for 2 dollars; the answer will evidently be of a yard for 1 dollar, and for 2 dollars. It is easy to see that any number may be divided into as many parts as it contains units, and that the number of units used will be so many of the parts of that number. Hence if it be asked, what part of 5, 3 is, we say, of 5, because 1 is į of 5, and 3 is three times as much. We can now answer the question proposed above, viz. How many yards of cloth, at 6 dollars a yard, may be bought for 45 dollars ? 42 dollars will buy 7 yards, and the other 3 dollars will buy of a yard. Ans. 7 yards, which is read 7 yards and of a yard. A man hired a laborer for 15 dollars a month ; at the end of the time agreed upon, he paid him 143 dollars. How many months did he world? Operation. 143 (15 Price of 9 months 135 9 months. Remainder 8 The wages of 9 months, is 135 dollars, which subtracted from 143, leaves 8 dollars. Now 1 dollar will pay for is of a month, consequently 8 dollars will pay for its of a month. Ans. 9 months. Note. The number which remains after division, as 8 in this example, is called the remainder. At 97 dollars a ton, how many tons of iron may be bought for 2467 dollars ? Operation. 2467 (97 250 tons. After paying for 25 tons, there are 42'dollars left. 1 dollar will buy o'y of a ton, and 42 dollars will buy *** of a ton. How many times is 324 contained in 18364 ? Operation. 564 times. of winou Remainder 220 It is contained 56 times and 220 over. 1 is 324, and 220 is of 324, Ans. 56 times and 1 o of another time. From the above examples, we deduce the following general rule for the remainder : When the division is performed, as far as it can be, if there is a remainder, in order to have the true quotient, write the remainder over the divisor in the form of a fraction, and annex it to the quotient. XI. We observed in Art. V. that when the multiplier is 10, 100, 1000, &c. the multiplication is performed by annexing the zeros at the right of the multiplicand. In like manner when the divisor is 10, 100, 1000, &c. division may be performed by cutting off as many places from the right of the dividend as there are zeros in the divisor. At 10 cents a pound, how many pounds of meat may be bought for 64 certs? The 6 which stands in ten's place shows how many times ten is contained in 60, for 60 signifies 6 teos, and the 4 shows how many the number is more than 6 tens, therefore 4 is the remainder. The operation then may be performed thus 6.4. The answer is 610 pounds. Å man has 2347 lbs. of tobacco, which he wishes to put into boxes containing 100 lbs. each ; how many boxes will it take? It is evident that 100 is contained in 2300, 23 times, consequently it will take 23 boxes, and there will be 23106 47 lbs. left, which will fill 16 of another box. The operation may be performed thus, 23.47. Answer In general if one figure be cut off from the right, the tens will be brought into the units' place, and hundreds into the tens' place, &c. If two figures be cut off, hundreds are brought into the units' place, and thousands into the tens' place, &c. And if three figures be cut off, thousands are brought into the units' place, &c. that is, the numbers will be made 10, 100, or 1000 times less than before. Hence to divide by 10, 100, 1000, &c. cut off from the right of the dividend as many figures as there are zeros in the divisor. The remaining figures will be the quotient, and the figures cut off will be the remainder, which must be written over the divisor, and annexed to the quotient. XII. We observed in article X, that any two numbers being given, it is easy to tell, what part of the one the other is. Thus : . 10, and 3 is x of 10. What part of 237 barrels is 82 barrels ? Ans. 1 is zi, of 237, and 82 is zo of 237. Fractions are properly parts of a unit, but by extension the term fraction is often applied to numbers larger than unity. This happens when the numerator is larger than the denominator, in which case there are more parts taken than are sufficient to make a unit. All fractions in which the oumerator is equal to the denominator, as j, š, $, i}, &c. are equal to unity; all in which the numerator is less than the denominator are less than unity, and are called proper fractions; all in which the numerator is greater than the denominator, are more than unity, and are called improper fractions. Thus, *, *, are improper fractions. The process of finding what part of one number another is, is called finding their ratio. What is the ratio of 5 bushels to 3 bushels, or of 5 to 3? This is the same as to say, what part of 5 is 3? The answer is š. The ratio of 5 to 3 is What part of 3 is 5? Answer . The ratio of 3 to 5 is What is the ratio of 35 yards to 17 yards ? Answer To find what part of one number another is, make the number which is called the part (whether it be the larger or the smaller) the numerator of a fraction, and the other number, the denominator. Also to find the ratio of one number to another, make the number which is erpressed first the denominator, and the other the numerator. XIII. A gentleman gave } of a dollar each to 17 poor persons; how many dollars did it take? It took * of a dollar. But of a dollar, make a dollar, consequently as many times as 5 is contained in 17, so many dollars it is. 5 is contained 3 times in 17, and 2 over. That is * make 3 dollars, and there are of another dollar. Ans. 3f dollars. If 1 man consume z's of a barrel of flour in a week, how тапу barrels will an army of 537 men consume in the same time? They will consume . t of a barrel make a barrel, therefore as many times as 35 is contained in 537, so many barrels it is. 537 (35 12 35 is contained 15. times in 537 and 12 over, which is of another barrel. Numbers like 3, 154%, which contain a whole number and a fraction, are called mixed numbers. The above process by which was changed to 3}, and to 153}, is called reducing improper fractions to whole or mixed numbers. Y 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 anner it to the quotient and it will form the mixed number required. X 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. Tai is, 3 dollars are equal to of a dollar. A man distributed 184 bushels of wheat among some poor persons, giving them of a bushel each; how many persons were there? This question is the same as the following: In 18ş bushels, how many — of a bushel ? That is, how many 7ths of a bushel ? In 1 bushel there are, consequently in 18 bushels there are 18 times 7 sevenths; that is, 146, and more make 1. Ans. 129 persons. Reduce 283 to an improper fraction. That is, in 283how many ? Since there in 1, in 28 there must be 28 times aj many. 28 times 25 are 700, and 17 more are 717. Ans. 713. 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. are 2 5 |