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EXAMPLES.

(1) If 100 lb. of London be equal to 1131b. of Marseilles, and 100 lb. at Marseilles be equal to 81 lb. of Amsterdam; how many pounds at London are equal to 60 lb. of Amsterdam?

(2) If 104 lb. of English be equal to 841⁄2 lb. of Geneva, and 100 lb. of Geneva be equal to 108 lb. at Rouen; how many pounds English are equal to 64 lb. of Rouen? (3) Suppose 100 yards English to be equal to 78 ells French, and 78 ells French equal to 1334 ells at Amsterdam; how many yards English are equal to 100 ells at Amsterdam?

(4) If 100 canes of Genoa be equal to 191 ells of England, and 78 ells of England be equal to 131 of Brussels; how many canes of Genoa are equal to 100 ells of Brussels?

Case 2. When it is required to find how many of the last sorts, of weight or measure mentioned, are equal to a given number of the first.

RULE.

1. Place the numbers alternately, beginning at the left hand as before, and set the last number on the right hand.

2. Multiply the first row for a divisor, and the other for a dividend.

EXAMPLES.

(5) Suppose 100 lb. of Portugal be equal to 02 lb. of Antwerp, and 100 lb. of Antwerp be equal to 110lb. of Lyons; how many pounds at Lyons are equal to 60 lb. of Portugal?

(6) If 74 yards of English be equal to 100 brasses of Florence, and 100 brasses of Florence be equal to 30 canes of Marseilles; how many canes of Marseilles are equal to 100 yards English?

XXXII. POSITION,

OR,

THE RULE OF FALSE,

IS so called, because we suppose some uncertain or false numbers; in order that, by reasoning from them, according to the nature thereof, we may, by those false supposed numbers, find the true number sought.

This Rule is divided into two parts, commonly called the Single Rule and Double Rule.

SINGLE POSITION.

By Single Position are answered all such questions as require only one supposition to discover the true result.

RULE.

Make choice of your position: work with that supposition, according to the nature of the question, as if it were the true number; and if you find, after ordering your position, the result either too much or too little, you may then find the true answer, by this proportion, viz.

As the result of your position: is to the position :: so is the given number: to the number sought.

PROOF.

Add the several parts of the sum together, and if the sum agree with the given number, it is right.

EXAMPLES.

(1) Three persons, A. B. and C. conversing respecting their ages; B. said to A. I am as old and half again as old as you: then said C. to B. I am twice as old as you : now said A. to them both, I am sure, if our ages be added together, the sum will be 132. I demand each man's age.

(2) A man overtaking a maid driving a flock of geese, said to her, Where are you going with these 40 geese? No, Sir, said she, I have not forty; but if I had as many more, half as many more, and 10 geese besides, I should have 40. How many geese had she?

(3) A. B. C. and D. were in company together: A. told C. that he was older than him by 4 years; B. told them that he was as old as both of them together, and 9 years older; D. hearing them, said, I am just 45 years old, and that is equal to the sum of your ages added to gether. How old was each of them severally?

(4) Three persons, viz. Andrew, Benjamin, and Christopher, are to go a journey of 469 miles; of this journey Andrew is to go a certain number of miles unknown; Benjamin is to go three times as many miles as Andrew, and one league more: and Christopher is to go twice as many miles as Benjamin, and 16 miles more. How many miles must each of these persons travel severally ?

(5) Admit three merchants, A. B. and C. to build a ship, which costs them 2000l. of which A. paid a certain part unknown; B. paid 31⁄2 as much, wanting 451. 15s.; and C. paid as much as both A. and B. together, and 261. 10s. more. How much did each person pay?

(6) A cistern, with three unequal cocks, contains 60 pipes of water; the greater cock will empty the cistern in one hour, the second in two, and the third in three. In what time will they empty the cistern, supposing they all be set open at once?

(7) A general being asked the number of men his army consisted of, answered that of amounted to 900. What number of men had he?

(8) A schoolmaster being asked how many scholars he had, answered, If I had as many, as many, as many, and as many, I should have 333. How many had he?

XXXIII. DOUBLE POSITION

IS when two suppositions are used; and if we miss in both, as it generally happens, observe the nature of the errors, whether they be greater or less than the given number; and accordingly they must be made use of thus:

RULE.

1. Place the error against its respective position, and multiply them crosswise.

2. If the errors be alike, that is, both greater, or both less than the given number, take their difference for a divisor, and the difference of their products for a dividend.

But if unlike, that is, one too much, and the other too little, then take their sum for a divisor, and the sum of their products for a dividend; the quotient will be the

answer.

EXAMPLES.

(1) A gentleman has two horses of good value, and a saddle worth 50l. which, if set on the back of the first horse, will make his value double that of the second; but if set on the back of the second horse makes his value triple that of the first horse. I demand the value of each horse.

(2) Double my money for me, said A. to B. and I will give you 6d. out of the stock: with the remainder he applied in the like manner to C. with equal success, and gave him also 6d. He repeated this proposal to D. and then 6d. was all he had to give. Pray, what sum had he to begin with?

(3) Three gentlemen, A. B. and C. playing at hazard together, the money staked was 112 guineas, but disagreeing, each seized as many as he could; A. got a certain quantity, B. as many as A. and 16 more; but C. got only a sixth part of their sum. Ilow many had each?

(4) A boy stealing apples was taken by Mad Tom, and to appease him gave half he had, and Tom gave him back 10; in his return home he was met by Raving Ned, 11 who took from him one half of what he had left, and returned him back 4; after that, unluckily, Positive, Jack met him, when he gave him one half of what he had left, and he returned him back 1; at last getting safe away, he found he had 18 left. How many had he at first?

(5) A son asked his father how old he was; his father replied, Your age is now of mine; but 4 years ago, your age was only of what mine is now. What were their ages?

(6) The head of a certain fish is nine inches long, the tail as long as the head and half the body, and the body is as long as both the head and the tail. I demand the whole length of the said fish.

(7) To find a number, which, if added to itself, and the sum multiplied by the same, and the same number still subtrated from the product; and, lastly, the remainder divided by the same; that it may produce 13.

(8)

QUESTIONS for Exercise at leisure Hours.

When first the marriage-knot was ty'd
Betwixt my wife and me,
My age did hers as far exceed

As three times three does three;

But when ten years, and half ten years,
We man and wife had been,
Her age came up as near to mine
As eight is to sixteen.

Ladies, or gentlemen, now tell, I pray,
What were our ages on the wedding-day?

(9) A gentleman finding several beggars at his door, gave to each four-pence, and had sixteen-pence left; but if he had given to each sixpence, he would have wanted twelve-pence. How many beggars were there?

(10) To find a number which, being multiplied by 3, subtract 5 from the product; and the remainder divided by 2, if the number sought be added to the quotient, that the sum may be 40.

(11) Two companions having a parcel of guineas, A. said to B. "If you will give me one of your guineas, I shall have as many as you will have left."-" Nay," replied if you will give me one of your guineas, gun I shall have twice as many as you will have left." How many guineas had each of them?

B.

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(12) A son asked his father how old he was? His father answered him thus: If you take away 5 from my years, and divide the remainder by 8, the quotient will be of your age; but if you add 2 to your age, and multiply the whole by 6, and then subtract 7 from the product, you will have the number of the years of my age. What was the age of the father and son?

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