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Example 25 may be solved, most readily, by assuming
From which the values of 20, y, and 2n may readily be found.
1. Find two numbers whose sum is difference is b.
Let denote the greater number,
Solving, by the preceding rules, we have,
2 Since a and b may be any numbers whatever, we have these general principles by means of which all similar cases can be solved :
1. The greater number is equal to the half sum irecreased by the half difference.
2. The lesser number is equal to the half sumi diminished by the half difference.
2. If 2 be added to the numerator of a certain fraction, its value will become ; but if 2 be added to the denominator, its value will be ž. What is the fraction ? Let @ denote the numerator,
denote the denominator. From the given conditions, we have the equations of the problem, 2 + 2
y whence, .= 7, and y= 12: Hence, the fraction is iz:
Y + 2
3. The hands of a clock are together at 12 o'clock. When are they next together?
Let 2 denote the number of minute spaces passed over by the minute hand, and y denote the rumbei of minute spaces passed over by the hour hand. From the natnre of the problem, we have,
= y + 60
2C = 12y.
51: Hence, they are together at ih 50 m.
This problem has been solved already by means of a single unknown quantity, many of the following problems can also be thus solved.
4. A person has 22000 dollars at interest, which yields him 1220 dollars annually; a part bears interest at 5 per cent., and the remainder at 6 per cent. How many dollars in each part ?
Lct denote the number of dollars in the first part,
and y denote the number of dollars in the second part.
From the conditions,
2 + y = 22000 2 X 160 + y X To = 1220.
... 2 = 10000, y = 12000.
5. A's age is equal to twice B’s age; 20 years ago, A's age was 4 times B’s age. What are their ages ?
Ans. A's 60; B's 30. 16. There are two numbers : the first added to half the second gives 35; the second added to half the first gives 40. What are the numbers ? Ans. 20 and 30.
7. A man has three sons; the sum of the ages of the first and second is 27, that of the first and third is 29, and that of the second and third is 32.
What are the ages of each ?
Ans. 12, 15, and 17. 8. Two men are in trade; the stock of the first increased by one third that of the second, is $1700; the stock of the second increased by one fourth that of the first, is $1800. What is the stock of each?
Ans. $1200 and $1500
9. Find two numbers such that į the first plus the second shall equal 45, and } the second plus } of the first shall equal 40.
Ans. 50 and 60,
10. The sum of the first and second of three numbers is 13, that of the first and third 16, and that of the second and third 19. What are the numbers ?
Ans. 5, 8, and 11. 11. Bought 100 lbs. of sugar and 80 lbs. of coffee for $28, and afterwards bought at the same rates 200 lbs. of sugar and 60 lbs. of coffee for $36. What did each cost per pound?
Ans. Sugar 12 cents, and coffee 20 cents.
12. There are three numbers; the first increased by twice the second and three times the third, makes 74; the second, increased by twice the third and three times the first, makes 90; the third, increased by twice the first and three times the second, makes 100. What are the numbers ?
Ans. 20, 18, and 6.
13. A butcher bought of one person 12 sheep and 20 lambs for 44 dollars, and of a second person 7 sheep and 13 lambs for 27 dollars, at the same rates. How inany dollars did he give a piece ?
Ans. 2 for sheep, and i for lambs
14, Divide the number 1152 into three parts such, that 9 times the sum of the first and second shall be equal to 7 times the sum of the second and third and if 8 times the first be subtracted from 8 times tho sccond, the remainder shall be equal to the sum of the first and third.
Ans. 288, 384, and 480.
15. A farmer mixed rye and oats, forming 100 bushels