Preparatory Explanations for Exercises Group III. This third group of exercises will be a miscellaneous set, relating to the subjects in general which have been brought forward in the present chapter on fractional multiplication and division, including treatment of complex fractions. The teaching already given throughout the chapter should be attended to as preparation for this set of exercises. Remark.-Much saving of arithmetical labour in the bringing out of required results may often be effected by making at an early stage a scrutiny of the mutual relations of the operations at first proposed, and abandoning any operations which may be found mutually counteractive, and simplifying others if any easy means of simplification be noticed. Exam. 1. Required to find the product of 1, 2, and. 19 27 7 45' = 19 19 x 27 x 7 45 × 28 × 19 1 x 3 x 1 1= 5 x 4 x 1 3 20' answ. Here, when the proposed operations are at first indicated by sigus, but not fully carried into effect, we may notice that 19 stands as a factor in both numerator and denominator, and may either be struck out, or 1 may be put instead of it in the writing down of an equal but simpler fractional expression. The 19 in the numerator is a multiplier, and the 19 in the denominator is a divisor, and the two operations by this number, 19, would be mutually counteractive, and so both may be abandoned. Again, we may notice that the factors 27 and 45 in the numerator and denominator are each divisible by 9, and so, dividing them by 9, we find that we may put instead of them 3 and 5. And, further, we may notice that, as the 7 and 28 in numerator and denominator are each divisible by 7, we may put instead of them 1 and 4. So the required product comes to be. If we had at first proceeded to multiply together the factors 19, 27, and 7 in the numerator, and to multiply together the factors 45, 28, and 19 in the denominator, we would have got a true expression for the required product; but it would have been in very large numbers, 35, and would have required much trouble for the reducing of it to its lowest terms. It is much easier to find modes of simplification before the factors in numerator and denominator are multiplied together than after. Remark. When mixed numbers are given or otherwise occur as factors, they are usually more easily managed by being reduced to fractional form. Exam. 2. To find the product of 21, 27, 11, and. 1 31 Here, reducing mixed numbers to fractional form, for 2 we put 5, for 1 we put 10, and for 3 1 31 = 27 10 X 20 = 5 x 27 x 10 x 4 2 x 20 x 9 × 13 5 x 3 x 1 x 4 = 2 x 2 x 1 x 13 5 x 3 13 15 = or 1, answ. 13' = Group III, of Exercises. Exer. 58. A sum of money, amounting to £159 - 10 - 6, is to be divided among four persons so that the first three shall receive equal shares, and that the fourth person shall receive of one of those equal shares. Find one of the three equal shares; and also find the smaller share. The following suggestions may be helpful:-If one of the equal shares be called X, we may see that 3 of X will amount to the whole sum of £159-106. We may write this as an equation thus: Then dividing both sides of this equation by 35, we get So in order to find one of the equal shares we have to divide £159 - 10 6 by 35; and, to do so, we may multiply that sum of money by the inverse of 35. 59. If the circumference of a carriage wheel be 13 feet, how often will it turn in going a mile, the mile being 5280 feet? The following suggestions may give aid :-If x be put to denote the number of turns (or rather the numeric, integral or fractional, expressing the number of turns and fraction of a turn) we havex × 13=5280; and dividing both sides of this equation by 134, we get x = 5280 ÷ 131. 60. A safety valve, arranged so that its area exposed to the action of the steam can be very exactly ascertained, is found to expose 29 square inches to the steam. It is loaded with 1350 lbs. What pressure per square inch will just suffice to balance this load and make the valve be on the point of rising? The following suggestions may be useful :-If x be the number (or rather the numeric) expressing the pressure in pounds on one square inch, then there will be x x 29 lbs. total pressure of steam on the 29 square inches, and this is to be equal to 1350 lbs. Or we may note that x × 293 = 1350; and therefore x=1350÷292. 61. A link in a chain of a suspension bridge has a cross-sectional area of 9 square inches. It is subjected to a pull of 42 tons. What is the pull in tons per square inch, on the supposition that the tensile stress is uniformly distributed throughout the cross-section? 422 The following suggestions may afford aid. We may state that there is a rate of 421 tons per 92 square inches; and then, dividing both terms of this rate by 93, we get as an equivalent rate per 1 square inch. tons 62. Find a sum of money such that of it shall be £89 - 14 - 6. 63. Find a sum of money such that 53 of it shall be £56 - 10 - 4. 61. Find a sum of money such that 5 of it shall be £77 - 3 - 5. 65. Disply £98 - 14 - 6 by . 66. Ply £3 - 10 - 6 by 4 of 31, and disply the result by 7. 67. Find in a simple form the ratio of to 12. 68. Find in a simple form the ratio of to 13. 69. Find in a simple form the numeric that is of 4. 70. Simplify the complex fraction 3 20 71. The capacity of a cistern is 623 gallons. water to it at the rate of 25 gallons per minute. will be required for the pipe to fill the cistern? 72. A pipe delivering water uniformly fills, in cistern whose capacity is known to be 26 gallons. of flow in gallons per minute? 100 73. Find in simple form the inverse of 75 A pipe supplies 153 minutes, a What is the rate 75. Six persons, A, B, C, &c., have joined originally as partners all alike in the purchase of a ship, with the arrangement that the profits accruing from time to time are to be divided among them in equal shares. Afterwards B, wishing to retire from the business, sells his share in the ship to A; and C, being in want of some ready money, sells of his share to A; so that A now owns 23 shares in the ship. In a division of the profits, the sum paid to A, as his 2 shares of the profit, was £585 - 86. What was the amount of one of the equal shares of the profit on that occasion? 