months he was turned away, and received his livery and 8. 10s. in money. What was the prime cost of his livery?

(44.) Shipped off 350 casks of butter, weight 516cwt. 2qr. 14lb., which cost me 27. 5s. per cwt., paid duty 6d. per cwt., cooperage 2l. 16s. 04d., boat-hire 188., porterage, &c. 21. 38. 7d., cellarage 31. 4s. 7d. What did lcwt. of the butter stand me in when on board?

(45.) A man and his wife found, by experience, that a barrel of beer, which lasted them both 12 days, would serve the man, when alone, 20 days: how long would it serve the wife in the absence of her husband, supposing when alone they drank the same quantity each as when they were together?

(46.) There is an island 73 miles in circumference, and three footmen all start together to travel the same way round it; A travels 5 miles a day, B 8 miles a day, and C 10 miles a day: in how many days will they all come together again, and how many times will each have travelled round the island?

(47.) The distance from London to York is 198 miles: two travellers set out at the same time in order to meet : A from London towards York, and B from York towards London. When they met, which was at the end of 6 days, A had travelled 3 miles a day more than B. How many miles did each travel per day?

(48.) A hare pursued by a greyhound was 86 yards before him at starting; whilst the hare ran 5 yards, the dog ran 7 yards. How far had the dog ran when he caught the hare?

(49.) I bought 60 yards of cloth at the rate of 5 yards. for a guinea, and 70 yards more at the rate of 7 yards for a guinea, and immediately sold the whole at the rate of 12 yards for two guineas. Whether did I gain or lose, and how much?

(50.) A tradesman increased his capital annually onefourth part, and at the end of three years, one year's interest thereon at 5 per cent. amounted to 1227. Is, 4ąd. What sum did he begin with?

(51.) If, when port-wine is 40 guineas per hhd., a company of 60 people will spend 20 guineas therein in a cer

tain time, what is wine a pipe, when 15 persons more will spend 65 guineas in twice the time, drinking at the same rate?

(52.) A merchant began the world with a capital of 10,000l.; he gained 10,000l. in 5 years by trading to Russia, and 10,000l. in 8 years by trading to America; but he spent 10,000l. every 2 years in gaming and extravagance. How many years did he go on at this rate before he lost all his property?

INVERSE PROPORTION.

1

Definition.-Inverse, or reciprocal, Proportion teaches by three given numbers, to find a fourth, which shall have the same ratio to the second, as the first has to the third; that is, if the first be greater than the third, the fourth will be greater than the second; and, if the first be less than the third, the fourth will be less than the second.

RULE.

State the question as in the direct rule. Multiply the first and second terms together, and divide the product by the third, the quotient will be the answer, and of the same denomination as you left the second number.

Note 1. Direct and inverse proportion are, properly, only parts of the same general rule; and, in a mathematical arrangement, it would be best to treat of them together. However, as inverse proportion is not of such extensive use in mercantile affairs as direct proportion, I have, according to custom, considered them separately, as being more intelligible to young students.

2. I shall here specify, by familiar examples, the difference between direct and inverse proportion in as clear and concise a manner as possible.--Observe, when the question is stated, that if the third term be greater than the first, and requires the fourth to be greater than the second; or, if the third term be less than the first, and requires the fourth to be less than the second, the proportion is direct. But, if the third term be greater than the first, and requires the fourth to be less than the second; or, if the third term be less than the first, and requires the fourth to be greater than the second, the proportion is in

verse.

Ex. 1st. If 3 yards of cloth cost 18s. what will 24 yards cost?
If *3 yards : 18s. ::. 24 yards: 144s, or £7 4s.

Here it is evident that 24 yards will cost more than 3 yards at the saine rate; hence the proportion is direct; for the third term is greater than the first, and requires the fourth to be greater than the second.

Ex. 2. If 112lb. of sugar cost 56s. what will 1lb. cost?
If #112lb. : 56s. :: 1lb. : 6d.

Here 1lb. of sugar will certainly cost less than 112lb., and consequently the proportion is direct: for, the third term is less than the first, and requires the fourth to be less than the second.

Ex. 3. If 4 men can do a piece of work in 80 days, how many days, of the same length, will 16 men require to do the same work?

If 4 men : 80 days :: *16 men : 20 days.

