FRACTIONS.
A fraction is a quantity which expresses a part, or parts of a unit, or integer. It is denoted by two numbers placed with a line between them.
A Simple fraction consists of two numbers, called the numerator, and denominator; thus,
| 3 | numerator, |
| 5 | denominator. |
The Denominator is placed below the numerator, and expresses the number of equal parts into which the integer is divided.
The Numerator expresses the number of parts of the broken unit, or integer; or shows how many of the parts of the unit are expressed by the fraction.
A Compound fraction is a fraction of a fraction, as ⅔ of ⅘.
A Mixed number consists of a whole number with a fraction annexed to it, as 4⅖.
An Improper fraction has the numerator greater than the denominator, as 6/5.
REDUCTION OF FRACTIONS.
is bringing them from one denomination to another.
To reduce a fraction to its lowest terms.
Rule.—Divide the numerator, and the denominator, by any number that exactly divides them, and the quotients by any other number, till they can be no longer divided by any whole number, when the fraction will be in its lowest terms.
Example.—Reduce 4032 6048 to its lowest terms.
Thus, 4 4032 6048 = 12 1008 1512 = 6 84 126 = 7 14 21 = 2 3. Answer.
To reduce an improper fraction to a whole, or mixed number.
Rule.—Divide the numerator by the denominator, the quotient will be the whole number; and the remainder (if any) the numerator of the fraction, having the divisor for the denominator.
Example.—Reduce 114 12 to a whole, or mixed number.
| 12 ) 114 | |
| 9 6 12 | Answer. |
To reduce a mixed number to an improper fraction.
Rule.—Multiply the whole number by the denominator, and add the numerator to the product, under which place the given denominator.
Example.—Reduce 17⅝ to an improper fraction.
| 17⅝ | |
| 8 | |
| 141 | |
| 8 | Answer. |
To reduce a compound fraction to a simple fraction.
Rule.—Multiply all the numerators together for the numerator, and all the denominators for the denominator.
Example.—Reduce ⅜ of ⅙ of ½ of 9 to a simple fraction.
| Numerators | 3 × 1 | × | 1 × 9 | = | 27 | = | 9 | Answer. |
| Denominators | 8 × 6 | × | 2 × 1 | 96 | 32 |
To reduce fractions of different denominators to equivalent fractions, 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.[42]
Example.—Reduce ⅜, ⅔, and ⅘ to fractions having a common denominator.
3 × 3 × 5 = 45
2 × 8 × 4 = 80
4 × 8 × 3 = 96
8 × 3 × 5 = 120 Answer, 45 120 , 80 120 , 96 120 .
ADDITION OF FRACTIONS.
Rule.—Bring compound fractions to simple fractions; reduce all the fractions to a common denominator, then add all the numerators together, and place their sum over the common denominator. When mixed numbers are given, find the sum of the fractions, to which add the whole numbers.
Example.—Add together ⅚, ¾, and 6½.
5 × 4 × 2 = 40 40 48 + 36 48 + 24 48 + 6 = 8 4 48 .
3 × 6 × 2 = 36
1 × 6 × 4 = 24 or, by cancelling, and dividing,[43]
6 × 4 × 2 = 48 10 12 + 9 12 + 6 12 + 6 = 8 1 12 Answer.
SUBTRACTION OF FRACTIONS.
Rule.—Prepare the quantities, as in addition of fractions. Place the less quantity under the greater. Then, if possible, subtract the lower numerator from the upper; under the remainder write the common denominator, and, if there be whole numbers, find their difference as in simple subtraction. But if the lower numerator exceed the upper, subtract it from the common denominator, and to the remainder add the upper numerator; write the common denominator under this sum, and carry 1 to the whole number in the lower line.
Example.—
From 54 5 6 or 54 25 30
Take 25 5 15 or 25 10 30
——
29 15 30 Answer.
MULTIPLICATION OF FRACTIONS.
Rule.—Reduce mixed numbers to equivalent fractions; then multiply all the numerators together for a numerator, and all the denominators together for a denominator, which will give the product required.
Example.—Multiply ⅚, ⅜, and 2½ together.
⅚ × ⅜ × (2½ or) 5 2 = 75 96 Answer.
DIVISION OF FRACTIONS.
Rule.—Prepare the fractions, as for multiplication; then divide the numerator by the numerator, and the denominator by the denominator, if they will exactly divide; but if they will not do so, then invert the terms of the divisor, and multiply the dividend by it, as in multiplication.
Example.—Divide 9 16 by 4½.
9 16 ÷ (4½ or) 9 2 = ⅛ Answer.
RULE OF THREE IN FRACTIONS.
Rule.—State the terms, as directed in “Simple proportion;” reduce them (if necessary) to improper, or simple fractions, and the two first to the same denomination. Then multiply together the second and third terms, and the first with its parts inverted, as in division, for the answer.
Example.—If 4⅕ cwt. of sugar cost £19⅞, how much may be bought for £59⅝?
As 19⅞ : 59⅝ :: 4⅕
Or, 159 8 : 477 8 :: 21 5 : 12⅗ Answer.
8 159 × 477 8 × 21 5 = 80136 6360 = 12⅗ cwt.