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Note. Take the sum of these six equations.
Problems involving Equations of the First Degree with several
Prob. 1. Find two numbers such that if the first be added to four times the second, the sum is 29; and if the second be added to six times the first, the sum is 36.
Prob. 2. If A's money were increased by 36 shillings, he would have three times as much as B; but if B's money were diminished by 5 shillings, he would have half as much as A. Find the sum possessed by each.
Prob. 3. A pound of tea and three pounds of sugar cost six shillings; but if sugar were to rise 50 per cent. and tea 10 per cent., they would cost seven shillings. Find the price of tea and sugar. Ans. Tea, 5s. per pound; Sugar, 4 pence. Prob. 4. What fraction is that to the numerator of which if 2 4 be added the value is one half; but if 7 be added to the denominator, its value is one fifth?
Prob. 5. A certain sum of money, put out at simple interest, amounts in 8 months to $1488, and in 15 months it amounts to $1530. What is the sum and rate per cent.?
Prob. 6. A sum of money put out at simple interest amounts in m months to a dollars, and in n months to b dollars. Required the sum and rate per cent.
Ans. The sum is
; the rate is 1200 x
Prob. 7. There is a number consisting of two digits, the second of which is greater than the first; and if the number be divided by the sum of its digits, the quotient is 4; but if the digits be inverted, and that number be divided by a number
greater by two than the difference of the digits, the quotient is 14. Required the number.
Let x represent the left-hand digit, and y the right-hand
Then, since x stands in the place of tens, the number will be represented by 10x+y.
Whence x=4, y=8, and the required number is 48.
Prob. 8. A boy expends thirty pence in apples and pears, buying his apples at 4 and his pears at 5 for a penny, and afterward accommodates his friend with half his apples and one third of his pears for 13 pence. How many did he buy of
Prob. 9. A father leaves a sum of money to be divided among his children as follows: the first is to receive $300 and the sixth part of the remainder; the second, $600 and the sixth part of the remainder; and, generally, each succeeding one receives $300 more than the one immediately preceding, together with the sixth part of what remains. At last it is found that all the children receive the same sum. What was the fortune left, and the number of children? Ans. The fortune was $7500, and the number of children 5. Prob. 10. A sum of money is to be divided among several persons as follows: the first receives a dollars, together with the nth part of the remainder; the second, 2a, together with the nth part of the remainder; and each succeeding one a dollars more than the preceding, together with the nth part of the remainder; and it is found at last that all have received the same sum. What was the amount divided, and the number of persons? Ans. The amount was a (n-1); the number of persons =n-1. Prob. 11. A wine-dealer has two kinds of wine. If he mixes
9 quarts of the poorer with 7 quarts of the better, he can sell the mixture at 55 cents per quart; but if he mixes 3 quarts of the poorer with 5 quarts of the better, he can sell the mixture at 58 cents per quart. What was the cost of a quart of each kind of wine?
Ans. 48 cents for the poorer, and 64 for the better. Prob. 12. A person owes a certain sum to two creditors. At one time he pays them $530, giving to one four elevenths of the sum which is due, and to the other $30 more than one sixth of his debt to him. At a second time he pays them $420, giving to the first three sevenths of what remains due to him, and to the other one third of what remains due to him. What were the debts?
Prob. 13. If A and B together can perform a piece of work in 12 days, A and C together in 15 days, and B and C in 20 days, how many days will it take each person to perform the same work alone?
This problem is readily solved by first finding in what time they could finish it if all worked together.
Prob. 14. If A and B together can perform a piece of work in a days, A and C together in b days, and B and C in c days, how many days will it take each person to perform the same work alone?
Ans. A in
days; B in
Prob. 15. A merchant has two casks, each containing a certain quantity of wine. In order to have an equal quantity in cach, he pours out of the first cask into the second as much as the second contained at first; then he pours from the second into the first as much as was left in the first; and then again from the first into the second as much as was left in the second, when there are found to be a gallons in each cask. How many
gallons did each cask contain at first?
Prob. 16. A laborer is engaged for n days on condition that he receives p pence for every day he works, and pays q pence for every day he is idle. At the end of the time he receives a pence. How many days did he work, and how many was he idle?
Ans. He worked nq+a
Prob. 17. A certain number consisting of two digits contains the sum of its digits four times, and their product three times. What is the number?
Prob. 18. A father says to his two sons, of whom one was four years older than the other, In two years my age will be double the sum of your ages; but 6 years ago my age was 6 times the sum of your ages. How old was the father and each
of the sons?
Ans. The father was 42, one son 11, and the other 7 years old. Prob. 19. It is required to divide the number 96 into three parts such that if we divide the first by the second the quotient shall be 2, with 3 for a remainder; but if we divide the second by the third, the quotient shall be 4, with 5 for a remainder. What are the three parts? Ans. 61, 29, and 6.
Prob. 20. Each of seven baskets contains a certain number of apples. I transfer from the first basket to each of the other six as many apples as it previously contained; I next transfer from the second basket to each of the other six as many apples as it previously contained, and so on to the last basket, when it appeared that each basket contained the same number of apples, viz., 128. How many apples did each basket contain before the distribution?
Ans. The first 449, the second 225, the third 113, the fourth 57, the fifth 29, the sixth 15, and the seventh 8 apples.
155. When we have only one equation containing more than one unknown quantity, we can generally solve the equation in an infinite number of ways. For example, if a problem involving two unknown quantities (x and y) leads to the single equa tion ax+by=c,
we may ascribe any value we please to x, and then determine the corresponding value of y. Such a problem is called indeterminate. An indeterminate problem is one which admits of an indefinite number of solutions.
156. If we had two equations containing three unknown quantities, we could, in the first place, eliminate one of the unknown quantities by means of the proposed equations, and thus obtain one equation containing two unknown quantities, which would be satisfied by an infinite number of systems of values. Therefore, in order that a problem may be determinate, its enunciation must contain as many different conditions as there are unknown quantities, and each of these conditions must be expressed by an independent equation.
157. Equations are said to be independent when they express conditions essentially different, and dependent when they express the same conditions under different forms.
158. If, on the contrary, the number of independent equa tions exceeds the number of unknown quantities, these equa tions will be contradictory.
For example, let it be required to find two numbers such that their sum shall be 8, their difference 2, and their product 20. From these conditions we derive the following equations:
From the first two equations we find
x=5 and y=3.
Hence the third condition, which requires that their product shall be equal to 20, can not be fulfilled.