DIVISION OF VULGAR FRACTIONS. We have seen that DIVISION of INTEGERS is the REVERSE of MULTIPLICATION and vice versa. To prove this it will only be necessary to show that these operations DESTROY the effect of each other. Thus, if 6 is multiplied by 5, and that product divided by 5, we shall have 6 again; for 6 times 5 are 30, and 30 divided by 5 equals 306. i. e. the EFFECT of MULTIPLICATION is destroyed by DIVISION; again, If 30 is DIVIDED by 5 we shall have 6 for the quotient, and if this QUOTIENT (6) be MULTIPLIED by 5, we shall have 30 again-i: e: the effect of DIVISION is destroyed by MULTIPLICATION. Hence MULTIPLICATION and DIVISION are the REVERSE of each other. Suppose (e. g.) we are required to divide 30 by 6, as before; the operation may be expressed thus: 30, i. e. 30 units, DIVIDED by 6 units. Now suppose we invert (turn upside down) the divisor (f) making it, and then multiply 30 by we shall have 30 1 30.1 30 5 111 1 6 1 6 6 1 5, the answer sought; the same would be true if the divisor had been any other * P. S. Whole numbers, as 7, 4, &c., when joined with fractions as multi pliers, should stand thus, 7, 1, &c. number, either integral or fractional. If the divisor is a MIXED number, it should be reduced to an IMPROPER fraction, and then inverted; as, Lastly, when the divisor is a COMPOUND fraction, it should be prepared as in MULTIPLICATION, i. e. the WHOLE numbers if any, should be represented in the FORM of a fraction, and the MIXED numbers should be reduced to an IMPROPER fraction. Suppose it is required to divide by 4 times of 31; the divisor in this case is 4.1.7 1.6.2 and this, Hence for DIVIDING quantities whether FRACTIONAL or INTEGRAL, we have the following General RULE. If the divisor is an INTEGER reduce it to the FRACTIONAL form; or if it is an IMPROPER fraction, it should be reduced to a MIXED NUMBER; then INVERT THE DIVISOR and proceed as in the multiplication of fractions; the PRODUCT will be the QUOTIENT required. NOTE.-We have seen in division of integers, that if the divisor is GREATER than a unit, the QUOTIENT will be LESS than the dividend; hence, it follows that when the divisor is LESS than a unit, the quotient will be GREATER than the dividend. We will now find the value of q., (quotient,) in the following EXAMPLES. }= 5.3.1 1. (8 of 3 of 4)÷=q. Thus: & of % of 6.5.4' (the dividend,) and 7 inverted, is ;. Now, if (the dividend,) 5.3.1 is multiplied by, and cancelled we shall have $.2.1.8 2. (of 3 of 3)-1=q. Thus: 1, which inverted. 923 3. 7÷(% of of 3)=q. Thus: 7=7; and (3 of % of 3) inverted, = 8.7.8.7 6.7.8 5.6.7 6.7.8.7 which, multiplied by 7-5.6.7.1 = 5···1=3=1=q. Answer. 4. (5+43)÷(10+{})=q. 5. 24-7=9. 8. (4×8 of 14)÷22=q. 6. (6×25)÷(9×5)=q. 25.6 7. (125584×132)÷÷} of 2)—q. 9. (§ of 11+4)÷(} of 13=q. DEFINITIONS. 1. A COMPLEX fraction is one which has a fraction for its NUMERATOR or DENOMINATOR, or BOTH; as 2. A PARENTHESIS, (plural, parentheses,) marked thus, (); a VINCULUM, (marked thus, -), or BRACKETS, (marked thus, []), signify that the quantities they INCLUDE, must be regarded as a SINGLE NUMBER, which is the result of adding, subtracting, multiplying, or dividing, ALL the rest, according to the signs prefixed to each; as, (8÷1), }+&, (}÷}),&c. 3. An INCREMENT, (from Lat. INCREMENTUM, increase, addition, is a quantity to be added. 4. A DECREMENT, (from Lat. DECREMENTUM, decrease, subtraction,) is a quantity to be SUBTRACTED. NOTE. The terms INCREMENT and DECREMENT, (pronounced dek-re-ment,) will be found convenient in reading symbolical quantities, as will be shown hereafter. DIVISION BY FACTORS. RULE. First. When the divisor is the EXACT product of two or more numbers less than 12, resolve it into factors. Second. If the dividend is an integer, divide it (the dividend,) by one of the factors, as in case I., page 50; but if the dividend is a MIXED number, divide the integers, as before, and reduce the remaining mixed number to an improper fraction, with which proceed according to the Rule for dividing vulgar fractions, (page 62,) and repeat this process till all the factors are used, the last quotient will be the answer. 561847 EXAMPLES. 1. Given 168=q., to find the value of q., proceed thus: 168 resolved into factors, equals 8x7x3; hence, by successive divisions, we have the following equations, 561847 561847 = 168 8.7.3 FIRST equation. Now reduce the right hand member (58.18.47), by cancelling 8 from the deno minator, and dividing the numerator (561847) by the same, 561847 70230 for the SECOND (8), and we have = 8.7.3 7.3 equation. Again, cancel 7 from the denominator of the right hand member of the SECOND equation, as before, and 7023010032 we have for the THIRD equation; lastly, cancel 3, and divide by it, (3), as before, and we have 100328 3 168 55 = 3344,55 for the FOURTH or last equation, the right hand member of which, viz. 3344 is the LAST QUOTIENT, or answer required. The best mode of writing this 561847 561847 process arithmetically, is as follows: 5493291 315 168 answer; which should be =q., to find the value of q. Proceed as 2. 87195,20 5 =1743920-q. Hence, q.=17439,2, the quo TIENT, or answer required. In the same manner find the value of q. in the following |