that is, £335. 1s. Od. is equal to 13 times £24. 16s. 4d. and this being regarded as a compound unit and represented by 1, the former will be represented by 131.

(5) A person possessed of ths of a coal mine, sells ths of his share for £2000; what is the whole mine worth?

Here, if the mine be considered the unit and be represented by 1,

the fraction of it sold for £2000: that is

therefore is worth of £2000, or £666. 13s. 4d.:

and 1, or the whole mine, is worth

(£666. 13s. 4d.) × 10= £6666. 13s. 4d.

(6) A can do a piece of work in 5 days, B in 6 and C in 7: how much of it can they jointly do in 2 days? Assuming the piece of work to be represented by the unit or 1, we have

is the part done jointly by A, B and C in 1 day: whence the work done by them jointly in 2 days, will be

that is, they could finish the whole work in 2 days, and 2 of the same work besides.

105

Hence also, the time in which they would exactly complete the work is

(7) One half of the trees in an orchard are apple trees, one fourth are pear trees, one sixth plum trees, and there are 50 cherry trees: what number of trees does it contain?

Representing the number of trees in the orchard by the unit or 1, we have

and the number of trees in the orchard = 600.

Examples for Practice.

of a lottery ticket cost £4. 10s., what is

of a ticket?

Answer: £4. 16s.

(2)

4

The owner of 17 of a ship, sold of of his

21

share for £12; what would off of it cost, at the same

rate?

41

Answer: £200.

(3) Express a degree of 69 miles in metres, where 32 metres are equal to 35 yards,

Answer: 111835 metres.

(4) If I import 5763 bushels of wheat for £1800. 18s. 9d., and pay an import duty of 10 per cent. on the money expended, what is the duty per bushel?

Answer: 73d.

(5) Find the value of the metre of France, in terms of the foot of Cremona, if 48 Cremonese feet = 56 English feet, and if the metre be 39 English inches.

371

1000

H

(6) What number is that, whereof the part expressed by++ is 45?

Answer: 60.

(7) A post has one-fourth of its length in the mud, one-third in the water, and 10 feet above the water: find its whole length.

Answer: 24 feet.

311 42

(8) A met two beggars B and C, and having of

10% 77 10 of of a moidore in his pocket, gave + of of it 7

540

to B, and of the remainder to C: what did each receive?

2

31

Answer: B received 6d., and C had 2s. 6d.

(9) A had at first £1. 8s.; and B, when he had paid

of £1. 11s. 6d. to A, found that he had remaining

of what A then had: what had B at first?

Answer: £7. 8s.

(10) If a cask be emptied by two taps in 4 and 6 hours respectively, in what time will it be emptied by both of them together, the rates of efflux remaining the same throughout?

Answer: 2 hrs. 24min.

(11) A, B and C can perform a piece of work in 12 hours: also A and B can do it in 16 hours, and A and C in 18 hours: what part of the work can B and C do in 9 hours?

Answer: .

(12) Ten excavators can dig 12 loads of earth in 16 hours, whilst 12 others can dig only 9 loads in 15 hours: find in what time they will jointly dig 100 loads.

Answer: 74 hours.

(13) A cistern is filled by two spouts in 20 and 24 minutes respectively, and emptied by a tap in 30 minutes: what portion of it will be filled in 15 minutes, when they are all left open together, the influx and efflux being uniform?

(14) In an orchard, of the trees are apple trees, pear trees,cherry trees, filbert trees, and there are 12 walnut trees: what is the number of each sort?

Answer: 80 apple trees, 60 pear trees, 48 cherry trees, 40 filbert trees, and 12 walnut trees.

(15) A person after paying away one-third of his money together with £10., finds that he has remaining £15. more than its half: what money had he?

Answer: £150.

(16) A farmer pays a corn-rent of 5 quarters of wheat and 3 quarters of barley, Winchester measure: what is his rent, wheat being at 60s., and barley at 54s. per quarter, imperial measure, it being assumed that 32 imperial gallons are equivalent to 33 Winchester gallons?

Answer: £22. 8s.

CHAPTER V.

THE THEORY OF DECIMALS,

COMMONLY CALLED DECIMAL FRACTIONS.

97. DEF. IN the articles upon the Notation of 고 Integers, it has been seen that the figures in the units' place alone retain their absolute values, whilst the local values of figures in other situations increase tenfold for every individual figure we advance towards the left hand from that place. Hence, therefore, in beginning at the left figure of any number and proceeding towards the right hand, it necessarily follows that the local value of every successive figure will be a tenth part of that which immediately precedes it: and if we suppose figures to be situated to the right of the units' place, and this kind of tenfold subdivision to be extended to them, it is manifest that the local values of such figures in order from the place of units' will be a tenth, a hundredth, a thousandth, &c., parts of their absolute values.

This consideration will therefore enable us to represent integers and fractions by one uniform system of notation, by merely fixing upon the place of units: and whilst Integers are expressed by figures in the units' place and in places to the left of it, Fractions will be represented by figures situated in places on the right of the units, called the places of tenths, hundredths, thousandths, &c.

In this manner originates the System of Decimal Notation, being merely an extension of the Notation of Integers and from the circumstance of its representing only tenths, hundredths, thousandths, &c., of the unit, all fractions belonging to it are termed Decimals, or, Decimal Fractions, in contradistinction to Vulgar Fractions, whereof the denominations may be any parts we please. Whence Decimals are sometimes defined to be Fractions whose denominators are 10, 100, 1000, &c.