sold the whole for: also how much he gained by the sale thereof, supposing he gave for it 4l per ounce ? રે He sold it for Ans. And gained 58251 8s 5d 1qr. 5777 8 5 1. 5 A crafty servant agreed with a farmer (ignorant in numbers) to serve him 12 years, and to have nothing for his service but the produce of a wheat-corn for the first year; and that product to be sowed for the second year; and so on from year to year, antil the end of the said time -I demand the worth of the whole produce, supposing the increase to be but in a tenfold proportion, and sold out at 4s per bushel? Answer 452112/ 4s. rejecting remainders. Note 1. 7680 Wheat or barley-corns are supposed to make a pint, and 64 pints a bushel. 2. If the first term in any series be either greater or less than the ratio (except unity) then multiply any two terms together, and their product divide by the first terms-that quotient will exactly correspond with the sum of their indices. 6 A thresher worked 20 days at a farmer's, and received for the first day's work barley corns, for the second 12 barley-corns, for the third 36 barley-corns, and so on in triple proportion geometrical-I demand what the 20 days labour came to, supposing the whole quantity to be sold for 2s 6d per bushel? Answer 17731 7s 6d rejecting remainders. 7 A merchant sold 30 yards of fine velvet, trimmed with gold very curiously, at 2 pins for the first yard, 6 pins for the second, 18 pins for the third, &c. in triple proportion, geometrical-I demand how muck the velvet produced, when the pins were afterwards sold at 100 for a farthing; also, whether the said merchant gained or lost by the sale thereof, and how much, supposing the said velvet to have been bought at 50% per yard? The velvet produced Answ. The merchant gained 2144699292 13s. 01 2144697792 13 0 OF PERMUTATION. Q. WHAT is Permutation? A. Changing the Order of things. Q. How do you find all the variations any number of things is capable of going through? A. Multiply all the given terms one into another continually; the last product is the number of changes re quired. EXAMPLES. 1. I demand how many changes may be rung upon 12 bells and also how long they would be in ringing but once over, supposing 24 changes might be rung in one minute, and the year to contain 365 days, 6 hours? Ans. the number of changes is 479001600, and the time is 37 years, 49 weeks, 2 days, 18 hours. 2. Seven gentlemen who were travelling met together by chance, at a certain inn upon the road, where they were so well pleased with their host, and each other's company, that in a frolick they offered him 301 to stay at that place so long as they, together with him, could sit every day at dinner in a different order: the host thinking that they could not sit in many different positions, because there were but few of them, and that himself would make no considerable alteration; he being but one, imagined that he should make a good bargain, and readily, (for the sake of a good dinner, and better company) entered into an agreement with them, and so made himself the eighth person; I demand how long they staid at the said inn, and how many different positions they sat in? Ans. the number of positions were 40320; and the time that they staid was 110 years, 142 days; allowing the year to consist of 365 days. 6 hours. Note. There is one thing in progression, and in varying the order of things, which is well worth our observation; and that is, the power of numbers, which is surprisingly great, and beyond common belief; and is no ways conceivable by a common practitioner, hardly by a very good artist; it being (in appearance) not so much against reason as above it. The first example in geometrical progression discovers what a prodigious sum of money a horse sold after that manner would produce, viz no less than four million, four hundred and seventy-three thousand nine hundred and twenty-four pounds, whereas if the same horse had been sold at the same rate, and but a 4th part of the nails, he would have brought to his owner no more than 5s. 3d. three farthings. The second example in Permutation does likewise discover the impossibility of the innkeeper's performing his promise: and in both, the simplicity of two men, who thinking they have got very good bar. gains do, instead thereof, find themselves severe sufferers. And although at the first appearance each question seems to produce a trifle; yet upon a mature consideration, there would not be found a man in the kingdom able to purchase the one, or long-lived enough to stand to the agreement with the other. Hence observe the great possibility of a man's being imposed on in this way by sharpers, without a careful examination into the affair, before any contract is made. THE SCHOOLMASTER'S ASSISTANT. PART II. OF VULGAR FRACTIONS. . OF FRACTIONS IN GENERAL: Q. WHAT is a Fraction? A. It is a broken number, and signifies the Part or parts of a whole Number. Q. How many kinds of Fractions are there? A. Two: Vulgar and Decimal. OF NOTATION OF VULGAR FRACTIONS. Q. What is a Vulgar Fraction? A. Any two numbers placed thus make a Vulgar Fraction. Q. What is the upper number of such a Fraction called? A. It is called Numerator, and is the Remainder after Division. Q. What is the lower Number called? A. It is called Denominator, and denotes any Whole divided into Parts: and is the Divisor in Division. Q. How many sorts of Vulgar Fractions are there? A. When the Numerator is less than the Denominator, as 7. Q. How far may a proper Fraction be expressed? A. Without end: as may be called 2 or 3 or 4, &c. but the lowest Term is always desired. Q. What is an improper Fraction ? A. When the Numerator is greater than the Denominator, as . Q. What is a Compound Fraction? A. It is the Fraction of a Fraction, as of 3. &c. OF REDUCTION OF VULGAR FRACTIONS. Q. How CASE 1. OW are Vulgar Fractions reduced to a common denominator? A. 1. Multiply each Numerator into all the Denominators but its own, for a new Numerator. 2. Multiply all the Denominators for a common Denominator. 1 Reduce & to a common denominator. Facit &. 2 Reduce Z. aud to a common denominator. Facit 360 86 and 880 960. 3 Reduce 6 Facit 2024 2520 560 30 0 5040 3040 4 Reducer 7. Facit 16 and, to a common denominator. and 4320 and to a common denominator. 1188 and 693 1386 1386 138 2 1386 5 Reduce 9, 4, and 7 to a common denominator. 8 Facit 1,432,504 and 1323. Reduce Facit 480 480 384 240 400 20 480 CASE 2. Q. How do you reduce a Vulgar Fraction to its lowest Terms A. Find a common Measure by dividing the lower Term by the Upper; and that Divisor by the Remainder following, till nothing remains: The last Divisor is the common Measure. 2. Divide both parts of the Fraction by the common Measure, and the quotient will make the Fraction required. Note 1. If the common Measure happens to be 1, the given Fraction is already in its lowest Terms. 2. When a Fraction hath Cyphers at the right hand, it may be abbreviated by cutting them off; thus, 8. Facit . . It is composed of a whole Number and a Fraction, thus 72 Q. How is a mixt Number reduced to an improper Fraction. A. 1. Multiply the whole Number into the Denominator of the Fraction. 2. To the Product, add the Numerator for a new Numerator 1 Q. How is an improper Fraction reduced to its proper Terms. A. Divide the upper Term by the lower. 48 EXAMPLE3. 1 Reduce 2 to its proper Terms Facit 121. 961 CASE 5. Q. How do you reduce a Compound Fraction to a single one? A. 1. Multiply all the numerators for a new numerator. 2. Multiply all the denominators for a new denominator. Q. How do you reduce the Fraction of one denomination to the fraction of another, but greater, retaining the same value? A. 1. Reduce the given fraction to a compound fraction by comparing it with all the denominations between it, and that denomination, which you would reduce it to. 2. Reduce that compound fraction to a single one, by Case 5. EXAMPLES. 1. Reduce of a Penny to the Fraction of a Pound. Facit i |