+(plus) the character for addition: Thus 2+3 = 5, 2 added to 3 are equal to 5. 4 + 6 =7+ 3, 4 added to 6 are equal to 7 and 3. -(minus) signifies subtraction: As 5-32 3 subtracted from 5 is equal to (or leaves) 2. 4 — 3 — 2 — 1, the difference of 4 and 3 is equal to that of 2 and 1. the character for multiplication : 2 x 3 = 6, 2 multiplied by 3 is equal to (or produces) 6. 2 × 3 × 4 = 24, the continual product of 2, 3, and 4, is equal to 24. 37. But the proper method of abbreviating division is to set down the quotient in the form of a fraction by placing the divisor under the dividend; thus, 3 divided by 4 gives for the quotient; 5 divided by 2 gives the quotient; and 1 divided by 4 produces, or a quarter. In general, every fraction should be considered as the quotient arising from the division of the numerator by the denominator. REDUCTION OF VULGAR FRACTIONS. 38. REDUCTION of Vulgar Fractions principally consists in changing them to a more commodious form for the operations of addition, subtraction, &c. CASE 1. To abbreviate or reduce fractions to their lowest terms. 39. If the terms of a fraction are multiplied or divided by any number, its value will evidently remain the same as before; thus, the numerator and denominator of multiplied by 4 produces the fraction, or divided by 3 gives (or half), the same asor. Therefore to reduce a fraction to its lowest terms, divide the terms of the fraction by any number that will leave no remainder, and the quotients again by the same, or any other number, and so on, till 1 is the greatest divisor; then the fraction will be in its lowest terms. Ex. 1. Reduce 1408 to its lowest terms. This fraction may be reduced by a continual division by 2: When 2 fails as a divisor, try 3, 5, or 7, because if a number is divisible by any digit, (1 excepted) it must be divisible by either 2, 3, 5, or 7. 40. If the numerator and denominator are large numbers, find their greatest divisor, or common measure, by the following rule: Divide the greater by the less, and the last divisor by the last remainder, and so on, till nothing remains; then the last divisor is the greatest common measure required. If 1 remains for the last divisor, the numerator and denominator (having 1 for their greatest common measure) are said to be prime to each other; and the fraction is already in its lowest terms. Ex. 4. Reduce 753 to its lowest terms. 7631) 26415 (3 22893 3522) 7631 (2 587)3522 (6 3522 Therefore the last divisor 587 is the greatest number that will divide 7631 and 26415 without leaving any remainder. In like manner the greatest divisor or common measure of three or more numbers may be found. For having found the greatest common measure of two of them, as above, find the greatest divisor of that common measure and another of the numbers, and so on. Thus 15 is the greatest common measure of 1995, 840, and 600. The foregoing rule for finding the greatest common divisor of two numbers is founded on the following axiom; if a number measures another number, and also a part of that number, it will measure the remaining part. Thus 5 measures 40 (or 5 is contained in 40 an exact number of times), and it also measures 25 (a part of 40), therefore it measures 15 the other part. That the operation brings out the greatest divisor may be shewn from the 4th example, thus:-The denominator 26415 is equal to the numerator 7631 × 3 + 3522 (by the method of proving common division): now if there is a greater divisor than 587 which measures 7631, and 7631 × 3 + 3522, it must (by the preceding axiom) measure 3522. And for the like reason, if it measures 3522, it must measure 3522 × 2. And if it measures 7631 and 3522 × 2, it must (by the same axiom) measure their difference, or 7631—3522 × 2, or 587, viz, the greater measures the less, which is absurd. CASE 2. To reduce an improper fraction to its equivalent whole or mixt number. 41. THIS is evidently nothing more than common division. Therefore divide the numerator by the denominator, and the quotient will be the answer. CASE 3. To reduce a mixt number to its equivalent improper fraction. 42. THIS operation is the reverse of the former; therefore multiply the whole number by the denominator of the fraction, and add the numerator to the product, then place the sum over the denominator for the fraction required. Example. Reduce 2213 to an improper fraction. Hence to reduce a whole number to an improper fraction having a given denominator :-multiply the said number by the proposed denominator, and make the product the numerator of the required fraction. Example. Let 13 be reduced to a fraction whose denominator is 7. 13 X 791 the numerator. Answer . For 9 13 by the preceding article. 7 CASE 4. To reduce a compound fraction to an equivalent simple one. 43. MULTIPLY all the numerators together for the numera tor, and all the denominators together for the denominator of the fraction required. If part of the compound fraction be a mixt, or a whole number, reduce the former to an improper fraction, and make the latter a fraction by placing 1 under it as a denominator. 44. When a like number of like factors are found in the nu merator and denominator, cancel them in both. Ex. 3. Reduce of 1 of 3 of 4 of 3 to a simple fraction. 2 X 1 X 3 x 5 X 3 3 X 2 X 5 X 7 X 4 here cancelling 2, 3, and 5, in both numerator and 3 1 x 3 denominator, the fraction becomes = the answer. This is redu 7 X 4 28 cing the fraction to lower terms by means of the divisors 2, 3, and 5. (39) The rule for reducing compound fractions may be derived as follows: Suppose a shilling to be the integer; then because 48 farthings make 1 shilling, the simple fraction denoting 3 farthings is and the compound 3 48, (or of a penny), and the respective products of 3 fraction will beof( ог - the simple fraction. 4 X 12 48 |