divide; but if not, invert the terms of the divisor, and multiply the dividend by it, as in multi plication. EXAMPLE.-Divide 25 by 5. 25 ÷ 3 = 3=13 Ans. To find the Value of a Fraction in Parts of a whole Number. RULE.-Multiply the whole number by the numerator, and divide by the denominator; then, if anything remains, multiply it by the parts in the next inferior denomination, and divide by the denominator, as before, and so on as far as necessary; so shall the quotients placed in order be the value of the fraction required. EXAMPLE.-What is the value of of of $9 ? EXAMPLE. Reduce of a pound to avoirdupois ounces. 1 4) 3 (0 lbs. 16 ounces in a lb. 4) 48 12 ounces, Ans. EXAMPLE.—Reduce of a day to hours. 3×24=18=72 hours, Ans. 10 To reduce a Fraction from one Denomination to another. RULE.-Multiply the number of parts in the next less denominator by the numerator if the reduction is to be to a less name, but multiply by the denominator if to a greater. EXAMPLE.-Reduce of a dollar to the fraction of a cent. 4x100 = 100-25, the answer. EXAMPLE. Reduce of an avoirdupois pound to the fraction of an ounce. 16 = 16 = 3, the answer. EXAMPLE.-Reduce 2 of a cwt. to the fraction of a lb. 2×3×28224=32, the answer. EXAMPLE.-Reduce of of a mile to the fraction of a foot. 3 of 3528031680 12 2640 the answer. EXAMPLE. Reduce of a square foot to the fraction of an inch. 1×144 — 144 - 36 Ans. For Rule of Three in Vulgar Fractions, see page 29. DECIMAL FRACTIONS. 85 100 A DECIMAL FRACTION is that which has for its denominator a UNIT (1), with as many ciphers annexed as the numerator has places; it is usually expressed by setting down the numerator only, with a point on the left of it. Thus, is .4, 10 is .85, 0075 is .0075, and 125 is .00125. When there is a deficiency of figures in the numerator, prefix ciphers to make up as many places as there are ciphers in the denominator. 10000 100000 Mixed numbers consist of a whole number and a fraction; as, 3.25, which is the 325 same as 3. 25 or 100' 100 Ciphers on the right hand make no alteration in their value; for .4, .40, .400 are decimals of the same value, each being, or . ADDITION OF DECIMALS. RULE. Set the numbers under each other according to the value of their places, as in whole numbers, in which state the decimal points will stand directly under each other. Then, beginning at the right hand, add up all the columns of numbers as in integers, and place the point directly below all the other points. EXAMPLE.-Add together 25.125, 56.19, 1.875, and 293.7325. 25.125 56.19 1.875 293.7325 376.9225 the sum. SUBTRACTION OF DECIMAL FRACTIONS. RULE.-Place the numbers under each other as in addition; then subtract as in whole numbers, and point off the decimals as in the last rule. EXAMPLE.-Subtract 15.150 from 89.1759. 89.1759 15.150 74.0259 Rem. MULTIPLICATION OF DECIMALS. RULE. Place the factors, and multiply them together the same as if they were whole numbers; then point off in the product just as many places of decimals as there are decimals in both the factors. But if there be not so many figures in the product, supply the deficiency by prefixing ciphers. EXAMPLE.-Multiply 1.56 by .75. 1.56 .75 780 1092 1.1700 Prod. BY CONTRACTION. To contract the Operation so as to retain only as many Decimal places in the Product as may be thought necessary. RULE.-Set the unit's place of the multiplier under the figure of the multiplicand whose place is the same as is to be retained for the last in the product, and dispose of the rest of the figures in the contrary order to what they are usually placed in. Then, in multiplying, reject all the figures that are more to the right hand than each multiplying figure, and set down the products, so that their righthand figures may fall in a column straight below each other; and observe to increase the first figure in every line with what would arise from the figures omitted; thus, add 1 for every result from 5 to 14, 2 from 15 to 24, 3 from 25 to 34, 4 from 35 to 44, &c., &c., and the sum of all the lines will be the product as required. EXAMPLE.-Multiply 13.57493 by 46.20517, and retain only four places of decimals in the product. 13.574 93 71 502.64 |