Ex. 1. Divide the number 56 into two such parts, that their product shall be 640. By the question, Let x one part, then 56-x= the other part. x (56-x)=640; or x2-56x-640; which solved gives x=40, or 16; and 56-x=16 or 40; .. 40 and 16 are the parts required. Ex. 2. There are two numbers whose difference is 7, and half their product plus 30 is equal to the square of the smaller number. the numbers. Let x= the smaller number, x+7= the greater number. Find By the question Ex. 3. Divide the number 30 into two such parts, that their product be equal to 8 times their difference. Now 40 cannot be a part of 30, and therefore we take 6 and 24 as the parts. Ex. 4. A labourer dug two trenches, one of which was 7 yards longer than the other, for £14. 9s., and for the digging of each he had as many shillings per yard as there were yards in its length. Find the length of each. Let x=no. of yards in length of shorter trench, x+7= = longer ......... Then x.x, or x2= money received for shorter trench, in shillings, (x+7) (x+7), or (x+7)2 longer By the question, x2+(x+7)2=289; which solved gives x=8, or -15; .. the lengths were 8, and 15 yards. Ex. 5. A person bought a certain number of sheep for £120. If there had been 8 more, each sheep would have cost him 10s. less. Find the number of sheep. cost of each sheep in the second case + 10s. or £= cost of each sheep in the first case; whence x=40, or −48, which latter root is rejected by the nature of the problem. But, if the conditions of the question be modified, then the value 48 will be applicable in the equation for a substitute -x, then we have which shews that the condition of buying 8 more must be changed into that of buying 8 less; and the price of each sheep instead of being 10s. less will now be 10s. more; and the question to which -48 answers is this. A person bought a certain number of sheep for £120. If there had been 8 less, each sheep would have cost him 10s. more. Find the number of sheep." Ex. 6. A and B set off at the same time to a place 300 miles distant. A travels 1 mile an hour faster than B, and arrives at his journey's end 10 hours before him. Find the rate per hour at which each person travelled. Let x= number of miles per hour that B travels, then x+1= number of miles per hour that A travels. whence x=5, or -6, but -6 is rejected by the nature of the problem. Therefore number of miles per hour travelled by B=5, number of miles per hour travelled by A=6. Ex. 7. From two places, distant from each other 320 miles, A and B set out at the same time to meet each other. B travelled 8 miles a day less than A, and the number of days in which they met, was equal to half the number of miles travelled by B in the day. Find the number of miles travelled by A and B respectively in each day; and also the whole number of miles travelled by each. Let x=number of miles which B travelled cach day, x Then number of days during which each were travelling; Therefore rates are 16, and 24 miles; A went 192 miles and B 128 miles. Ex. 8. Divide the number 10 into two such parts, that their product shall be equal to 30. which shews that the problem is impossible, or no such numbers can be found. Ex. LIII. Examples for practice. 1. 2. Find two numbers whose difference is 8, and their product 240. Divide 33 into two such parts that their product shall be 162. 3. The difference of two numbers is 6; and if 47 be added to twice the square of the lesser, the resulting number will be equal to the square of the greater. Find the numbers. 4. Of two numbers one is equal to the square root of 16 times the other, and the sum of their squares is 225; find the numbers. 5. The difference of two numbers is 2, and the sum of their squares is 244, find them. 6. The floor of a room contains 40 square yards; its height is 5 yards, and the length is 3 yards more than the breadth; find the number of quare yards in the 4 walls. 7. Divide the quantity m into two such parts that the product of the whole and one of the parts shall be the square of the other part. 8. Find a number from which 4 being subtracted and the remainder multiplied by the number sought, the product shall be 45. the 9. Divide 420 into two such parts that the greater may be equal to square of the less. 10. Find two numbers, whose product=96, and the difference of their squares 80. 11. A man buys a horse which he sells again for £56, and gains as many pounds in £100 as the horse cost him; how much did he give for the horse? 12. A man playing at Hazard won at the first throw just as much money as he had in his pocket; at the second throw he won 5 shillings more than the square root of what he then had; at the third throw he won the square of all he then had; and then he had £112. 16s. What had he at first? 13. A company dining at an hotel find their bill amounts to £8. 15s.; if there had been two more of the party, each of them would have had 10s. less to pay than what he had now to pay. Find the number of guests. 14. A and B set out from two towns, distant from each other 247 miles. A travelled at the rate of 9 miles a day, and the number of days at the end of which they met was greater by 3 than the number of miles which B went in a day. Find the number of miles each travelled. 15. A vintner sold 7 dozen of sherry and 12.dozen of claret for £50, and finds that he has sold 3 dozen more of sherry for £10 than he has of claret for £6. Find the price of each. 16. A grazier bought a certain number of oxen for £240, and after losing 3, sold the remainder for £8 a head more than they cost him, and gained thereby £59. How many oxen did he buy? 17. A person bought sheep for £33. 15s. which he sold again at £2. 8s. a head, gaining thereby as much as one sheep cost him. How many sheep did he buy? 18. A person bought some wine for £60; if he had bought 3 dozens more for the same sum he would have paid £1 less per dozen. How many dozen did he buy? Interpret the negative answer. 19. A body of men are formed into a hollow square 3 deep, it is observed that with the addition of 25 men, a solid square could be formed |