Since a goose is worth 2 ducks,

.. y=2z......

.(ii).

And, since 14 ducks are worth 7 shillings more than a pig,

From (i) and (ii) we have the values of x and z in terms of y; and, substituting these values in (iii), we have

If y=7, x= =42 from (i), and z = from (ii).

If y=1, x=0 from (i), and z= from (ii). These values are however inadmissible, since pigs cannot be bought for nothing.

Hence a pig cost 42s., à goose 7s., and a duck 3s. 6d.

1.

EXAMPLES XV.

Divide 50 into two parts, such that twice one part is equal to three times the other.

2. A has £5 less than B, C has as much as A and B together, and A, B, C have £50 between them. How much has each ?

3.

One man is 70 and another is 45 years of age; when was the first twice as old as the second?

4. How much are eggs a score, if a rise of 25 per cent. in the price would make a difference of 40 in the number which could be bought for a sovereign?

5. A bag contains 50 coins which are worth £14 altogether. A certain number of the coins are sovereigns, there are three times as many half-sovereigns, and the rest are shillings. Find the number of each.

6. A can do a piece of work in 20 days, which B can do in 12 days. A begins the work, but after a time B takes his place, and the whole work is finished in 14 days from the beginning. How long did A work?

7. A man buys a certain number of eggs at two a penny, four times as many at 5d. a dozen, five times as many at 8d. a score, and sells them at 3s. 8d. a hundred, gaining by the transaction 3s. 6d. How many eggs did he buy?

8. A bill of £63. 5s. was paid in sovereigns and half-crowns, and the number of coins used was 100; how many sovereigns were paid?

9. A man walking from a town A to another B at the rate of 4 miles an hour, starts one hour before a coach which goes 12 miles an hour, and is picked up by the coach. On arriving at B he observes that his coach journey lasted two hours. Find the distance from A to B.

10. Two passengers have altogether 600 lbs. of luggage and are charged for the excess above the weight allowed 3s. 4d. and 11s. 8d. respectively. If all the luggage had belonged to one person he would have been charged £1. How much luggage is each passenger allowed free of charge?

11. A piece of work can be done by A and B in 4 days, by A and C in 6 days, and by B and C in 12 days: find in what time it would be done by A, B and C.

12. A father's age is equal to those of his three children together. In 9 years it will amount to those of the two eldest, in 3 years after that to those of the eldest and youngest, and in 3 years after that to those of the two youngest. Find their present ages.

13. A and B start simultaneously from two towns to meet one another: A travels 2 miles per hour faster than B and they meet in 3 hours: if B had travelled one mile per hour slower, and ▲ at two-thirds his previous pace they would have met in 4 hours. Find the distance between the towns.

14. A traveller walks a certain distance: if he had gone half a mile an hour faster, he would have walked it in of the time: if he had gone half a mile an hour slower he would have been 24 hours longer on the road. Find the distance.

S. A.

12

15. Divide 243 into three parts such that one-half of the first, one-third of the second, and one-fourth of the third part, shall all be equal to one another.

16. A sum of money consisting of pounds and shillings would be reduced to one-eighteenth of its original value if the pounds were shillings, and the shillings pence. Shew that its value would be increased in the ratio of 15 to 2 if the pounds were five-pound notes, and the shillings pounds.

17. £1000 is divided between A, B, C and D. B gets half as much as A, the excess of C's share over D's share is equal to one-third of A's share, and if B's share were increased by £100 he would have as much as C and D have between them; find how much each gets.

18. Find two numbers, one of which is three-fifths of the other, so that the difference of their squares may be equal to 16.

19. Find two numbers expressed by the same two digits in different orders whose sum is equal to the square of the sum of the two digits, and whose difference is equal to five times the square of the smaller digit.

20. A man rode one-third of a journey at 10 miles per hour, one-third more at 9 miles per hour, and the rest at 8 miles per hour. If he had ridden half the journey at 10 miles per hour and the other half at 8 miles per hour, he would have been half a minute longer on the journey. What distance did he ride?

21. Two bicyclists start at 12 o'clock, one from Cambridge to Stortford and back, and the other from Stortford to Cambridge and back. They meet at 3 o'clock for the second time, and they are then 9 miles from Cambridge. The distance from Cambridge to Stortford is 27 miles. When and where did they meet for the first time?

22. Divide £1015 among A, B, C so that B may receive £5 less than A, and C as many times B's share as there are shillings in A's share.

23. On a certain road the telegraph posts are at equal distances, and the number per mile is such that if there were one less in each mile the interval between the posts would be increased by 21 yards. Find the number of posts in a mile.

24. The sum of two numbers multiplied by the greater is 144, and their difference multiplied by the less is 14: find them.

25. A and B start simultaneously from two towns and meet after five hours; if A had travelled one mile per hour faster and B had started one hour sooner, or if B had travelled one mile per hour slower and A had started one hour later, they would in either case have met at the same spot they actually met at. What was the distance between the towns?

26. A battalion of soldiers, when formed into a solid square, present sixteen men fewer in the front than they do when found in a hollow square four deep. Required the number of men.

27.

A number of two digits is equal to seven times the sum of the digits; shew that if the digits be reversed, the number thus formed will be equal to four times the sum of the digits.

28. A sets out to walk to a town 7 miles off, and B starts 20 minutes afterwards to follow him. When B has overtaken A he immediately turns back, and reaches the place from which he started at the same instant that A reaches his destination. Supposing B to have walked at the rate of 4 miles an hour: find A's rate.

29. A starts to bicycle from Cambridge to London, and B at the same time from London to Cambridge, and they travel uniformly: A reaches London 4 hours, and B reaches Cambridge 1 hour, after they have met on the road. How long did B take to perform the journey?

30.

The

A number consists of 3 digits whose sum is 10. middle digit is equal to the sum of the other two; and the number will be increased by 99 if its digits be reversed. Find the number.

31. Two vessels contain each a mixture of wine and water. In the first vessel the quantity of wine is to the quantity of water as 1: 3, and in the second as 3: 5. What quantity must be taken from each in order to form a third mixture, which shall contain 5 gallons of wine and 9 gallons of water?

32. Supposing that it is now between 10 and 11 o'clock, and that 6 minutes hence the minute hand of a watch will be exactly opposite to the place where the hour hand was 3 minutes ago: find the time.

33. A, B and C start from Cambridge, at 3, 4 and 5 o'clock respectively to walk, drive and ride respectively to London. C overtakes B at 7 o'clock, and C overtakes A 4 miles further on at half-past seven. When and where will B

overtake A?

A train 60 yards long passed another train 72 yards long, which was travelling in the same direction on a parallel line of rails, in 12 seconds. Had the slower train been travelling half as fast again, it would have been passed in 24 seconds. Find the rates at which the trains were travelling.

35. A distributes £180 in equal sums amongst a certain number of people. B distributes the same sum but gives to each person £6 more than A, and gives to 40 persons less than A does. How much does A give to each person?

36. Three vessels ply between the same two ports. The first sails half a mile per hour faster than the second, and makes the passage in an hour and a half less. The second sails three-quarters of a mile per hour faster than the third and makes the passage in 24 hours less. What is the distance between the ports?

37. Two persons A, B walk from P to Q and back. A starts 1 hour after B, overtakes him 2 miles from Q, meets him 32 minutes afterwards, and arrives at P when B is 4 miles off. Find the distance from P to Q.