This shows that something must be subtracted from the present age; that is, the son was a fourth part as old as the father some years before. This equation gives x = 3. Therefore he was one fourth part as old 3 years before, when the father was 52, and the son 13. 2. A man when he was married was 45 years old, and his wife 20 How many years before, was he twice as old as she? There is a wrong supposition in this question. Put- into the equation it becomes ting This shows that she was not half as old as he when they were married, but that it was to happen 5 years afterward, when the man was 50, and the wife 25. 3. A labourer wrought for a man 15 days, and had his wife and son with him the first 9 days, and received $14.25. He afterwards wrought 12 days, having his wife and son with him. 5 days, and received $13.50. How much did he receive per day himself, and how much for his wife and son? 4. A labourer wrought for a man 11 days, and had his wife with him 4 days, and received $17.82. He afterwards wrought 23 days, having his wife with him 13 days, and received $38.78. How much did he re ceive per day for himself, and how much did he pay per day for his wife ? 5. A labourer wrought for a gentleman 7 days, having his wife with him 4 days, and his son 3 days, and received $7 89. At another time he wrought 10 days, having his wife with him 7 days, and his son 5 days, and received $11.65. At a third time he wrought 8 days, having his wife with him 5 days, and his son 8 days, and received $7.54. How much did he receive per day himself, and how much for his wife and son severally? 6. What number is that, whose fourth part exceeds its third part by 16? The question as it was proposed involves some contradiction. Putting in it becomes x 4 Changing all the signs 814 x = 192. 16 This shows that the question should have been as follows; What number is that, whose third part exceeds its fourth part by 16? 7. What number is that, of which exceeds of it by 18? 8. What fraction is that, to the numerator of which if 1 be added, its value will be 3, but if I be added to its denominator, its value will be ? 9. What fraction is that, from the denominator of which, if 2 be subtracted, its value will be 14, but if 2 be subtracted from its numerator, its value will be ? 10. It is required to divide the number 20 into two such parts, that if the larger be multiplied by 3, and the smaller by 5, the sum of the products will be 125. 11. It is required to find two numbers, whose difference is 25, and such that if the larger be multiplied by 7, and the smaller by 5, the difference of their products shall be 215. XXV. Explanation of Negative Exponents. It was observed above, that when the dividend and the divisor were different powers of the same letter, division is performed by subtracting the exponent of the divisor from that of the dividend: thus a <-1 Now = 1. By the above principle a1—1= a°; a therefore a° = 1. a3 b a α Also = a3-3 = ao = 1; / =b1-1 = b° = 1; b a3 That is, any quantity having zero for its exponent is The quantities a3, a3, a1, ao, a-1, a-3, a-3, &c. have the same value as a3, a2, a1, 1, 1 1 1 · a a &c. On this principle the denominator of a fraction, or any factor of the denominator may be written in the numerator with the sign This mode of notation is often very convenient; I shall therefore give a few ex1 amples of its application. * Exponents may be used for compound quantities as well as for simple; and multiplication and division may be performed on those which are similar, by adding and subtracting the exponents. |