7. clearing of fractions be+aca bx—a2 x+bc — α c Solve the 7th Ex. by these formulas; also try other numbers. 9. When a company at a tavern came to pay their reckoning, they found that if there had been three per sons more, they would have had a shilling apiece less to pay; and if there had been two less, they would have had to pay a shilling apiece more. How many persons were there, and how much had each to pay ? 10. A sum of money is to be divided equally among a certain number of persons. Now if there were 3 claimants less, each would receive 150 dollars more; and if there were 6 more, each would receive 120 dollars less. How many persons are there, and how much is each to receive? Ans. There are 9 persons, and they receive 300 dollars each. 11. What fraction is that, to the numerator of which if 1 be added, its value will be, but if 1 be added to its denominator its value will be . Ans. T 12. What fraction is that, to the numerator of which a if ɑ be added, its value will be, but if a be added to n 13. What fraction is that, from the numerator of which if a be subtracted, its value will be be subtracted from its denominator, its value will be N. B. The answers to the 12th and 13th differ only in the signs of the denominators. The learner will find by endeavoring to solve particular examples from these formulas, that he will not always succeed. If in making examples for the 12th, he selects his numbers, so that n p is greater than mq, the formula will fail; but if he takes the same numbers, and applies them according to the conditions of the 13th, they will answer those conditions. When m q is greater than np the numbers will not suit the conditions of the 13th, but they will answer to those of the 12th. The numbers in example 11th will not form an example ac、 cording to the 13th. The following numbers will form an example for the 13th but not for the 12th. 14. What fraction is that, from the numerator of which if 3 be subtracted, its value will be, but if 3 be subtracted from its denominator its value will be? The reason why numbers chosen indiscriminately will not satisfy the conditions of the above formulas will be explained hereafter. Equations with several unknown Quantities. XXIII. Questions involving more than two unknown Quantities. Sometimes it is necessary to employ, in the solution of a question, more than two unknown quantities. In this case, the question must furnish conditions enough to form as many distinct equations as there are unknown quantities. 1. A market woman sold to one man, 7 apples, 10 pears, and 12 peaches, for 63 cents; and to another, 13 apples, 6 pears, and 2 peaches, for 31 cents; and to a third, 11 apples, 14 pears, and 8 peaches for 63 cents. She sold them each time at the same rate. What was the price of each? The second being multiplied by 6, the z will have the same coefficient as in the first. If the second be multiplied by 4, the z will have the which contain only two unknown quantities. These may now be reduced in the same manner as others with two unknown quantities. Multiplying the 5th by 5, and the 7th by 13, the coefficient of y will be the same in both. We have now found an equation containing only one unknown quantity. 178 x = 178 x= 1. Putting the value of x into the 7th, it becomes 41+10 y 61 10 y = 20 |