DEM.-Inverting the first term is exactly the same as multiplying the second and third terms together and dividing by the first, as plainly appears from Division of Vulgar Fractions, therefore the same reasoning might be offered here as in the Rule of Three Direct in whole numbers, so that the reason of our rule is obvious. 2. If of a ship cost 150 pounds; what are of her worth? Ans. £17 17s. 1d. 2 qrs. 3. If 1 yard cost of a pound; what will 42 yards cost? Ans. £35. 4. If of a yard cost of a shilling; what will of a yard cost? 5. If of a yard cost cost? Ans. 1s. 2d. of a dollar; what will 44 yards Ans. $79,11 cents 1 mills. 5. If 3 yards cost 24 pounds; what will 143 yards cost? Ans. £13 15s. 4d. Rule of Three Inverse in Vulgar Fractions. RULE.-Prepare the fractions and state the question as in the RULE OF THREE DIRECT; then invert the third term, and multiply all the three terms together; the product will be the answer in the same name of the second term. EXAMPLES. 1. How much cloth of a yard wide, will line 15 yards of cloth, 1 yard wide? Ans. 30 yards. :: yd. yds. yd. Invert the third term thus, as yd: 15yds.:: yd.: 30 yds. the Answer. DEM.--Inverting the third term is the same as multiplying the first and second terms together and dividing the product by the third; hence the reason of the rule is obvious. 2. What length of board 7 inches wide, will make a square foot, or as much as another piece of 12 inches long and 12 broad? Ans. 18 inches. 3. A regiment of soldiers, consisting of 980 men, are to be clothed anew; each coat to contain 22 yards of cloth that is 1 yard wide, and lined with shalloon yard wide; how many yards of shalloon will it take to line them? Ans. 4550 yards. 4. If a suit of clothes can be made of 43 yards of cloth, 13 yard wide; how many yards of coating of a yard wide, will Ans. 6yds. 1qr. 3 n. hours, in how Ans. 17h. it require for the same person? many 6. How much in length, of a piece of land that is 111 poles broad, will make an acre? Ans. 13 poles. QUESTIONS ON VULGAR FRACTIONS. NOTE. In asking the questions on Vulgar Fractions, the teacher should commence with those following the introduction to Vulgar Fractions, page 123, and end with the following questions: How do you reduce fractions to a common denominator? A. By multiplying each numerator into all the denominators except its own, for the new numerators, and multiplying all the denominators together for a common denominator. Why does not that operation alter the value of the fractions? A. Because it is multiplying each numerator and its denominator by the same numbers, and consequently the value of each fraction remains the same. What is the use of reducing fractions to a common denominator? A. In order to prepare fractions for adding or subtracting, as the case may be. Until the fractions are reduced to a common denominator, they cannot be added or subtracted, because an integer is unequally divided, as our denominators show, consequently the unequal parts cannot be placed in one column for adding or subtracting. What is the effect, if you multiply the numerator of a fraction by any number, the denominator remaining unchanged? A. It increases the value of a fraction as many times as the multiplier expresses a unit. If you multiply the denominator of a fraction by any number, the numerator remaining unchanged, what is the effect? A. It diminishes the value of the fraction as many times as the multiplier expresses a unit. Multiplying or dividing the numerator and denominator of the fraction by the same number; what is the effect? A. It alters the terms of the fraction, not its value. If you divide the numerator of a fraction by any number, the denominator remaining unchanged; what is the effect? A. It lessens the value of the fraction as many times as the divisor expresses a unit. If you divide the denominator of a fraction by any number, the numerator remaining unchanged; what is the effect? A. It increases the value of the fraction as many times as the divisor expresses a unit. ARITHMETICAL PROGRESSION, Is any rank or series of numbers, more than two, increasing or decreasing by a common difference. When the numbers increase by a continual addition of the common difference, they form an ascending series; but when they decrease by a continual subtraction of the common difference, they form a descending series. 2, 4, 6, 8, 10, 12, &c. is an ascending series. Thus, {13, 10, 8, 8, 4; 2, &c. is a descending series. The numbers which form the series are called the TERMS of the progression. THE FIRST and LAST terms are the EXTREMES, and the other terms are called the MEANS. In Arithmetical Progression there are five terms, three of which are always given in the question, 1st. The first term. 3d. The number of terms. 2d. The last term. 4th. The common difference. 5th the sum of all the terms. Any three of which being given, the other two may be found. CASE I-The first term, common difference, and number of terms given, to find the last term. RULE.-Multiply the number of terms, less 1, by the common difference, and add the first term to the product, the sum will then be the last term, or answer sought. EXAMPLES. 1. A merchant sold 200 yards of cloth; for the first yard he received 4 cents, 7 cents for the second, 10 cents for the third, and so on, with the common difference of three cents; what did he receive for the last yard? Ans. 601 cents $6,01 cent. 199 the number of terms less 1. 3 the common difference. 597 4 the first term added. Ans. 601 cents, he received for the last yard. DEM.