PARTNERSHIP. 69. A field is rented by two persons for £35; the former keeps in it 5 oxen for 10 weeks; the latter, 10 oxen for 2 weeks; find the rent each paid. If a person keeps 5 oxen in a field for 10 weeks, it is equivalent to keeping 50 oxen for 1 week, and similarly the other person will have an equivalent to the keep of 20 oxen for the same time: hence their payments should be in the ratio of 50: 20. .. 50+ 20 : 50:: £35: 1st man's share = £25. A, B, C are partners in a business. A puts in £3000, B £2500, and C £2000. After 3 months, A withdraws £1000, and B £500. At the end of 3 months more, A withdraws another £1000, B and C each add £1000 to theirs. At the end of 3 months more, A puts in £2000. The profits are £590; how must they be divided? M. A is represented by M. M. M. 3000 × 3 + 2000 × 3 + 1000 × 3 + 3000 × 3 315: 300 = 70. Ex. 7 apples are worth as much as 5 pears; 3 pears as much as 8 plums; 12 plums as much as 45 nuts; 100 nuts as much as 14 oranges; how many oranges is an apple worth? SQUARE ROOT. 71. The arithmetical process of extracting the square root is based upon the algebraical. We know that the square of a + b is a2 +2ab+b2. Now a+b is deduced from this expression in the following way: a2 +2ab+b2 (a+b a 2 2a + b | 2ab+b2 Arrange in descending powers of some letter. The square root of the first term is a place the square of a under the first term and subtract-the remainder is 2ab+b2. Double a and divide 2ab by it-the quotient is b. Add b to 2a and place it as a divisor on the left of the remainder. Multiply 2a+b by b and place the product under 2ab+b2 -add o to the quotient. Thus, a + b has been obtained from a2 +2ab+b2. Hence the rule for extracting the square root of a whole number. Begin at the unit's figure and mark off the figures in pairs— find the largest number whose square can be subtracted from the first period (which may consist of one or two figures)write this as the first figure of the root, subtract its square from the first period, and to the remainder bring down the next pair of figures. Double the first figure of the root, and take this as a trial divisor into the figures of the new dividend-place the trial quotient to the figure (or figures) already forming the divisor, and multiply the number so formed by the trial quotient, placing the product under the dividend, and subtract. To the remainder bring down the next pair of figures-double the quotient, which now consists of two figures, and proceed as before. Should the number consist of a whole number and a decimal, mark off in pairs, right and left, from the decimal point. Should the number be a decimal only, begin from the decimal point and mark off in pairs. 72. Note 58 is the common sense way of working out examples in Compound Proportion-other methods have been adopted, one of which will be sufficient to append. Taking the same example "If 9 men, &c." Read the question, first making the number of days unity, thus: "If 9 men in one day consume 12 loaves; how many loaves will 6 men consume in one day?" The statement is 9 : 6 :: 12 So making the number of men unity it would read "If one man in 4 days consume 12 loaves; how many loaves will one man consume in 6 days?" Days Days Loaves : 6 Men 12 Men 73. Ex. Find the value of 3lbs. 4oz. 9dwts. 131grs. of gold at £3 17s. 6d. an ounce. If the example had been:-find the value of 341lbs. 40%. 9dwt. 13grs. at £3 17s. 6d. per ounce, we must first reduce 341lbs. 40%. to ounces=4096oz. and find, by simple practice, the cost of 4096oz. at the price named: thus 74. The arithmetical process is based upon the algebraical. Begin at the unit's figure and mark off the figures in trios. Should the number, whose cube root is required, consist of a whole number and a decimal, mark off in trios right and left from the decimal point. Should the number be a decimal only, begin from the decimal point and mark off in trios. The following example will show the process: 690 is 30 times the quotient already obtained: 158700 is 300 times the square of the quotient. Instead of squaring, the numbers 189, 1389, and 9, have been added together. |