76. Ply 25 by, and ply the result by the inverse of 21, and disply the new result by 18 77. Multiply 34 by 5, and multiply the result by the reciprocal of 17%, and divide the new result by 1000 78. What is the fraction, or what is the numeric, that 35 is of 84? 79. What is the numeric that 84 is of 35? 84. Five dozen eggs are packed equally into three baskets. State by number or fraction of dozen (or, in better words, state by numeric of dozen) the eggs in each basket. Next let each basket go for 5 persons. How many eggs will there be for each person? 85. Divide 22 by 41. 86. Multiply 23 by 41. 87. Divide by . 88. Divide by g. 89. Divide by 2. 90. Supposing gun-metal to be composed of 90 parts of copper to 9 parts of tin by weight, find how much tin (stated as a fraction of a pound) there is per pound of copper. 91. Also find how much tin and how much copper there is per pound of the gun-metal. 100 92. The metre is 328 feet British approx. Express 1 foot as a fraction of a metre. DECIMAL FRACTIONS. A DECIMAL FRACTION, often for brevity called a DECIMAL, is a fraction whose denominator is 10, or some number produced by the continued multiplication of 10 as factor, such as 100, 1000, &c.* * When a number is multiplied by itself, so that it enters just twice as a factor in making a product, the product is called the second power of the number. When a number is multiplied by itself, and the product again is multiplied by the number to make a final product, so that the number enters just three times as a factor into the product, the product is called the third power of the number. In like manner the product made by taking one same number four times as a factor is called the fourth power of the number; and so on. Thus 10 x 10, or 100, is the second power of 10; 10 x 10 x 10, or 1000, is the third power of 10; 10x 10 x 10 x 10, or 10,000, is the fourth power of 10; and so on. Instead of the word number throughout the above statement, the word numeric might properly have been used. It is to be understood that in that statement the word "number" has been employed in the wider but less proper sense in which it is customarily often employed, being taken to mean, not only any proper number, but also any numeric, whole or fractional, greater or less than unity, or unity itself. Thus x 2, or 4, is the second power of ; × ×, or is the third power of; and so on. Further information on the subject of powers will be given in the chapter on Powers and Roots, farther on in this treatise. 3 475 100 75 Thus 10 100 100000 105, and 2475 are decimal fractions. The last two of these, it will be noticed, are numerics of the kind commonly called improper fractions, being each of them greater than unity. They might otherwise be noted as 45 and 24,75. In this way there is exhibited in each of them an integral part; in the one it is 4, and in the other it is 24, and in each there is exhibited a fractional part, 75, which is less than unity. Often, in commonly used language, the entire fractional expression, though it be greater than unity, is spoken of as a decimal fraction, or a decimal; and sometimes the integral part is regarded and spoken of as a whole number or integer, and only the fractional part less than unity is called the decimal fraction. Our ordinary language is rather imperfectly expressive in this matter, not only in reference to decimal fractions but also to fractions in general. Usually, however, the meaning of the writer or speaker may be gathered sufficiently from collateral statements. All the rules for the management of fractions in general are, of course, applicable in regard to decimal fractions. A specially simple and easy mode of notation, however, is available for decimal fractions, which renders their management usually much easier than it would be by employment of those methods alone which are applicable to fractions in general. This mode of notation, which may be called the decimal notation as applied not only to numbers proper but to fractions, may be explained thus:-If we write down a line of consecutive figures, 542 for instance, we have the figure 2 at the right hand meaning simply 2 of whatever objects or units are reckoned, while the figure 4 next on the left means 4 tens, or 40 of them; and the figure 5 next farther towards the left means 5 groups of a hundred each, or 500 of the objects or units. We may thus notice that in passing from figure-place to figure-place in the direction from left to right, a figure in any place expresses a tenth of what the same figure would do in the place before; and we have to observe that in the usual notation for expressing a "whole number" we recognize the units place merely by its being the extreme place at the right-hand side. But, further, in order to allow of an extension of the same decimal system of notation being made, so that the extended system may serve to express decimal fractions alike with decimally grouped numbers proper, it is only necessary to provide some other way of indicating the units place, which shall not make that place necessarily be the last or terminal place in proceeding from left to right; and then to arrange that the row of figures may be extended as far as we please to the right of the units place, subject to the same convention as before, that a figure in any place shall express a tenth of what it would express in the place next on the left of it. The units place might be marked in any convenient way. It might, for instance, be indicated by a crescent marked over it. Thus the expression 54236 would mean 500 + 40+2++, or, 542,36, or For the purpose of suggesting readily the corresponding relations of figures equally distant to the left and to the right of the units figure, a mark placed over or under the units figure would be 54236 100 |