A loss of £149 is to be made good by three persons, A, B, and C. Had there been a gain, A would have gained 4 times as much as B, and C as much as A and B together. How much of the loss must each bear? _Answer_, A pays £59. 12, B £14. 18, and C £74. 10. 256. It may happen that several individuals employ several sums of money together for different times. In such a case, unless there be a special agreement to the contrary, it is right that the more time a sum is employed, the more profit should be made upon it. If, for example, A and B employ the same sum for the same purpose, but A’s money is employed twice as long as B’s, A ought to gain twice as much as B. The principle is, that one pound employed for one month, or one year, ought to give the same return to each. Suppose, for example, that A employs £3 for 6 months, B £4 for 7 months, and C £12 for 2 months, and the gain is £100; how much ought each to have of it? Now, since A employs £3 for six months, he must gain 6 times as much as if he employed it one month only; that is, as much as if he employed £6 × 3, or £18, for one month; also, B gains as much as if he had employed £4 × 7 for one month; and C as if he had employed £12 × 2 for one month. If, then, we divide £100 into 6 × 3 + 4 × 7 + 12 × 2, or 70 parts, A must have 6 × 3, or 18, B must have 4 × 7, or 28, and C 12 × 2, or 24 of those parts. The shares of the three are, therefore, 6 × 3 × 100 4 × 7 × 100 £----------------------, £----------------------, 6 × 3 + 4 × 7 + 12 × 2 6 × 3 + 4 × 7 + 12 × 2 12 × 2 × 100 and £----------------------. 6 × 3 + 4 × 7 + 12 × 2 EXERCISES. A, B, and C embark in an undertaking; A placing £3. 6 for 2 years, B £100 for 1 year, and C £12 for 1½ years. They gain £4276. 7 How much must each receive of the gain? _Answer_, A £226. 10. 4; B £3432. 1. 3; C £617. 15. 5. A, B, and C rent a house together for 2 years, at £150 per annum. A remains in it the whole time, B 16 months, and C 4½ months, during the occupancy of B. How much must each pay of the rent?[56] _Answer_, A should pay £190. 12. 6; B £90. 12. 6; C £18. 15. [56] This question does not at first appear to fall under the rule. A little thought will serve to shew that what probably will be the first idea of the proper method of solution is erroneous.