That is, make 3 doliars, and there are of another dollar. Ans. 3 dollars. If i man consume z's of a barrel of flour in a week, how many barrels will an army of 537 men consume in the same time? They will consume 5. of a barrel make a barrel, therefore as many times as 35 is contained in 537, so many barrels it is. 537 (35 15}} barrels. Ans. 12 35 is contained 15 times in 537 and 12 over, which is the of another barrel. Numbers like 33, 154, which contain a whole number and a fraction, are called mixed numbers. The above process by which was changed to 3%, and 3 to 15}}, is called reducing improper fractions to whole or mixed numbers. Since the denominator always shows how many of the parts make a whole one, it is evident that any improper fraction may be reduced to a whole or mixed number, by the following rule : Divide the numerator by the denominator, and the quotient will be the whole number. If there be a remainder, write it over the denominator, and annex it to the quotient, and it will form the mixed number required. XIV. It is sometimes necessary to change a whole or a mixed number to an improper fraction. A man distributed 3 dollars among some beggars, giving them } of a dollar apiece; how many received the money ? That is, in 3 dollars, how many fifths of a dollar ? Each dollar was divided equally among 5 persons, consequently 3 dollars were given to 15 persons. That is, 3 dollars are equal to us of a dollar. A man distributed 183 bushels of wheat among some poor persons, giving them 1 of a bushel each ; how many persons were there? This question is the same as the following: In 183 bushels, how many 4 of a bushel ? That is, how many 7ths of a bushel ? In 1 bushel there are 7, consequently in 18 bushels there are 18 times 7 sevenths ; that is, 146, and more make 14°. Answer 129 persons. Reduce 281} to an impriper fraction. That is, in 2811 how many as Since there are 5 in 1, in 28 there must be 28 times as many. 28 times 25 are 700, and 17 more are 717. Ans. 717. Hence to reduce a whole number to an improper fraction with a given denominator, or a mixed number to an improper fraction : multiply the whole number by the denominator, and if it is a mixed number add the numerator of the fraction, and write the result over the denominator. XV. A man hired 7 labourers for 1 day and gave them of a dollar apiece; how many dollars did he pay the whole ? If we suppose each dollar to be divided into 5 equal parts, it would take 3 parts to pay 1 man, 6 parts to pay 2 men, &c. and 7 times 3 or 21 parts, that is, * of a dollar to pay the whole. * of a dollar are 41 dollars. Ans. 4} dollars. A man bought 13 bushels of grain, at of a dollar a bushel ; how many dollars did it come to ? of a dollar are 5 shillings. 13 bushels at 5 shillings a bushel, would come to 65 shillings, which is 10 dollars and 5 shillings. In the first form, 13 times of a dollar are 6 of a dollar ; that is 105 dollars, as before. A man found by experience, that one day with another, his horse would consume jî of a bushel of oats in a day; how many bushels would he consume in 5 weeks or 35 days? If we suppose each bushel to be divided into 37 equal parts, he would consume 13 parts each day. In 35 days he would consume 35 times 13 parts, which is 455 parts. But the parts are 37ths, therefore it is = 12 H bushels. 35 13 105 35 437* = 1237 455 This process is called multiplying a fraction by a whole number. Multiply by 48. The fraction a signifies that 1 is divided into 1372 equal parts, and that 253 of those parts are used. To mul. tiply it by 48, is to take 48 times as many parts, that is, to multiply the numerator 253 by 48. 253 9 = 811 12144 The product of 253 by 48 is 12144 ; this written over the denominator is 9344*, which being reduced is 8119. Ans. To multiply a fraction then, is to multiply the number of parts used; hence the rule: multiply the numerator and write the product over the denominator. Note. It is generally most convenient, when the numerator becomes larger than the denominator, to reduce the fraction to a whole or mixed number. It is sometimes necessary to multiply a mixed number. Bought 13 tons of iron, at 9744 dollars a ton; what did it come to? In this example the whole number and the fraction must be multiplied separately. 13 times 97 are 1261. 13 times 14 are , equal to 437; this added to 1261 makes 126544 dollars. Ans. 1261 182 1261 +434= 126534 dolls. Hence, to multiply a mixed number : multiply the whole number and the fraction separately ; then reduce the fraction to a whole or mixed number, and add it to the product of the whole number. That is, XVI. We have seen that single things may be divided into parts, and that numbers may be divided into as many parts as they contain units ; that is, 4 may be divided into 4 parts, 7 into 7 parts, &c. It now remains to be shown, how every number may be divided into any number of equa! parts. If 3 yards of cloth cost 12 dollars, what is that a yard? It is evident that the price of 3 yards must be divided into 3 equal parts, in order to have the price of 1 yard. of 12 must be found. We know by the table of Pythagoras, that 3 times 4 are 12, therefore of 12, or 4 dollars is the price of 1 yard. If 5 yards of cloth cost 45 dollars, what is that a yard ? 1 yard will cost of 45 dollars. 5 times 9 are 45, therefore 9 is į of 45, or the price of 1 yard. The two last examples are similar to the first example Art. 9. If we take 1 dollar for each yard, it will be 5 dollars, then one for each yard again, it will be 5 more, and so on, until the whole is divided. The question, therefore, is to see how many times 5 is contained in 45, and the result will be a number that is contained 5 times in 45. 5 is conlained 9 times, therefore 9 is contained 5 times in 45. This is evident also from Art. III. When a number, therefore, is to be dividel into parts, it is done by division. The number to be divided is the dividend, the number of parts the divisor. and the quotient is one of the parts. A man owned a share in a bank worth 136 dollars, and sold of it; how many dollars did he sell it for ? 136 (2 Ans. 68 dollars. 2 is contained 68 times in 136, therefore 2 times 68 are 136, consequently 68 is of 136. A ticket drew a prize of 2,845 dollars, of which A owned }; what was his share? 2845 (5 Ans. 569 dollars. Since 5 is contained 569 times in 2,845, 5 times 569 are cqual to 2,845, therefore 569 is } of 2,845. Division may be explained, as taking a part of a number. In the above example I say, į of 28(00) is 5(00) and 3(00) over. Then supposing 3 at the left of 4, I say, } of 340) is 6(0) and 4(0) over. Then of 45 is 9. Writing all together it makes 569, as before. The same explanation will apply when the divisor is a large number. Bought 43 tons of iron for 4,171 dollars ; how much was it a ton ? 1 ton is as part of 43 tons, therefore the price of 1 ton will be as part of the price of 43 tons, 4171 (43 97 dollars. Two men A and B traded in company and gained 456 dollars, of which A was to have and B%; what was the share of each? The name of the fraction shows how to perform this example. of 456 signifies that 456 must be divided into 8 equal parts, and 5 of the parts taken. of 456 is 57, 5 times 57 are 285, and 3 times 57 are 171. A's share 285, and B's 171 dollars. |