2. A and B gained by trading $182. A put into stock $300, and B $400. What is the gain of each? Ans. A's gain, $78, B's gain, $104. 3. A merchant failing in trade, owes A $600, B $760, C $840, D $800. His effects are sold for $2275. What will each receive of the dividend? TO FIND THE GAIN OR LOSS OF INDIVIDUALS DOING BUSINESS IN PARTNERSHIP, WHEN THEIR STOCK CONTINUES IN TRADE DURING UNEQUAL TIMES. RULE.-Multiply each man's share of the stock by the time it continued in trade, and proceed according to the preceding rules. 1. Three graziers hired a piece of land for $60,5. A put in 5 sheep for 4 months, B put in 8 for 5 months, and C put in 9 for 6 months; how much must each pay of the rent? 5X41 22,5 8X5=40, 9X61=58,5 121, 60,51,5 ,5X22,5=$11,25, A's share. ,5X40, $20, B's share. ,5X58,5 $29,25, C's share. 2. Two persons hired a coach to go 40 miles, for $20, with permission to take in two more wl:en they pleased. Now, when they had gone 15 miles, they admit C, who wished to go the same route; and on their return, within 25 miles of home, they admit D for the remainder of the journey. Now, as each person was to pay in proportion to the distance he rode, it is required of you to settle the coach-hire between them. Ans. A and B, $6,4 each, C, $5,2, and D, $2. 3. Two merchants enter into partnership for 18 months; A put into stock at first $200, and at the end of 8 months he put in $100 more; B put in at first $550, and at the end of 4 months took out $140. Now, at the end of the time they find they have gained $526; what is each man's share? Ans. A's, $192,957, B's, $333,041184 4. A, with a capital of $1000, began trade, Jan. 1, 1776, and meeting with success in business, he took in B a partner, with a capital of $1500, on the 1st of March following. Three months after that, they admit C as a third partner, who brought into stock $2800; and after trading together till the first of the next year, they find the gain, since A commenced business, to be $1776,5. How must this be divided among the partners? A's $457,40264. Ans. B's $571,83222. XXV. Square Root. The product of any number multiplied by itself, is called When considered in relation to this product, the a square. given number is called a square root. Thus, 12X12-144, the square of 12. But 12, considered as one of the factors of 144, are the square root of 144. Square numbers are found by multiplying any numbers by themselves. It will be found by experiment, that the square of any number contains twice as many figures as the root, or one less than twice as many. Thus, 9X9 81, the square of 9. But 10X10=100, the square of 10. The square of a fraction is found by multiplying the numerator by itself, for the square of the numerator, and the denominator by itself, for the square of the denominator. Thus, the square of is 49 TOO &c. To find the square root of any number, is to find a number which, multiplied by itself, will produce the given number. TO FIND THE SQUARE ROOT. RULE-Beginning at the unit figure, distinguish the sum into periods of two figures each, by placing a dot over the unit figure, another over hundreds, and so on. If the sum contain decimals, place a dot over hundredths and tenths of thousandths, and so on. Find, by trial, the greatest square of the left-hand period, and place its root at the right hand of the given sum. Subtract the greatest square of the left-hand period from that period, and to the remainder annex the second period, and call them the first dividend. Divide this dividend (except the right-hand figure) by twice the root of the greatest square of the left-hand period, and place the quotient at the right hand of the root before found, and also at the right hand of the divisor. Multiply the divisor, with the figure annexed, by the last figure of the root, and subtract the product from the first dividend. To the remainder annex the third period, and call them the second dividend. Divide the second dividend (except the right-hand figure) by the first figure of the root and twice the last figure, (as one divisor,) and place the quotient figure at the right hand of the figures in the root already found, and also at the right hand of the divisor, and proceed as before, until the whole root is found. 1. What is the square root of 1936? 1938 44, Ans. 16 84|336 336 3. What is the square root of 103041? 103041 321, Ans. 9 62130 first dividend. 124 641 641 second dividend. 4. What is the square root of 28,7296? 28,729615,36, Ans. 25 103 3,72 first dividend. 309 1066 6396 second dividend. 6396 5. What is the square root of 13,3225? Ans. 3,65. Ans. 898. Ans. Ans. After the square root of a fraction is found, the value of that root may sometimes be expressed in lower terms than those of the root. But the fraction must not be reduced to its lowest terms before finding the square root. Although the value of the root would be the same in each case, yet the expression would be very different. 9. What is the square root of IJ? Ans. 31. In finding the root of a mixed number, reduce the mixed number to an improper fraction, and find the root of the numerator and denominator, and then reduce the fraction to a whole or mixed number. I 10. What is the square root of 74? Ans. 81 Ans. 61. Ans. 3,25. Ans. 4,5. Ans. 8,5. Ans. 3 Ans. 625. Ans. 875. Ans.,0625. Ans. ,04. Ans. 6,004. Ans. 3,03. Ans. 4,08. Ans. 325. Ans. 3,25. Having shown the scholar how sums are performed in the square root, I wish him now to attend carefully to the following explanation of the rule, that he may understand the nature of the process, and thus be able to give the why in relation to the several steps of the rule. Suppose a man is to receive 3249 square rods of land in one square lot; what will be the length of one side of the field? 3249 First point the sum to distinguish the periods. I point it off into periods of two figures each, because the square of a number contains twice as many figures as the root, or one less than twice as many. If the left-hand period contain two figures, then the given sum contains twice as many figures as the root. The periods serve to show how many figures will be contained in the root. Having found the |