(4) A nobleman dying, left 10 sons, to whom he left a certain sum of money to be divided among them, viz. The youngest son to have 500l. the second to have as much and half as much, and so on, every one to exceed the next youngest in the same ratio of 14. What is the share of the eldest? PROPOSITION III. When the first term, ratio, and number of terms are given, to find the sum of all the terms. RULE. Find the last term as before, from which take the first; divide the remainder by the ratio, less one, and to that quotient add the last term; which gives the sum required. EXAMPLES. (5) On New-year's day a gentleman married, and received of his father-in-law a guinea, on condition that he was to have a present on the first day of every month for the first year, which should be double still to what he had the month before. What was the young lady's portion? (6) One, at a country fair, had a mind to a string of 20 fine horses: but not caring to take them at 20 guineas per head, the jockey consented that he should, if he thought good, pay but a single farthing for the first, doubling it only to the 19th, and he would give the 20th into the bargain. This being presently accepted, how were they sold per head? (7) A laceman, well versed in numbers, agreed with a gentleman to sell him 20 yards of rich gold brocaded lace, for 2 pins the first yard, 6 for the second, 18 for the third, and so on in triple proportion. I demand how much the lace produced. The pins afterwards sold at a farthing per 100. I demand whether the laceman gained or lost by the sale thereof, supposing the said lace to have been bought at 8l. 1s. 8d. per yard. (8) A servant agreed with a master unskilled in numbers to serve him 11 years without any other reward for his service but the produce of a wheat corn for the first year, and that product to be sown the second year, and so on from year to year, until the end of the time, allowing the increase to be but tenfold proportion. I demand what the 11 years' service came to, supposing the sum of the whole produce to be sold at 4s. per bushel. Note.-7680 wheat corns, round and dry out of the middle of the ear, are computed to fill a statute pint. PROPOSITION IV. Of any decreasing series in to find the sum of those series. RULE. whose last term is a cipher, Divide the square of the first term by the difference between the said first term and the second term in the series; the quotient will be the sum of the series. EXAMPLES. (9) A great ship pursues a little one, steering the same way, at the distance of four leagues from it, and sails twice as fast as the small ship. It is asked how far the great ship must sail before it overtakes the lesser. (10) Suppose a ball to be put in motion by a force which drives it 12 miles the first hour, 10 the second, and so on, continually decreasing in proportion of 12 to 10, to infinity. What space would it move through? XXXVI. PERMUTATION, OR, VARIATION, IS the changing or varying the order of things, in respect of their places. RULE. Multiply all the given terms in a series of Arithmetical Progressionals continually, whose first term or common difference is unity or 1, and the last term the number of things proposed to be varied together: and the product will be the number of changes or variations required. EXAMPLES. (1) Six gentlemen that were travelling, met together by chance at a certain inn upon the road, where they were so pleased with their landlord, and each other's company, that in a frolic they made a contract to stay at that place so long as they, together with their landlord, could sit every day in a different order or position at dinner. Quere, the time they staid. (2) I demand the number of changes that may be rung on 12 bells. Also, in what time they may all be rung, allowing 3 seconds to every round, and 365 days 6 hours to the year. (3) An accomptant told a gentleman, who had constantly 8 persons at his table, that he would gladly make a ninth, and was willing to give 20 guineas for his board, so long as he could place the said company at dinner differently from any one day before. This being accepted, what did his entertainment cost him per year? END OF BOOK II. A FRACTION is a part or parts of something considered as a unit or integer, and consists of two parts or quantities, one written over the other, with a line between them, as, , &c. The number placed below the line is called the denominator of the fraction, because it denominates or shows how many parts the unit is broken or divided into; and the number above the line is called the numerator, because it enumerates or shows how many of those parts are contained in the fraction. t A vulgar fraction is either proper, improper, compound, or mixed. A proper fraction is such whose numerator is less than the denominator, as,,,, &c. An improper fraction is when the numerator is equal to, or greater than, its denominator, 18 247 as, 물, 물, &c. A compound fraction is the fraction of a fraction, and known by the word of, as of off, &c. A mixed number is composed of a whole number and a fraction, as 44, 123, 142, &c. XXXVIII. REDUCTION OF VULGAR FRACTIONS. Case 1. To reduce a vulgar fraction to its lowest terins. RULE. Divide the greater term by the less, and that divisor by the remainder following, till nothing remain: then by the last remainder divide both parts of the fraction, and the quotients will give the fraction required. If the remainder be 1, the fraction is already in its least terms. 336 EXAMPLES. (1) Reduce to its lowest terms. (2) Reduce 28838 to its lowest terms. (3) Reduce 숙종 to its lowest terms. (4) Reduce to its lowest terms. 337 (5) Reduce to its lowest terms. 1. When the numerator and denominator each of them end with ciphers, strike off an equal number of ciphers in both, and the remaining figures will be a fraction of the same value, which reduce to its lowest terms. EXAMPLES. (6) Reduce 888 to its lowest terms. (7) Reduce400 to its lowest terms. 2. When you discern any number will equally divide both numerator and denominator, you may abbreviate the fraction thereby. 24 60 and, to their lowest terms. (8) Reduce 324 144 Case 2. To reduce a compound fraction to a single one. RULE. Multiply all the numerators together for a new numerator, and the denominators for a new denominator. Reduce the new fraction to its lowest terms by the last case. When it can be done, you may cancel the fractions, by dividing the numerator and denominator of any two terins by the same number, and use the quotient instead thereof. EXAMPLES. (9) Reduce of of to a single fraction. (10) Reduce of of 4 to a single fraction. (11) Reduce of of to a single fraction. Case 3. To reduce whole or mixed numbers into an improper fraction. |