565. Eliminate x from the equations a2x2 + bx+c=0, and h2x2+kx+t=0.

566. If ax3=by3=cz3 and x-1+y-1+z-1=d-1, find the value of ax2+by2+cz2 in terms of a, b, c and d.

√x+Ny

567. Given

=a, to find

√x-Ny √2x+ √34

PROBLEMS IN SIMPLE EQUATIONS.

568. What number is that to which if 16 be added, four times this sum will be equal to ten times the number increased by I? 569. What number is that which exceeds its seventh part by 12? 570. Find that number to which if its third part be added, the sum will equal 4 times the number diminished by 8.

571. Find a number such that if increased by 16 it will become 7 times as great as the third part of the original number.

572. Two coaches start at the same time from York and London, a distance of 200 miles. The one from London travels at 91⁄2 miles an hour, and that from York at 83. Where will they meet, and in what time from starting?

573. A and B begin to play with equal sums; A won £5, and then 3 times A's money was equal to II times B's: what had each at first?

574. A messenger starts on an errand at the rate of 4 miles an hour; another is sent an hour and a half after to overtake him: the latter walks at the rate of 4 miles an hour; when will he overtake the former?

575. A garrison of 500 men was victualled for 48 days; after 15 days it was reinforced and then the provisions were exhausted in II days required the number of men in the reinforcement.

2

7

576. A person performs ths of a piece of work in 13 days; he then receives the assistance of another person, and the two together finish it in 6 days; in what time could each do it separately?

577. A cistern is filled in 24 minutes by 3 pipes, one of which conveys 8 gallons more, and another 7 gallons less than the third, every three minutes. The cistern holds 1050 gallons. How much flows through each pipe in a minute?

578. A person buys four houses: for the second he gives half as much again as for the first; for the third, half as much again as for the second; and for the fourth, as much as for the first and third together he pays £8000 for them all. What is the cost of each?

:

579. Find the fraction, which, if I be added to its numerator, becomes; but if I be added to its denominator, becomes

3

I

4

580. Find the time in which A, B and C can together perform a piece of work, which requires 7, 6 and 9 days respectively when done singly.

581. A and B start to run a race to a certain post and back again. A returning meets B 90 yards from the post and arrives at the starting place 3 minutes before him. If he had returned

I

6

immediately to meet B he would have met him at of the distance between the post and starting place. Find the length of the course and the duration of the race.

582. A waggon is furnished with a mechanical contrivance by which the difference of the number of revolutions of the fore- and hind-wheels is registered. The circumference of the fore-wheel is 7 feet, that of the hind-wheel is 9 feet; what is the distance gone over when the fore-wheel has made 500 revolutions more than the hind-wheel?

583. From a certain sum of money I took away a third part, and put in its stead £50; next from the sum thus augmented I took away a quarter, and put again in its stead £70; I then counted the money and found £120; what was the original sum?

584. Bacchus having caught Silenus asleep by the side of a full cask, seized the opportunity of drinking, which he continued for two-thirds of the time that Silenus would have taken to empty the whole cask. After that Silenus awakes and drinks what Bacchus had left. Had they both drank together it would have been emptied two hours sooner, and Bacchus would have drunk only half of what he left for Silenus: required the time in which each would have emptied the cask separately.

585. A bill of 25 guineas was paid with crowns and half-guineas, and twice the number of half-guineas exceeded three times that of the crowns by 17; how many were there of each?

586. A farm was rated at 3s. an acre, and the tenant on receiving back at his rent-day 10 per cent. of his rent, found that the sum returned amounted to £6 more than the whole rate. The next year the rates were increased 40 per cent., and he received back 12 per cent. of his rent, but the sum returned only just paid the whole rate. What was the rent of the farm, and of how many acres did it consist?

587. Find two numbers such, that if the first be added to four times the second, the sum is 29; and if the second be added to six times the first, the sum is 36.

588. A and B engaged in play; when A had lost £20 he had

only rd the money which B had, but by continuing to play he not only won back his £20 but also £50 more with it, and then found he possessed half as much again as B: with what sum did they respectively begin?

589. A farmer mixes barley at 2s. 4d. a bushel with rye at 38. a bushel, and wheat at 4s. a bushel, so that the whole is 100 bushels, and worth 3s. 4d. per bushel. Had he put twice as much rye, and 10 bushels more of wheat, the whole would have been worth exactly the same per bushel: how much of each kind was there?

590. A man and his wife could drink a barrel of beer in 15 days; after drinking together 6 days, the woman alone drank the remainder in 30 days: in what time would either alone drink it?

591. Find that number of two figures to which, if the number formed by changing the places of the digits be added, the sum is 121; and if the same two numbers be subtracted the remainder is 9. 592. A shopkeeper, on account of bad book-keeping, knows neither the weight nor the prime cost of a certain article which he had purchased; he knows however that if he had sold the whole at 30s. per lb. he would have gained £5 by it, and if he had sold it at 228. per lb. he would have lost £15 by it: what was the weight and prime cost of the article?