-It is plain, that each term exceeds that preceding it by the common difference; it is then evident, that the last term exceeds the first by as many times the common difference, as there are terms after the first; therefore the last term must equal the first, and the number of terms, less one, repeated by the common difference, as our example plainly shows. 2. A man put out $100, at 7 per cent, simple interest, which amounted to $107 in a year, $114, in 2 years, and so on, in arithmetical progression, with the common difference of $7; what was the amount due at the expiration of 50 years? Ans. $450. 3. John owes William a certain sum, to be paid in arithmetical progression; the first payment is 6 pence, the number of payments 52, and the common difference of the payments is 12 pence; what is the last payment? Ans. £2 11s. 6d. CASE II.-The first term, common difference, and the number of terms given, to find the sum of all the terms. RULE.-Multiply half the sum of the extremes by the number of terms, and the product will be the answer or sum of all the terms. EXAMPLES. 1. A person bought 16 yards of cloth; for the first yard he gave 5 pence, for the second 9 pence, and so on, in arithmetical progression; what did the last yard cost him, and what did he pay for the whole ? [ceding case. Ans. 65 pence the cost of the last yard, obtained by the pre5 the cost of the first yard. 2)70 the sum of the extremes. 35 half the sum of the extremes. 16 the number of terms. 210 35 DEM. The price of each creased by a constant exsucceeding yard being incess, it is plain that the average price is as much less than the price of the last yard, as it is greater than the price of the first yard; hence it is evident, that half the sum of the first and last is the average price, and it is also obvious, that the average price multiplied by the whole number of yards, that is, terms, gives the price of the whole number of yards. 12)560 the sum of all the terms. 2/0)4/6 8d. A. £2 6s. 8d. the cost of 16 yards. 2. A man bought 17 yards of Irish linen; for the first yard he gave 2 shillings, for the last, 10 shillings, the price of each yard increasing in arithmetical progression; how much did Ans. £5 2s. the whole amount to? 3. How many times does the hammer of a clock strike in 12 hours? Ans. 78 times. NOTE. It is supposed that every scholar knows the first term and common difference of the hammer of a clock's striking. CASE III.-The extremes and number of terms being given, to find the common difference. RULE.-Divide the difference of the extremes by the number of terms, less 1, and the quotient will be the common difference. EXAMPLES. 1. A merchant sold 200 yards of cloth; for the first yard he received 4 cents, for the last, 601 cents; the common difference is required. Ans. 3, common difference. price of the last yard. 199) 597(3 common difference. 597 DEM.-It is plain, that the difference between the price of the first yard,and the price of the last yard, that is, the difference of the extremes, is equal to the additions which were made to the price of the first yard till it became equal to the last yard, and these additions were one less than the number of terms, therefore it is evident, that the sum of these additions, that is, the difference of the extremes, divided by the number of terms, less one, that is, the number of additions, gives the common difference. A 2. The first term is 3, the last term 48, and the number of terms 10; what is the common difference? Ans. 5. 3. A gentleman has 7 sons, whose several ages differ alike; the youngest is 5 years old, the eldest 47; what is the common difference of their ages? Ans. 7 years. QUESTIONS ON ARITHMETICAL PROGRESSION. What is Arithmetical Progression? A. It is any rank or series of numbers, increasing or decreasing by a common difference. What are numbers called when they increase by a continual addition? A. An ascending series. What are numbers called when they decrease by a continual subtraction? A. A descending series. When the first term, the common difference, and the number of terms are given, how do you find the last term? A. By multiplying the number of terms, less one, by the common difference, and adding the first term to the product for the last term. When the first term, common difference, and number of terms are given, how do you find the sum of all the terms? A. Multiplying one half the sum of the extremes by the number of terms, gives the answer or sum of all the terms. How do you find the common difference, when the extremes and number of terms are given? A. Dividing the difference of the extremes by the number of terms, less one, gives the common difference. GEOMETRICAL PROGRESSION, Is any rank or series of numbers increasing by one common multiplier as 2, 4, 8, 16, 32, &c. or decreasing by a common divisor; as 32, 16, 8, 4, 2, &c. First, the multiplier, 2, by which the series is increased, is called the ratio; secondly, the divisor, 2, by which the series is diminished, is called the ratio. When any number of terms is continued in Geometrical Progression, the product of the two extremes will be equal to the product of any two means equally distant from the extremes, or when the terms are odd, equal to the square of the middle term; thus, 2, 4, 8, 16, 32; 2X32=64, 4×16=64, 8×8=64. There are five terms given in Geometrical Progression, the same as in Arithmetical Progression. viz. : 1. The first term. 2. The last term. 3. The number of terms. 4. The ratio or common difference. 5, The sum of all the terms, NOTE. The first and last terms are called the extremes. Any two terms equally distant from the extremes, are called the means. |