(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 together. 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. 4. 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 501, 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. How 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, 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. Betwixt my wife and me, My age did hers as far exceed As three times three does three; Her age came up as near to mine Ladies, or gentlemen, now tell, I pray, (9) A gentleman finding several beggars at his door, gave to (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 B. "if you will give me one of your guineas, I shall have twice as many as you will have left." How many guineas had each of them? (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? 1 (13) Two men have a mind to purchase a house rated at 12001. A. said to B. If you give me of your money, I can purchase the house alone; but B. replied to A. If you will give me of yours, I shall be able to purchase the house. How much money had each of them? (14) Suppose the number 50 were to be divided into two parts, so that the greater part divided by 7, and the lesser multiplied by 3, the sum of this product, and the former quotient, might make the same number proposed, which was 50. (15) A certain man hired a labourer on this condition, that for every day he worked he should receive 12 pence, but for every day he was idle he should be fined 8 pence. When 390 days were passed, neither of them were indebted to one another. How many days did he work, and how many days was he idle ? (16) A person being asked how old he was, answered, if I quadruple of my years, and add of them +50 to the product, the sum will be so much above 100 as the number of my years is now below 100. (17) A certain person bought two horses, with the trappings, which cost 100%. ; which trappings, if laid on the first horse A. both the horses will be of equal value; but if the trappings be laid on the other horse, he will be double the value of the first. How much did the horses and trappings cost? (18) A young gentleman, at the age of 21 years, was told by his guardian, that his fortune consisted, in cash, of 7400l. and that his father died when he was but 10 years old: And for the money your father left, said the guardian, I have allowed you 5 per cent. per ann. for simple interest, only I have deducted 100l. per ann. for your education, &c. What was the son's fortune that was left by the father? XXXIV. PROGRESSION, Consisting of Two Parts, ARITHMETICAL AND GEOMETRICAL. ARITHMETICAL PROGRESSION IS when a rank or series of numbers increase or decrease by a common difference, or by a continual adding or subtracting some equal numbers. As { 1, 2, 3, 4, 5, 6, 7, 8. 8, 7, 6, 5, 4, 3, 2, 1. Or, 1, 3, 5, 7, 9, 11, 13. Here the common difference is 1. Here the common difference is 2. Also 35, 30, 25, 20, 15, 10, 5. Here the common difference is 5. 1. In any series of numbers in arithmetical progression, when the number of terms are even, as 1, 3, 5, 7, 9, 11, or the like, the sum of the two extremes will be equal to the sum of any two means that are equally distant from the extremes : Viz. 1, 3, 5, 7, 9, 11. 1+11=5+7=3+9=12. 2. When the number of terms are odd, as 2, 4, 6, 8, 10, the double of the middle figure or term will be equal to the the sum of the extremes, or any two means equally distant from the middle term: Viz. 2, 4, 6, 8, 10. 6 × 2=2+10=12 and 6 × 2=4+8=12. In arithmetical progression there are five things to be observed, viz. 1. The first term. 2. The last term. 3. The number of terms. 4. The common excess, or difference. 5. The aggregate sum of all the terms. Any three of which being given, the other two may be found. PROPOSITION I. When two extremes and the number of terms are given, to find the sum of all the series of terms. |