4. What is meant by inverse proportion? 5. What is meant by the Single Rule of Three ? What is the general rule for stating questions in the Rule of Three ? How is the answer then found? If the first and third terms be of different denominations, what is to be done? What, if there are different denominations in the second term? Of what denomination will the quotient be? What, if the quotient be not of the same denomination of the required answer? What is the method of proof in this rule? 6. What is compound proportion? By what other name is it called? What is the rule for stating questions in compound proportion for performing the operation ? 7. What is Fellowship? What is meant by capital or stock? What by dividend? What is the rule when the times are equal? What, when they are unequal? What is the method of proof? 8. What is Alligation? What is Alligation Medial?-Alligation Alternate ? What is the rule for finding the proportional quantities to form a mixture of a given rate? Explain by analysis of an example. When the whole composition is limited to a certain quantity, how would you proceed? How, when one of the simples is limited to a certain quantity? How is Alligation proved 9. What is Barter? What is meant by a tax? What is the common method of making out taxes? SECTION VII. Fractions. DEFINITIONS. 214. 1. Fractions are parts of a unit, or of a whole of any kind. If any number, or particular thing, be divided into two equal parts, those parts are called halves; if into 3 equal parts, they are called thirds; if into 4 equal parts, they are called fourths, or quarters (11); and, generally, the parts are named from the number of parts into which the thing, or whole, is divided. If any thing be divided into 5 equal parts, the parts are called fifths; if into 6, they are called sixths; if into 7, they are called sevenths; and so on. These broken, or divided quantities are called fractions. Now if an apple be divided into five equal parts, the value of one of those parts would be one fifth of the apple, and the value of two parts two fifths of the apple, and so on. Thus we see that the name of the fraction shows, at the same time, the number of parts into which the thing, or whole, is divided, and how many of those parts are taken, or signified by the fraction. Suppose 1 wished to give a person two fifths of a dollar; I must first divide the dollar into five equal parts, and then give the person two of these parts. A dollar is 100 cents-100 cents divided into 5 equal parts, each of those parts would be 20 cents. Hence, one fifth of 100 cents, or of a dollar, is 20 cents, and two fifths, twice 20, or 40 cents. The tediousness and inconvenience of writing fractions in words has led to the invention of an abridged method of expressing them by figures. One haf is written, one third,, two thirds,, &c. The figure below the ne shows the number of parts into which the thing, or whole, is divided, and the figure above the line shows how many of those parts are signified by the fraction. The number below the line gives name to the fraction, and is therefore called the denominator; thus, if the number below the line be 3, the parts signified are thirds, if 4, fourths, if 5, fifths, and so on. The number written above the line is called the numerator, because it enume rates the parts of the denominator signified by the fraction. As there are no limits to the number of parts into which a thing, or whole, may be divided, it is evident that it is possible for every number to be a numerator, or a denominator of a fraction. Hence the variety of fractions must be unlimited. 2. Fractions are of two kinds, Vulgar and Decimal, which differ in the form of expression, and the modes of operation. 3. A Vulgar Fraction is expressed by two numbers, called the nume rator and denominator, written the former over the latter, with a line between, as, the former before the latter, as 3-8-1. 4. A Decimal Fraction, or a Decimal, is a fraction which denotes parts of a unit which become ten times smaller by each successive division (113), and is expressed by writing down the numerator only. (See Part II. Sect. III). A decimal is read in the same manner as a vulgar fraction; thus 0.5 is read 5 tenths, 0.25 25 hundredths, and it is put into the form of a vulgar fraction by drawing a line under it, and writing as many ciphers under the line as there are figures in the decimal, with a 1 at the left hand; thus, 0.5 becomes f, 0.25, f, and 0.005, TOOO VULGAR FRACTIONS 215. 1. A proper fraction is one whose numerator is less than its denominator; as, 1, }, &c. (23). 2. An improper fraction is one whose numerator is greater than its denominator; as,,,, &c. (24). 3. The numerator and denominator of a fraction are called its terms (30). 4. A compound fraction is a fraction of a fraction; as, of. 5. A mixed number is a whole number and a fraction written together, as 121, and 61 (23). 6. A common divisor, or common measure, of two or more numbers, is a number which will divide each of them without a remainder. 7. The greatest common divisor of two or more numbers, is the greatest number which will divide those numbers severally without a remainder. 8. Two or more fractions are said to have a common denominator, when the denominator of each is the same number (25). 9. A common multiple of two or more numbers is a number, which may be divided by each of those numbers without a remainder. The least common multiple is the least number, which may be divided as above. 10. A prime number is one which can be divided without a remainder, only by itself, or a unit. 11. An aliquot part of any number, is such part of it, as being taken a certain number of times, will exactly make that number. 12. A perfect number is one which is just equal to the sum of all its aliquot parts. The smallest perfect number is 6, whose aliquot parts are 3, 2, and 1, and 3+2+1-6; the next is 28, the next 496, and the next 8128. Only ten perfect numbers are yet known. 216. WHOLE NUMBERS, CONSIDERED UNDER THE FORM OF FRACTIONS. ANALYSIS. 1. Change 16 to a whole or mixed number. 