enclosing the three coefficients connected by their signs in a parenthesis. This will be plain if we put it in numbers. 4x+6x+8 x is the same as (4 + 6 +8) x, that is, 18 x. (a+b+c) x=d If we had we should divide by 18, 18x= d In the same manner divide by (a + b + c), Particular Ans. 5 bushels. This general formula is expressed in words as follows: Divide the price of the whole by the price of a bushel of each sort added together, and it will give the number of bushels of each sort. 9. A father dying left $25000 (or a) to be divided between his wife, son, and daughter; his son was to have 3 (or b) times as much as the daughter, and the wife 2 (or c) times as much as the son. What was the share of each? In this example observe that x is taken 1 time, and 6 times, and b c times. When a letter is written without a coefficient, it is always understood to have 1 for its coefficient; thus x is the same as 1 x. Having found the share of the daughter, it is easy to find the shares of the other two. The son's share is 3 x = 7500, or b x = The wife's do. is 6 x = 15000, or b c x = a b 1 + b + b c The learner may now generalize some of the examples in Art. I. in this manner. 10. A gentleman, distributing some money among some beggars, found, that in order to give them 8 (or a) cents apiece, he should want 5 (or b) cents; he therefore gave them 7 (or c) cents, and he had 4 (or d) cents left. How many beggars were there? 11. There is a cistern which is supplied by two pipes; the first will fill it alone in 7 (or a) hours, the second will fill it alone in 5 (or b) hours. In what time will it be filled if both run together? Let the number of hours in which both together will fill it. The first will fill or of it in one hour, and the second will 1 α fill or of it in one hour; both together will fill † + ¦ or 1 b +of it in one hour. In a hours they will fill x times as But a hours is the whole time, therefore, the cistern being 1, Suppose one pipe would fill the cistern in 8 hours, and the other in 42 hours, and find the answer by the general formula. Ans. 3 hours. 12. Suppose it were required to make a rule for Fellowship. First take a particular case. Three men, commencing trade together, furnished money in the following proportions; A $8 as often as B $5, and as often as C $3. They gained $800. What is each man's share of the gain? It is evident that they must receive in the proportion of the capital that they respectively furnished. Now, instead of 8, 5, and 3, suppose they furnished in the proportion of m, n, and p; and let the whole gain be a. 1 m ma by m. Since a fraction is divided by dividing its numerator, the m + n + p will be found by dividing the numerator a multiplied by m is m a, therefore, m a divided by Hence to find the share of either, multiply the whole sum to be divided, by the proportion of the stock which he furnished, and divide the product by the sum of their proportions. The propriety of this rule is easily seen. For, putting in the numbers instead of the letters, A's share is $800, B's share is 3 5 8 8+5+3 or of or of it, and C's share is or of it. That is, the sum of all their propor 8+5+3 13. Let it be required to find what sum, put at interest at a given rate, will amount to a given sum in a given time; that is, to find a rule, by which the principal may be found, when the rate, time, and amount are given. First take a particular case. A man lent some money for 3 years, interest at 6 per cent, and received for interest and principal $472. What was the sum lent ? It is a custom established among mathematicians to use the first letters of the alphabet for known quantities, and some of the last letters for unknown quantities. It is, however, frequently convenient to choose letters, that are the initials of the words for which they stand, whether the quantities be known or unknown. |