how many parts the thing or number is divided, and the number above the line shows how many of the parts are used. Thus of an orange signifies, that the orange is divided into three equal parts, and that two of the parts or pieces are used. of a yard of cloth, signifies that the yard is supposed to be divided into five equal parts, and that three of these parts are used. The number below the line is called the denominator, because it gives the denomination or name to the fraction, as halves, thirds, fourths, &c. and the number above the line is called the numerator, because it shows how many parts are used. We have applied this division to a single thing, but it often happens that we have a number of things which we consider as a bunch or collection, and of which we wish to take parts, as we do of a single thing. In fact it frequently happens that one case gives rise to the other, so that both kinds of division happen in the same question. If a barrel of cider cost 2 dollars, what will į of a barrel cost ? To answer this question, it is evident the number two must be divided into two equal parts, which is very easily done. į of 2 is 1. Again, it may be asked, if a barrel of cider cost 2 dollars, what part of a barrel will one dollar buy? This question is the reverse of the other. But we have just seen that 1 is į of 2, and this enables us to answer the question. It will buy į of a barrel. If a yard of cloth cost 3 dollars, what will { of a yard cost ? What will of a yard cost ? If 3 dollars be divided into 3 equal parts, one of the parts will be 1, and two of the parts will be 2. Hence s of a yard will cost 1 dollar, and ș will cost 2 dollars. If this question be reversed, and it be asked, what part of a yard can be bought for 1 dollar, and what part for 2 dollars; the answer will evidently be į 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 in it be asked, what part of 5, 3 is, we say, of 5, because 1 is } 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 labourer for 15 dollars a month ; at the end of the time agreed upon, he paid him 143 dollars. How many months did he work? 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 its of a month, consequently 8 dollars will pay for $5 of a month. Ans. 9,8 months. Notc. 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. 2544 tons. Remainder 42 dollars. Operation. 563times It is contained 56 times and 220 over. I is it of 324, and 220 is 32: of 324. Ans. 56 times and jij of another time. From the above examples, we deduce the following gene z al 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 cents ? The 6 which stands in tens' place shows how many times ten is contained in 60, for 60 signifies 6 tens, 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 64 pounds. A man has 2347 lb. of tobacco, which he wishes to put into boxes containing 100 lb. 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 47 lbs. left, which will fill 4% of another box. The operation may be performed thus, 23.47. Answer 2347. 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, fc. 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 annered to the quotient. XII. We observed in Art: X, that any two numbers being given, it is easy to tell what part of the one the other is. Thus : 15, &c. What part of 10 yards are 3 yards ? Ans. 1 is it of 10, and 3 is to of ten. What part of 237 barrels is 82 barrels ? Ans. 1 is zit nf 237, and 82 is sy 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 numerator is equal to the denominator, as ž, s, š, 15, 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 ], 44, 45, are improper fractions. of finding what part of one number another number 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 . The ratio of 5 to 3 is . What part of 3 is 5? Answer: The ratio of 3 to 5 The process 1 is What is the ratio of 35 yards to 17 yards. Answer 17. What is the ratio of 17 to 35 ? Answer 34. To find what part of one number another is, make the number which is called the part (whether it be the larger or smaller) the numerator of a fraction, and the other number the Jenominator. Also to find the ratio of one number to another, make the rumber which is expressed 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 ovex That is, * make 3 dollars, and there are of another dolJar. Ans. 3 dollars. If 1 man consume 3'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 537. of a barrel nak 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 is of another barrel. Numbers like 3, 151%, which contain a whole number and a fraction, are called mixed numbers. The above process by which was changed to 3, and to 15}}, is called reducing improper fractions to whole or mired numbers. Since the denominator always shows how many of the parts make a whole one, it is evident that any improper frac tion may be reduced to a whole or mixed number, by the fol lowing 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 1 of a dollar. A man distributed 18 bushels of wheat among some poor persons, giving them 1 of a bushel each ; how many persons were there? |