(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. (8) (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 100l.; 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, ARITHMETICAL AND GEOMETRICAL. As {; 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. 2, 3, 4, 5, 6,7,8. 8,7, 6, 5, 4, 3, 2, 1. Or, 1, 3, 5, 7, 9, 11, 13. 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 Here the common difference is 1. 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. RULE. Multiply the sum of the two extremes into the number of terms, and divide the product by 2. The quotient will be the sum of all the series. Or multiply the sum of the two extremes by half the number of terms. EXAMPLES. (1) How many strokes do the clocks at Venice (which go on to 24 o'clock) strike in the compass of a natural day? (2) How many strokes does the hammer of a clock strike in 12 hours? (3) The length of a garden is 94 feet: now if eggs be laid along the pavement a foot asunder, and be fetched up singly to a basket, removed one foot from the first, how much ground does he traverse that does it? (4) Suppose 100 stones were placed in a right line, a yard distant from one another, and the first stone were one yard from a basket; I demand how many miles he must travel that gathers them singly into the basket. (5) A butcher bought 100 sheep, and gave for the first sheep 1s. and for the last 91. 19s. I demand what he gave for the 100 sheep. PROPOSITION II. When the two extremes and number of terms are given, to find the common difference. RULE. The difference of the two extremes divided by the number of terms less a unity or 1, the quotient will be the common difference. EXAMPLES. (6) One had 20 children that differed alike in their ages: the youngest was 5 years old, the eldest 43. What was the difference of their ages, and the age of each ? (7) A running footman, for a wager, is to travel from London to a certain place northwards in 19 days, and to go but 6 miles the first day, increasing every day's journey by an equal excess, so that the last day's journey may be 60 miles. I demand each day's journey, and what distance the place he goes to is from London. (8) A debt is to be discharged at 10 different payments in Arithmetical Progression: the first payment is to be 51. and the last 50l. What is the whole debt, and what must each payment be? PROPOSITION III. When the two extremes and the common difference are given, to find the number of terms. RULE. Divide the difference of the two extremes by the common excess or difference; add unity or 1 to the quotient, and the sum will be the number of terms. EXAMPLES. (9) A man being asked how many children he had, answered, that his youngest child was 5 years old, and the eldest 43, and that he had increased one in his family every two years. How many children had he? (10) A person travelling from London northward, went 6 miles the first day, and increased every day's journey 3 miles, till at last he went 60 miles in one day. How many miles did he travel? PROPOSITION IV. When the last term, the common difference, and the number of terms are given, to find the first term. RULE. Multiply the number of terms, less unity or 1, by the common difference; the product subtracted from the last term leaves the first. EXAMPLES. (11) A man in 19 days went from London to a a certain place in the country; every day's journey was greater than the preceding one by 3 miles: his last day's journey was 60 miles. What was the first? (12) A person takes out of his pocket, at 10 different times, so many different numbers of guineas, every one exceeding the former by two; the last was 23. What was the first? PROPOSITION V. When the number of terms, common difference, and the sum of all the terms are given, to find the first term. RULE. Divide the sum of all the series by the number of terms, and from that quotient subtract half the product of the common difference multiplied by the number of terms less one, gives the first term. 1. (13) A person is to receive 275l. at 10 different payments, each payment to exceed the former by 5l, he is willing to bestow the first payment on any one that can tell what it is. What must the arithmetician have for his pains? (14) Suppose it be 100 leagues between London and Edinburgh, two couriers set out from each place on the same road; that from London towards Edinburgh travelling every day two leagues more than the day before; that from Edinburgh to set off one day after the other, travelling every day three leagues more than the preceding one, and that they meet exactly half way, the first at the end of five days, and the other at the end of four: how many leagues did éach travel per day? PROPOSITION VI. When the first term, number of terms, and the common difference are given, to find the last term. |