3)76 As the denominator denotes the number of parts 25 into which the whole, or unit, is divided, and the numerator shows how many of those parts are contained in the fraction (22), there are evidently as many wholes, as the number of times the numerator contains the denominator; or, otherwise, since every fraction denotes the division of the numerator by the denominator (129), where the numerator is greater than the denominator, we have only to perform the division which is denoted. 217. To change an improper fraction to an equivalent whole or mixed number. RULE.-Divide the numerator by the denominator, and the quotient will be the whole, or mixed number required. 1. Change 25 to an improper fraction. 25X3+1 3 denotes the division of 1 by 3, (129); if now we multiply 25 by 3, and add the product to 1, making (25×3+1=) 76, and then write the 76 over 3, thus, 76 6, we evidently both multiply and divide 25 by 3; but as the multiplication is actually performed, and the division only denoted, the expression becomes an improper fraction. A whole number is changed to an improper fraction, by writing 1 under it, with a line between. 218. To change a whole or mixed number to an equivalent improper fraction. RULE.-Multiply the whole number by the denominator of the fraction, add the numerator to the product, and write the sum over the denominator for the required fraction. QUESTIONS FOR PRACTICE 2. Change 26 to a mixed number. 3. Change 24 to a mixed number. 4. In 236s. shillings, how many shillings? 5. In 24 of a week, how many weeks? 2. Change 83 to an improper fraction. 3. Change 273 to an improper fraction. 4. In 1988. how many 12ths? 5. In 3 weeks, how many 7ths? 219. MULTIPLICATION AND DIVISION OF FRACTIONS BY WHOLE NUMBERS. ANALYSIS. 1. James had of a peck of plums, and Henry had twice as many; how many had Henry? 1. Henry had of a peck of plums, which were twice the quantity James had; how many had James? Here we have evidently to divide Here we have evidently to multiply by 2; but two times is ; hence, to multiply by 2, we multi-into 2 equal parts; but & divided into 2 parts, one of them is ; then to divide by 2, we must divide the numerator by 2, and write the quotient, 1, over 4, the denominator; or, otherwise, if we multiply 4, the denominator, by 2, and write the product, 8, under 2, the numerator, thus, &, the fraction becomes divided by 2; for while the number of parts remains the same, the multiplication has rendered the parts only half as great; and these results, ply the numerator 2 by 2, and write the product, 4, over 8, the denominator; or, otherwise, if we divide 8, the denominator, by 2, and write the quotient, 4, under 2, the numerator, thus,, the fraction becomes multipled; for while the number of parts signified remains the same, the division has rendered those parts twice as great; and these results, and, are evidently the same in value, though differing in the magnitude of the terms. There fore 220. To multiply a fraction by a whole number. RULE.-Multiply the nume and, are evidently the same in value, though expressed in different terms. Hence 221. To divide a fraction by a whole number. RULE.-Divide the numera rator, or divide the denomina-tor, or multiply the denominator, of the fraction by the whole number; the result will be the product required. tor, of the fraction by the whole number; the result will be the required quotient. QUESTIONS FOR PRACTICE. 2. What is the product of| 2. How many times 24 in by 24?-of by 32?-of 12?-32 in 160 ?-36 in 18a? by 36?-of by 42?-of-42 in 136-9 in 27? by 3? 3. How many times 5 in 3. How many are 5 times?-3 in ?-14 in 1?—7 ?-3 times ?-14 times in f, or 5? ?-7 times ? 4. If 5 lb. of rice cost of a dollar, what will 1 lb. cost? 5. If 6 bushels of wheat of a dollar, what is it a cost bushel ? MULTIPLICATION BY FRACTIONS. 222. worth? ANALYSIS. If a load of hay be worth $12, what are of it Here 12 and are evidently two factors, which, multiplied together, will give the price; and since the result is the same, whichever is made the multiplier (86), we may make the multiplicand, and proceed (220) thus, X 1228 dollars. Ans. Otherwise, since in the multiplication by a whole number, the multiplicand is repeated as many times as the multiplier contains units, if therefore the multiplier be 1, the multiplicand will be repeated one time, and the product will be just equal to the multiplicand; if the multiplier be, the multiplicand will be repeated half a time, and the product will be half the multiplicand; if the multiplier be, it will be repeated one third of a time, and the product will be one third of the multiplicand, and generally, multiplying by a fraction is taking out such a part of the multiplicand as the fraction is part of a unit. Hence the product of 12 by , is of 12; and to find of 12, we must first find of 12, by dividing 12 by 3, and then multiply this third by 2; thus, 12-3-4, and 4x2=8; 88 then are of $12, or the product of $12 by, as by the former method. Therefore, 223. To multiply a whole number by a fraction. RULE. Divide the whole number by the denominator of the fraction, and multiply the quotient by the numerator, or multiply the whole number by the numerator, and divide the product by the denominator. QUESTIONS FOR PRACTICE. 2. What is the product of 4 multiplied by ?-of 7 multiplied by ?of 9 by? of 17 by ? 3. If a barrel of rum cost $24, what cost of it? Ans. $18. 4. What cost 18 bushels of corn, at of a dollar a bushel ? 224. Ans. $6. 5. If a bushel of pears cost 75 cents, what cost of them? Ans. 15 cts. 6. What is the product of 16 by ?—256 by ?—of 12 by ? NOTE. It will be observed from the above examples, that multiplication by a proper fraction gives a product which is less than the multiplicand (121). MULTIPLICATION OF ONE FRACTIONAL QUANTITY 1. A person owning of a gristmill, sold of his share; what part of the whole mill did he sell? Here we wish to take out of, which has been shown (222) to be the same as multiplying by; but to multiply by a fraction, we must divide the multiplicand by the denominator, and multiply the quotient by the nu merator; is divided by 3, by multiplying the denominator 4 by 3 (121), |