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Suppose the small wheel to be x feet, and the large wheel y feet in circumference.

780 780 In a distance of 260 yards the two wheels make and

revolutions respectively.

780 780 Hence

y
1 1 9

(1). y

52 Similarly from the second condition, we obtain

210 210

=27, x+1

1 1 9 3+1

70

or

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Y+1

or

(2).

y+1

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70'

9y +52

1 9 Substituting in (2),

614 +52 y+1
70 x 9yo = 9 (61y+52) (y+1),

9y2 - 113y - 52=0,
(y – 13) (9Y+4)=0,

4 .. y=13, or

9'

Putting y=13 we find that x=4. The other value of y is inadmissible; hence the small wheel is 4 feet, the large wheel 13 feet in circumference.

Example 5. On a river there are two towns 24 miles apart. By rowing one half of the distance and walking the other half, a man performs the journey down stream in 5 hours, and up stream in 7 hours. Had there been no current, each journey would have taken 5hours: find the rate of his walking, and rowing, and the rate of the stream.

Suppose that the man walks æ miles per hour, rows y miles per hour, and that the stream flows at the rate of z miles per hour.

With the current the man rows y +z miles, and against the current Y

- 2 miles per hour.

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(3).

y +2

1

- Z

12 12
+ 53

у

1 1 1 From (1) and (3) by subtraction,

y

18 1

1 Similarly, from (2) and (3)

(5).

y y From (4) 182=y (y+z).

(6); and from (5) 92=y (y z)

(7). From (6) and (7) by division, 2

y whence

y=3z; i. from (4) 2=1}; and hence y=41,

X=4. Thus the rates of walking and rowing are 4 miles and 44 miles per hour respectively; and the stream flows at the rate of 14 miles per hour.

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EXAMPLES XXIX.

1. Find a number whose square diminished by 119 is equal to ten times the excess of the number over 8.

2. A man is five times as old as his son, and the sum of the squares of their ages is equal to 2106 : find their ages.

3. The sum of the reciprocals of two consecutive numbers is 15

; find them. 56

4. Find a number which when increased by 17 is equal to 60 times the reciprocal of the number.

5. Find two numbers whose sum is 9 times their difference, and the difference of whose squares is 81.

6. The sum of a number and its square is nine times the next highest number; find it.

7. If a train travelled 5 miles an hour faster it would take one hour less to travel 210 miles: what time does it take ?

8. Find two numbers the sum of whose squares is 74, and whose sum is 12.

9. The perimeter of a rectangular field is 500 yards, and its area is 14400 square yards : find the length of the sides.

10. The perimeter of one square exceeds that of another by 100 feet; and the area of the larger square exceeds three times the area of the smaller by 325 square feet: find the length of their sides.

11. A cistern can be filled by two pipes running together in 22 minutes : the larger pipe would fill the cistern in 24 minutes less than the smaller one: find the time taken by each.

12. A man travels 108 miles, and finds that he could have made the journey in 4 hours less had he travelled 2 miles an hour faster : at what rate did he travel ?

13. I buy a number of foot-balls for $100; had they cost a dollar apiece less, I should have had five more for the money: find the cost of each.

14. A boy was sent out for 40 cents' worth of eggs. He broke 4 on his way home, and the cost therefore was at the rate of 3 cents more than the market price for 6. How many did he buy ?

15. What are the two parts of 20 whose product is equal to 24 times their difference ?

16. A lawn 50 feet long and 34 feet broad has a path of uniform width round it; if the area of the path is 540 square feet, find its width.

17. A hall can be paved with 200 square tiles of a certain size; if each tile were one inch longer each way it would take 128 tiles: find the length of each tile.

18. In the centre of a square garden is a square lawn ; outside this is a gravel walk 4 feet wide, and then a flower border 6 feet wide. If the flower border and lawn together contain 721 square feet, find the area of the lawn.

19. By lowering the price of apples and selling them one cent a dozen cheaper, an applewoman finds that she can sell 60 more than she used to do for 60 cents. At what price per dozen did she sell them at first ?

20. Two rectangles contain the same area, 480 square yards. The difference of their lengths is 10 yards, and of their breadths 4 yards; find their sides.

21. There is a number between 10 and 100; when multiplied by the digit on the left the product is 280; if the sum of the digits be multiplied by the same digit the product is 55: required the number.

22. A farmer having sold at $75 each, horses which cost him 3 dollars a piece, finds that he has realised x per cent. profit on his outlay: find x.

23. A merchant bought a number of yards of cloth for $100; he kept 5 yards and sold the rest at $2 per yard more than he gave, and received $20 more than he originally spent: how many yards did he buy?

24. If a carriage wheel 143 ft. in circumference takes one second more to revolve, the rate of the carriage per hour will be 2 miles less : how fast is the carriage travelling ?

25. A broker bought as many shares of stock as cost him $1875; he reserved 15, and sold the remainder for $1740, gaining $4 a share on their cost price. How many shares did he buy?

26. A and B are two stations 300 miles apart. Two trains start simultaneously from A and B, each to the opposite station. The train from A reaches B nine hours, the train from B reaches A four hours after they meet: find the rate at which each train travels.

27. A train A starts to go from P to Q, two stations 240 miles apart, and travels uniformly. An hour later another train B starts from P, and after travelling for 2 hours, comes to a point that A had passed 45 minutes previously. The pace of B is now increased by 5 miles an hour, and it overtakes A just on entering Q. Find the rates at which they started.

28. A cask P is filled with 50 gallons of water, and a cask Q with 40 gallons of brandy; a gallons are drawn from each cask, mixed and replaced; and the same operation is repeated. Find a when there are 87 gallons of brandy in P after the second replacement.

29. Two farmers A and B have 30 cows between them; they sell at different prices, but each receives the same sum. If A had sold his at B's price, he would have received $320; and if B had sold his at A's price, he would have received $245. How many had each?

30. A man arrives at the railroad station nearest to his house 14 hours before the time at which he had ordered his carriage to meet him. He sets out at once to walk at the rate of 4 miles an hour, and, meeting his carriage when it had travelled 8 miles, reaches home exactly 1 hour earlier than he had originally expected. How far is his house from the station, and at what rate was his carriage driven?

CHAPTER XXX.

INDETERMINATE EQUATIONS OF THE FIRST DEGREE.

230. IN Art. 153 we saw that if the number of unknown quantities is greater than the number of independent equations, there will be an unlimited number of solutions, and the equations will be indeterminate. By introducing conditions, however, we can limit the number of solutions. When positive integral values of the unknown quantities are required, the equations are called simple indeterminate equations.

The introduction of this restriction enables us to express the solutions in a very simple form.

Example 1. Solve 7x+12y=220 in positive integers.
Transpose and divide by the smaller coefficient; thus

3-5y
x=31-y+

7

3-5y

.. x+y-31=

7 Since x and y are to be integers, we must have

3-5y

integer.

7 Now multiplying the numerator by such a number that the division of the coefficient of y may give a remainder of unity, in this case 3, we have

9-157

= integer; 7

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