2.-If five men can do a piece of work in three days, how many days would it take one man to do it? How many days would it take three men? It is plain, that it would take one man five times as long as it would five men; and three, one third as long as it would one man. 3.-In twelve farthings, how many pence? How many in sixteen? How many in twenty? How many in thirtytwo? 4. If a man build forty rods of wall in eight days, how many rods will he build in one day? How many in three days? How many in five days? 5.-A man gave 144 dollars for twelve loads of hay; what did he pay for four loads ? 6.-An orchard consists of one hundred apple-trees, an equal number in each row; how many are there in each row? How many are there in four rows? LESSON TWENTY-THIRD. QUEST. 1.-In two shillings and fourpence, how many pence? How many shillings in forty-eight pence? How many pence in forty-eight farthings ? 2. In a school of 96 scholars, an equal number study arithmetic, grammar, geography, philosophy, chemistry, and history. How many are there in each class? 3.-The parents of a family are each as old as the united age of all the children, six in number; the average age of the children is four more than their number. What is the age of the parents ? 4.-A certain farm cost four hundred dollars, at twenty dollars per acre; how many acres are there? 5.-If two bushels of rye are equal to one bushel of wheat, and two bushels of oats are equal to one bushel of rye, how many bushels of oats are equal to twenty bushels of wheat? SECTION V. FRACTIONS. IF a unit, or a thing, be divided into parts, each of these parts is a fraction of the whole, and takes its name from the number of parts into which the thing is divided. A fraction, therefore, is a part of a thing. Two figures are necessary to express a fraction; they are written, one directly over the other. One half is written thus,; one third thus, . The figure below the line shows into how many parts the thing is divided. It is therefore called the denominator, or namer. The figure above the line shows the number of parts contained in the fraction. It is therefore called the numerator, or numberer. Thus, the 2 shows that the unit is divided into two parts; and the 1, that one of these parts is contained in the fraction. The numerator values the fraction; if, therefore, we increase it, the denominator remaining the same, we increase the value of the fraction. Thus,,; the value of the latter fraction is twice as great as that of the former, because the numerator is twice as large. If we increase the denominator, 2, the numerator remaining the same, we diminish the value of the fraction. Thus,,; the value of the latter is one-half the value of the former, because the denominator is twice as large. If the numerator of a fraction be equal to, or greater than the denominator, the fraction is called an improper fraction, because it is equal to a whole number. Thus, is an improper fraction. The unit is divided into two parts, and these two parts are contained in the fraction. A fraction and a whole number are sometimes written together, thus, 21. This expression is called a mixed number. A compound fraction is written thus, of; it is a fraction of a fraction. LESSON FIRST. Teacher. If a thing, an apple, or a number, be divided into parts, by what general name would each of these parts be called? Scholar. Fraction. T. By what name would each part be called? S. It would depend upon the number of parts into which the thing was divided. T. What do you mean by that? S. That the number of parts into which a thing is divided, gives name to each of those parts. T. Illustrate your meaning. S. If an apple be divided into two parts, or halves, each part is a half of the whole. If it be divided into three parts, each part is a third of the whole. T. Can these halves be again divided into halves? T. What part of the whole would one of this last division be? S. One fourth. T. How do you know? S. The thing was, in the first place, divided into two parts, and then each of these parts was divided into two parts, and twice two are four; therefore, the whole would be divided into four parts, and one of these parts would be one-fourth of the whole. LESSON SECOND. Teacher. You have said that a thing may be divided into two parts, three parts, and four parts. Into how many parts may a thing be divided? Scholar. Into as many as you please. T. And how many of those parts would make a whole ? S. As many as the thing is divided into. T. How do you know? S. It is plain that all the parts into which a thing is di vided, are equal to the thing itself. T. What is the general name of the parts into which a thing is divided ? S. Fractions. T. What then is a fraction? S. It is a part of a thing. T. James divided an apple into three parts, and gave away two of the parts; what part of the apple did he give away? S. Two thirds. T. How do you know? S. If James divide his apple into three parts, these parts are called thirds; if he give away two of these parts, he gives away two thirds. T. James had two apples, and gave away one of them; what part of what he had did he give away? S. One half. T. But a fraction is a part of a thing, and James gave away a whole apple. How will you reconcile this with your definition of a fraction? S. Although he gave away a whole apple, yet as it was a part of what he had, it was a fraction. LESSON THIRD. T. If an apple cost three cents, what will one third of it cost? S. One cent. T. How do you know? S. If an apple cost three cents, one third of it would cost one third of three cents; one third of three is one; therefore one third of the apple would cost one cent. T. If three apples cost one cent, what will one apple cost? S. One third of a cent. T. Give the process. S. One apple will cost one third as much as three apples; if three apples cost one cent, one apple will cost one third of a cent. T. How many halves in one whole? How many wholes in three halves? How many wholes in six thirds ? S. If three thirds make one whole, in six thirds there would be as many wholes as three is contained times in six, which is twice; therefore in six thirds there would be two wholes. T. How many fourths in one whole ? Two is one half of what number? How do you know ? S. Four fourths make a whole, and four times two are eight; therefore eight is that number, of which two be one fourth. T. How many twelfths in four wholes ? One third of three is one fourth of what number? One fourth of four is what part of two? One fourth of four is what part of three ? One fourth of eight is one half of what number? Two fifths of five is how many fifths of ten? Give the process. LESSON FOURTH. T. Three is what part of nine? Three is what part of fifteen? |