Here it is plain that 16 men will do a piece of work sooner than 4 men; hence this proportion is inverse; for, the third term is greater than the first, end requires the fourth to be less than the second.

Ex. 4. If 21 pioneers make a trench in 18 days, how many days, of the same length, will 7 men require to make a similar trench?

If 21 pioneers : 18 days :: *7 pioneers : 54 days. Here 7 men will evidently require a longer time than 21 men to dig a trench; hence the proportion is inverse; for, the third term is less than the first, and requires the fourth to be greater than the second.

Examples.

(1.) If a field of grass be mowed by 10 men in 12 days in how many days would it be mowed by 20 men?

Note. Such is the quantity of grass that 10 men would mow it in 12 days, it is therefore obvious, that, if 20 men were employed, they would mow it in half the time.

(2.) A certain piece of grass was to have been mowed by 20 men in 6 days; an extraordinary occasion calls off half the workmen :-it is required to find in what time the rest will finish it? Answer, 12 days.

(3.) If the penny-loaf weighs 5oz. when flour is at 28. a peck, what should it weigh when flour is sold for 2s. 6d. the peck?

(4.) Provisions in a garrison are found sufficient to last 1800 soldiers for three months; but a reinforcement being wanted, that the provisions may last for one month only, what number of soldiers may be added to the garrison on this emergency?

(5.) If 3yds. 2qr. of cloth of lyd. 3qr. wide will make a suit of clothes, how many yards of stuff, of yard wide, will make a suit for the same person?

(6.) If I lend my friend 2007. for 12 months, on condition of his returning the favour, how long ought he to lend me 150l. to requite my kindness?

(7.) If a statute acre be 220 yards long, the breadth will be 22 yards; but, if the breadth of an acre be 40 yards, what will the length be then?

CLASS II.

(8.) If 720 men be placed in a garrison, and have provisions for 6 months; but hear of no relief at the end of 5 months, how many men must depart that the remaining provisions may last 5 months longer?

(9.) If 5 oxen, or 7 colts, eat up a certain quantity of grass in 87 days, in what time will 2 oxen and 3 colts eat up the same quantity of grass?

(10.) A regiment of soldiers, consisting of 1000, are to be new clothed; each coat to contain 2 yards of cloth of 14 yard wide, and to be lined with shalloon of yard wide; how many yards of shalloon will line them?

(11.) A lent his friend B 91 guineas from the 11th of December, 1817, till the 10th of May, 1818; B, on another occasion, let A have 66l. 13s. 4d. from September 3, 1818, to Christmas, 1819, how long ought the person obliged to lend his friend 407. to retaliate the favour?

(12.) If a ball of 18lb. be shot from a cannon with such a force as to send it 100 feet in a second, with what velocity would a ball of 24lb. move, were it impelled by the same force?

(13.) Provisions in a garrison were sufficient to last 1800 men for 12 months, but at the end of 3 months the garrison was reinforced by 600 men, and two months after that a second reinforcement of 400 men was sent to the garrison; how long did the provisions last in the whole?

(14.) How many pounds of sugar at 9 per lb. are equal in value to 24lb. of tea worth 9s. 6d. per lb.?

(15.) There are two equal parallelograms; the length of the one is 10 feet 6 inches, and its breadth 7 feet 3 inches, the breath of the other is 4 feet 2 inches; what is its length?

(16.) How many yards of paper, 27 inches wide, will hang a room that is 24 yards in circuit and 9 feet 4 inches high?

(17.) If 3 men or 4 women can do a piece of work in 34 days, how long will 2 men and 3 women be in finishing a similar piece of work?

(18.) If a board be 9 inches broad, what must be its length to contain 10 square feet?

COMPOUND PROPORTION.

Compound Proportion consists of 5, 7, 9, 11, or 13, &c. conditional terms given, to find a 6th, 8th, 10th, 12th, or 14th, &c. term respectively. When five terms are given to find a sixth, it is called the Rule of Five, or the Double Rule of Three, because all questions, in which the number of terms does not exceed five, may be answered by two statings in the Single Rule of Three.

RULE I.

1. Make as many statings in the Rule of Three as there are terms of supposition or demand; using that term for

If five numbers be given to find a sixth, there will be two statings, and for every two given numbers, above five, there will be one additional stating.