months, B put in $375 for 12 months, and C put in $1125 for 16 months. They gained $860. What was each man's share of the gain ? First Solution. Therefore, A should have, or, of the gain, = $215. of the gain, = $129. = $516. Second Solution. The use of $ 750 for 10 mo. is worth the use of $7500 for 1 mo. The use of $375 for 12 mo. is worth the use of $ 4500 for 1 mo. The use of $1125 for 16 mo. is worth the use of $18000 for 1 mo. Use of whole stock is worth the use of $30000 for 1 mo. Therefore, A should have, or, of the gain, = $215. When the stocks of the several partners are convenient multiples or fractional parts of each other, a very neat solution can be given. Thus, in the above example, by noticing that B's stock equals of A's, and that C's stock equals of A's, we may have the following: Third Solution. The use of A's stock 10 mo. use of 10 times A's stock for 1 mo. The use of B's, or of A's stock, 12 mo. use of 42, or 6 times A's stock for 1 mo. The use of C's, or of A's stock, 16 mo. use of 48, or 24 times A's stock for 1 mo. Use of whole = use of 10 + 6 + 24, or 40 times A's stock for 1 mo. Therefore A should have 18, or ; B, or 2; and C 24, or %, of the gain, which will give the same answer as before. 2. Charles French, Francis Baker, and Otis Atherton traded in company, under the name of Charles French & Co. French put in $1000 for 20 mo., Baker put in $800 for 16 mo., and Atherton put in $500 for 20 mo. They gained $1500. How many dollars of the gain ought each to receive? 3. George Jackson, William Leach, and Albert Buffington traded in company. Jackson put in $144 for 6 mo., Leach put in $72 for 7 mo., and Buffington put in $216 for 6 mo. 20 da. They gained $114. What was each man's share of the gain? 4. A, B, C, and D hired a pasture together, in which A pastured 4 cows 13 weeks, B pastured 5 cows 16 weeks, C pastured 8 cows 10 weeks, and D pastured 4 cows 16 weeks. The rent of the pasture was $102. How many dollars ought each man to pay? 5. Samuel Austin, Jacob Brown, and Moses Sumner formed a partnership for 2 years, under the name of Samuel Austin & Co. Austin at first paid in to the stock $1000, but after 8 mo. had elapsed he paid in $500 more. Brown at first paid in $1250, and 16 mo. afterwards he paid in $250 more. Sumner at first paid in $1500, but at the end of 16 mo. he took out $500. They gained $3600. What was each man's share of the gain? Solution. Interest of $1000 for 8 mo. =$ 40 Interest of $1500 for 16 mo. = $120 } = $160 int. Austin's stock. Interest of $1250 for 16 mo. — $100 = $160=int. Brown's stock. Interest of $1500 for 8 mo. - = $ 60 Interest of $1000 for 8 mo. = =$ 40 }} = $160 int. Sumner's stock. By this, it appears that the interests of their respective stocks, for the time they were in trade, were alike. equally, and each partner should have Hence, the gain should be divided of $3600, which is $1200. character to those given to the NOTE. Other solutions similar in first example might have been added; but as the pupil can readily discover them, they have been omitted. 6. Joseph Southwick, Francis Lowe, and Henry Taft formed a partnership for 3 years, under the name of Southwick, Lowe, & Taft. When they commenced business, each partner put in $3000; but at the end of the first year Southwick put in $3000 more, and Lowe withdrew $1500. At the end of the second year, Southwick withdrew $2000, and Lowe put in $4000, and Taft put in $2000. When the partnership expired, they found that they had gained $9000. What was each partner's share of the gain? 7. S. Gamwell, C. Grover, R. Wheelock, and W. Godding formed a partnership, under the title of Gamwell, Grover, & Co. Gamwell at first put in $8000, but at the end of 6 mo. he withdrew $2000, and at the end of 12 mo. he withdrew $1000 more. Grover at first put in $6000, but at the end of 10 mo. he put in $3000 more. Wheelock put in $7000. Godding at first put in $10,000; at the end of 6 mo. he withdrew $2000, and at the end of 14 mo. he put in $4000. At the end of 2 years they found that they had gained $12,000. What was each man's share of the gain? SECTION XVI. POWERS AND ROOTS. 211. Definitions. (a.) THE product of a number taken any number of times as a factor is called a POWER of the number.. See 105, (d.) (e.) (f) and Note. (b.) A ROOT of any number is such a number as, taken some number of times as a factor, will produce the given number. (c.) If the root must be taken twice as a factor to produce the number, it is the SQUARE ROOT, or the SECOND ROOT; if three times, it is the CUBE ROOT, or the THIRD ROOT; if four times, it is the FOURTH ROOT; &c. Thus, 2 is the square root of 4, the third root of 8, the fourth root of 16, &c., because 22 4, 23 = 8, 24= 16; &c. (d.) The character √, called the RADICAL SIGN, is used to indicate that the root of the number over which it is placed is to be extracted. (e.) The DEGREE of the root is indicated by a small figure, called an INDEX, which is placed a little above and at the left of the sign. When no index is written, the square root is required. Thus, 4, or 2/4, means the square root of 4. 243 means the fifth root of 243. 73 means the fourth root of the 3d power of 7. (f) We may also indicate that a root is to be extracted, by using a fractional exponent. Thus, 99; (125) = 125; 27/272; &c. (g.) The process of finding the powers of numbers is called INVOLUTION, and the process of finding their roots is called EVOLUTION, or the EXTRACTING OF ROOTS. 212. Relation which the Denominations of a Square bear to those of its Root. (b.) The above table shows that, First. There are below 100 but 9 entire numbers which are perfect squares. Second. The entire part of the square root of any number below 100 will be less than 10, and therefore contain but 1 figure; of any number between 100 and 10,000 will lie between 10 and 100, and therefore contain 2 figures; between 10,000 and 1,000,000 will lie between 100 and 1000, and therefore will contain 3 figures; &c. 1. How many figures are there in the entire part of the square root of 865698 ? Answer. Since 865698 lies between 10,000 and 1,000,000, its root must lie between the roots of those numbers, i. e., between 100 and 1000, and must therefore contain 4 figures in its entire part. How many figures are there in the entire part of the root of 2. 69748769 ? 4. 12496743297 ? 3. 486497950068 ? 5. 5847695329 ? 213. Division into Periods. (a.) As the square of 10 is 100, of 100 is 10,000, &c., it follows that the square of any number of tens will be some number of hundreds; of any number of hundreds will be some number of ten thousands, &c.; or, in other words, that the square of tens will give units of no denomi nation below hundreds; the square of hundreds will give units of no denomination below ten-thousands; &c. (b.) Hence, the two right hand figures of any number will contain no part of the square of the denominations of the root above units; the four right hand figures will contain no part of the square of those above tens, &c. (c.) Therefore, if we should begin at the right of any number, and separate it into periods of two figures each, the number of periods would be the same as the number of figures in its square root. The square of the highest denomination of the root would be found in the left hand period; the square of the two highest denominations would be found in the two left hand periods; &c. 1. Separate 8478695 into periods, and explain their uses. Answer. 8478695. The left hand period, 8, contains all of the square of the thousands of the root; the two left hand periods, 847, the square of the thousands and hundreds ; &c. Separate each of the following numbers into periods, and explain their uses : — 2. 5794865. 3. 89475948. 4. 375486792. 5. 32500675. 214. Method of forming a Square. (a) To find a law of universal application in squaring or extracting the square roots of numbers, we will use the letter a to represent any number whatever, and b to represent any other number. |