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Example 25 may be solved, most readily, by assuming
co/ & 07/12/
which gives for the group, 25.
From which the values of x, y, and z, may readily be found.
and z =
x + y 1
x' = 2.
y = b 5
which are the equations of the problem.
y' = 3.
z' = 4.
Let x denote the greater number,
and y denote the lesser number.
From the conditions of the problem, we have,
1. Find two numbers whose sum is a, and whose difference is b.
Solving, by the preceding rules, we have,
and y =
Since a and 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 increased by the half difference.
2. The lesser number is equal to the half sum 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. What is the frac tion ?
Let x denote the numerator,
and y denote the denominator.
From the given conditions, we have the equations of the problem,
7, and y=12: Hence, the fraction is 77.
y + 60
3. The hands of a clock are together at 12 o'clock. When are they next together?
x = 12y.
Let x denote the number of minute spaces passed over by the minute hand, and y denote the number of minute spaces passed over by the hour hand.
From the nature of the problem, we have,
... x = 65, y = 57.
Hence, they are together at 1 h 55m.
This problem has been solved already by means of a single unknown quantity, many of the following prob lems can also be thus solved.
From the conditions,
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?
Let x denote the number of dollars in the first part, and y denote the number of dollars in the second
x + y = 22000
x × 150 + y × 180
.*. x = 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.
6. 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 the second shall equal 45, and of the first shall equal 40.
the first plus the second plus Ans. 50 and 60.
10. The sum of the first and second of three num bers 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 many dollars did he give a piece?
Ans. 2 for sheep, and 1 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 the second, 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