593. To complete a certain work A requires m times as long a time as B and C together, B requires n times as long as A and C together, and C requires p times as long as A and B together; compare the times in which each would do it, and prove that

594. Find two numbers whose difference is to the greater as

2 to 9, and the difference of whose squares is 128.

595. Find three numbers in the proportion of sum of whose

squares is 724.

I 2

and 3; the

4

2' 3'

596. The sum of two numbers is 16; and the quotient of the greater divided by the less is to the quotient of the less by the greater as 25 is to 9: find them.

597. Required two numbers whose product is equal to the square

2

of of one of them, and the difference of their squares exceeds by 3

I the square of double the other.

598. A farmer bought some sheep for £72, and found that if he

had bought 6 more for the same money, he would have paid £1 less for each. How many sheep did he buy?

599. The sum of two numbers is 6, and the sum of their cubes is 72: find them.

600. A and B distribute £60 each among a certain number of persons: A relieves 40 persons more than B, and B gives to each person 5s. a-piece more than A. How many persons did they each relieve?

601. A detachment from an army was marching in regular column, with 5 men more in depth than in front; but upon the enemy coming in sight, the front was increased by 845 men; and by this movement the detachment was drawn up in 5 lines. Required the number of men.

602. A vintner draws a certain quantity of wine out of a full vessel that holds 256 gallons; and then filling the vessel with water, draws off the same quantity of liquor as before, and so on, for four draughts, when there were only 81 gallons of pure wine left. How much wine did he draw each time?

603. There is a number consisting of two digits, which, when divided by the sum of its digits, gives a quotient greater by 2 than the first digit. But if the digits be reversed, and then divided by a number greater by unity than the sum of the digits, the quotient is greater by two than the preceding quotient. Required the number.

604. What two numbers are those whose sum multiplied by the greater is 204; and whose difference multiplied by the less is 35? 605. There are two numbers such that the sum of the products of the first by 4 and of the second by 3 is 53; and the difference of their squares is 15. Required the two numbers.

606. Find that fraction whose terms consist of two consecutive numbers, and which, when added to twice its reciprocal, exceeds 2 by twice the reciprocal of the product of its terms.

607. The product of the sum and difference of the hypothenuse and a side of a right-angled triangle is equal to 2; and 4 times the sum of the squares of the hypothenuse and this side is equal to 5 times the sum of these two lines. Find the three sides of the triangle.

608. A and B travelled on the same road, and at the same rate, from N to M: at the 50th mile-stone from M, A overtook a flock of geese, which were proceeding at the rate of 3 miles in 2 hours: and 2 hours afterwards met a stage waggon, which was moving at the rate of 2 miles per hour. B overtook the same geese at the 45th mile-stone, and met the stage waggon exactly 40 minutes before he came to the 31st mile-stone. Where was B when A reached M?

609. Two boys set off in opposite directions from the right angle

of a triangular field, and ran along the sides without varying their velocities, which were in the ratio of 13:11. They met in the middle of the opposite side, and afterwards 30 yards from the point where they started. Required the lengths of the sides of the field. 610. A vintner sold 7 dozen of sherry and 12 dozen of claret for £50. He sold of sherry 3 dozen more for £10 than he did of claret for £6. Required the price of each.

611. There are three numbers, the difference of whose differences is 5; their sum is 44; and continued product 1950. What are the numbers?

612. A detachment of soldiers from a regiment being ordered to march on a particular service, each company furnished 4 times as many men as there were companies in the regiment; but these being found to be insufficient, each company furnished 3 more men; when their number was found to be increased in the ratio of 17 to 16. How many companies were there in the regiment?

613. A body of men are just sufficient to form a hollow equilateral wedge, three deep; and if 597 be taken away, the remainder will form a hollow square four deep, the front of which contains one man more than the square root of the number contained in a front of the wedge. What is the number of men?

614. There are two numbers such that 3 times the sum of their squares multiplied by the less is equal to 26 times the greater; and twice the difference of their squares multiplied by the greater is equal to 15 times the less. Required the two numbers.

615. The fore-wheel of a carriage makes 6 revolutions more than the hind-wheel in going 120 yards; but if the periphery of each wheel be increased one yard, it will make only 4 revolutions more than the hind-wheel in the same space. Required the circumference of each.

616. There are two sorts of metal, each being a mixture of gold and silver, but in different proportions. Two coins from these metals of the same weight are to each other in value as 11 to 17; but if to the same quantities of silver as before in each mixture double the former quantities of gold had been added, the values of two coins from them of equal weights would have been to each other as 7 to II. Determine the proportion of gold to silver in each mixture; the values of equal weights of gold and silver being as 13 to 1.

RATIO, PROPORTION, AND VARIATION.

617. Prove that a: b is a greater ratio than ax: bx+h; and a less ratio than ax: bx-h.

618. Of the ratios a-b4: a"-b", and as+b2: a+ — a3b+a2b2 - ab3+b4, which is the greater?