9 4 X 13 = 117 44 73 from the other, and you then proceed with the whole numbers, as in Simple Subtraction. Example 3.-From 4 take the } of 4. Reduce the compound fraction, 1 of 4, to a simple one, 13 * * = by expressing the 4 fractionally, thus, * ; you then multiply the numerators together for a new numerator, and the denomi- 11 X 13 143 nators for a new denominator, thus, 13 4= 13; then reduce the simple fractions Ag and to a common denominator, subtract the numerators, and the difference is the answer sought. (1.) What is the difference between 4 and 1 ? Ans. as (2.) What is the difference between and ? Ans. 31 195 (3.) From is take . Ans. 93 . 247 (4.) From 4; take Ans. 316 (5.) Take 64 from 9o. Ans. 27 (6.) From 197 take 94%. Ans. 102. MULTIPLICATION OF FRACTIONS. MULTIPLICATION of fractions is in itself so simple, that it will not be necessary to dwell upon it. I need only call to the pupil's mind that to multiply is to increase ; and as he already knows that the nearer the numerator approaches in amount to the denominator, the greater the value of the fraction is ; therefore it is evident that if we wish to multiply a fraction by a whole number, we have only to increase the numerator, and our work is effected. Thus multiplied by 2 written thus, } *2 = 1. Now we know that two halves of anything make a · whole ; and we know also that when the numerator and denominator ofafraction are equal, they represent a whole, or one of anything that may be the subject of our consideration ; we see that this is effected in the above instance by the rule laid down; and what is true as regards one fraction, is equally true as regards another; that is, this rule will answer when we wish to multiply a fraction by a whole number. Multiply i by 2. Å x 2 = å or 1}. It is plain that twice three quarters are six quarters; and as laid down in the rule for reducing improper fractions to mixed numbers, we divide the denominator four into the numerator six, and find that it is contained once with a remainder of two; now two being the half of four, we write its value at once and in the shortest term To multiply one fraction by another, you multiply the numerators together for a new numerator, and the denominators for a new denominator ; their products give the fraction required. Multiply 4 by 5 4 is, or when reduced to its lowest term, io. The product of the numerators 7 and 4, when multiplied together, is 28, and the product of the denominators multiplied together, is 40; thus, 28, which can be reduced by a common measure, 48 + 4 = to To multiply a mixed number by a compound fraction, reduce the mixed number to an improper fraction, and the compound fraction to a simple one, then proceed as before directed; the product of the numerators will be necessarily larger than the product of the denominators; you will therefore reduce the improper fraction to a mixed number, which will be the answer. Multiply 82 by i of 16. The mixed number 8 = 3, and of lo = 27. produced the impro x 1% = 146 = 5 33. Ans. per fraction 35, and the compound fraction produced the simple one 27; the 3 numerators of these two when multiplied produced 245, and the denominators produced 1609 which sum, when divided into 945, gives the quotient 5, and 145 remainder. Now 145 reduced to its lowest terms, is f?, therefore the answer is 53. Ex. (1.) Multiply by i Ans. i (2.) Multiply by (3.) Multiply by H. (4.) Multiply 3 by . (5.) Multiply 6 by the of 1. (6.) Multiply 9. by the of 4. Ans. 28. (7.) Multiply 2416 by of Ans. 153 Ans. 15 40° Ans. 36 Ans. 5 126 DIVISION OF FRACTIONS. Division being the reverse of Multiplication, you can attain your object, should you have to divide a fraction by a whole number, by dividing such whole number into the numerator of your fraction, taking the quotient for your new numerator, and writing the denominator of your former fraction under it. Divide 4 by four : 4 • 4 = 7. The fourth part of four sevenths is equal to one seventh; the denominator is seven more than the numerator. You can also attain your object by multiplying the denominator by the divisor, taking the product for a new denominator, and writing the numerator of the former fraction over it, thus : Divide 4 by four : 4 + 4 = 4. 7 the denominator, being multiplied by 4 the divisor, the product is 28, which 28 is the denominator ; over this is written the former numerator 4, making Ag. Now observe that this is precisely the same sum that was done in the last example; its value is maintained, the denominator being seven times more than the numerator; D 5 and not only that, but you can, if you wish, reduce it to exactly the same amount. As it often occurs when you are dividing a fraction by a whole number, that your divisor will not be contained evenly in the numerator of the fraction you are dividing, I therefore would advise the pupil on all such occasions to adopt the latter method, that is, of multiplying the denominator by the divisor, as he thereby avoids the trouble and intricacy of compound fractions. To divide one fraction by another, you merely invert the denominator, and then multiply the top figures for a numerator, and the bottom for a denominator. Divide by f ; *inverted = 4 = 4. The sis inverted, and appears. We then multiply the 8 and 3, which produces 24; and the 7 and 4, which produces 28 ; written thus, s, which being reduced to its lowest terms, is g, the answer. To divide a mixed number by a compound fraction ; the mixed number is reduced to an improper fraction, and the compound fraction to a simple one; then invert the divisor, or simple fraction, and proceed as before directed. Divide 97 by the } of 7. The one half of seven we know to be three and a half, 91 x or. 6 consequently we know that 1 expresses it. The mixed 5x = 10 = 2 . number 9, when reduced to an improper fraction, is 55 ; this being the sum we have to divide by the improper fraction , we invert the divisor, and find the ternis of our proposition to stand thus; 55 . The 55 being multiplied by 2, the product is 110; the 6 and 7 being multiplied together, their product is 42; we have thus the improper fraction 42. To reduce this : Ans. 1 Ans. 24. to a mixed number, we divide the denominator into the numerator, and find the quotient to be 229, which being reduced to its lowest terms, is the answer, 22: Ex. (1.) Divide by Ans. 25. (2.) Divide by o (3.) Divide 1 by (4.) Divide 97 by 34. Ans. 1-931 (5.) Divide 4 by (6.) Divide 94 by the of Ans. 495 (7.) Divide 8 by the į of 1 Ans. 221 (8.) Divide 16, 1 by 714. (9.) Divide 1. by. Ans. 11 (10.) Divide i by 7. 34 34 Ans. 125 Ans. 2,352. Ans. 1 COMPOUND ARITHMETIC. HAVING now worked-and, as I trust, understood whole and broken numbers, we are led to consider others that cannot properly be said to come under either of these heads; and these are the figures we use to express quantities, whether of money, weight, measure, or time; and as these things mix largely in every business, we cannot bestow too much trouble in acquiring a knowledge of them. The quantities expressed by figures vary so much, and quantities of the same thing, only differing in amount, have so frequently to be joined together or separated, that we call the working of them Compound Arithmetic, to distinguish it from the working of whole numbers, which we call Simple Arithmetic. Now the only difference between these two sorts of arithmetic, is in the notation; in Simple Arithmetic our accounts are kept in tallies of ten, just as a person that |