I bought sheep for seventy-two pounds, and sold them again at 3s. 6d. each, gaining the cost price of four sheep by the transaction: how many did I buy? Let x = number bought.

A and B set out on a walk of two hundred miles. A travels one mile an hour faster than B, and finishes the journey in ten hours less time: what was the rate of each person?

Let a rate per hour of A.

A number consists of two digits; their sum is thirteen, and their product is thirty-four less than the number: what is that number?

10x+13-x= the number.

a (13-x)= product of digits.

x (13-x) = 10x+13-x-34.

x=76.

Divide seventy-two into two such parts that their product may be 1296.

Let x one. Then 72-x= the other.

x (72-x)

x = 36.

= 1296.
72-x=36.

What two numbers are those, the product of which is 333, and the sum 46?

Let the greater.

Then 46-x = the less.

x (46-x) 333.

x = 37.

46-379.

PROBLEMS THAT LEAD TO QUADRATIC EQUATIONS

Examples.

1. Find two numbers whose sum is 12 and product is 35.

Ans. 7 and 5. 2. Find two numbers whose product is 72 and the difference of whose squares is 17.

Ans. 8 and 9, or -8 and -9.

3. Find two consecutive numbers, the difference of whose cubes is 217.

Ans. 8 and 9, or -8 and -9. 4. Find a number such that three times its square, plus 5 times the number itself, is equal to 100. Ans. 5 or

%9.

5. A buys a certain number of oranges for a shilling; if he got 8 more for his shilling he would be paying 3d. less per dozen. How many does he Ans. 16.

get ?

6. A dog runs 6 yards 2 feet per second faster than a hare; the hare has four seconds' start, and the dog catches it at the end of 100 yards. Find the number of yards per second run by the hare and the dog. Ans. 10 and 163.

7. The areas of two fields are 2000 and 3000 square yards respectively, but the first is 10 yards shorter than the second, and 10 yards narrower. Find the dimensions of the fields..

Ans. The first is 50 by 40, the second 60 by 50. 8. The product of the three next numbers above a certain number is greater by 204 than the product of the three next below it. Find the number.

Ans. 4.

9. The product of two numbers is 42, and the difference of their cubes is 127. Find the numbers. Ans. 6 and 7, or- -6 and -7. 10. A buys a certain number of sheep for £50; he loses 2, and by selling the remainder for 10s. each more than they cost, he gains £7 10s. Find the number of sheep bought. Ans. 25.

11. A buys a butt of wine for £150, he adds 30 gallons of a wine which costs 15 shillings per gallon, and by selling the whole for 5 shillings per gallon more than the original wine cost, he gains £52 10s. by the transaction. How many gallons were there at first in the butt? Ans. 120 or 150.

12. A party at an inn expend £6 between them equally if there had been 3 more people and each had spent 1 shilling more, the amount would have been £8 2s. How many people were there?

Ans. 15 or 24.

13. A cistern containing 1000 gallons can be filled by a certain pipe in a certain number of minutes, and by another separately in 5 minutes longer; the cistern being empty, both pipes are turned on and the cistern is filled in 1.13 minutes. How long would each pipe take to fill it separately?

Ans. 20 and 25 minutes. 14. A cistern containing 600 gallons can be filled by a certain pipe in a certain number of minutes, and can be emptied by another pipe in 10 minutes less; the cistern being full, both pipes are

opened, and the cistern takes 1 hour to become empty. Find the number of minutes in which the cistern would be filled by the first pipe.

Ans. 30 minutes.

15. Two couriers start from the same place, one 20 hours before the other; the latter travels 3 miles per hour faster than the other, and overtakes him at the end of 1200 miles. Find the number of miles per hour of the two couriers. Ans. 12 and 15.

16. The hind wheel of a coach is 3 feet more in circumference than the fore wheel; in going 10 miles the hind wheel has made 880 revolutions less than the fore wheel. Find the circumference of the wheels. Ans. 15 feet and 12 feet.

17. Two vessels containing together 20 gallons are filled, one with wine and one with water; the contents are mixed, and the vessels filled with the mixture; it is now found that the vessel which originally contained water has 41% gallons of wine in it. Find the number of gallons which the vessel originally containing wine will hold.

Ans. 11 or 9. 18. A picture is 5 inches longer than it is wide; if it measured 10 inches less each way it would have only of its present area. Find its dimensions.

Ans. 20 by 15 inches.

19. A block of stone is one inch longer than it is broad, and one inch broader than it is thick; another, which measures 4 inches less each way, contains 600 cubic inches less. Find length of first block.

Ans. 10 inches.

20. There is a path 10 feet wide round a rectangular field. The area of this path is 2000 square feet. The area of the field inside the path is 2200 square feet. Find the dimensions of the field inside the path. Ans. 50 feet by 40.