Again, let 5:7 be a ratio of less inequality; then by subtracting 2 from each term, we obtain a new ratio of 3: 5. the magnitude of the given ratio is and that of the new one Whence = 59 the quantity by which the magnitude of 5: 7 exceeds the magnitude of 3 : 5. CASE IV.-The product of all the antecedents of any number of ratios, and the product of all the consequents, will form a new ratio. Thus, if there be taken 3: 4, 5: 6, and 7:8; then will 3 X 5×7:4 x 6 x 8; or 105; 192; and this ratio, compounded of the others, is called their sum. EXAMPLES. 1. How much is the magnitude of the ratio of 7 : 5 diminished by adding 2 to each of its terms? Ans. उड 2. How much is the magnitude of the ratio of 5: 7 increased by adding 2 to each of its terms? Ans. उ : 3. How much is the magnitude of the ratio of 7 5 increased by subtracting 2 from each term? Ans. 4. How much is the magnitude of the ratio of 5 : 7 diminished by subtracting 2 from each quantity? Ans. 5 : 5. Find whether the ratios 6: 7, 9: 11, and 8 5 are increased or decreased by adding 2, 3, and 4 to their terms respectively. Ans. 1st increased by 5, 2nd, by 737, and 3rd diminished by + 6. Find whether the ratios 7 5, 8: 11, and 9:15 are increased or diminished by subtracting 2, 4, and 6 from their terms respectively. Ans. 1st increased by 4, 2nd decr. 44, and 3rd decreased. 7. What is the ratio arising from the composition of 4:5 and 6: 7? Ans. 24 35. 8. What is the ratio arising from the composition of, 3 and 4: 5? : : Ans. 1:44. PROPORTION. PROPORTION is the relation of equality between two ratios: Thus, if we take the two ratios of equality, 4 : 6 and 8: 12, the magnitude of the first is 4, and that of the second, and they are equal; that is 4: 68: 12, Therefore Whence 4 x 12: 6 x 8. In Proportion, the product of the means is equal to the product of the extremes; therefore 4: 6: 8: 12. In this proportion the 4 and 12 are the extremes, and the 6 and 8 the means. In every proportion, the first term is greater, equal to, or less than the second, according as the third term is greater, equal to, or less than the fourth. Two ratios not having the same magnitude, the four terms are not proportionals. If four numbers constitute a proportion, the product of the means is equal to the product of the extremes; hence, if the four terms of the proportion admit of a common divisor, each term may be divided by it, without destroying the proportion. Thus in the proportion 16: 24:: 32 : ; 48 ; Dividing by 8, then 2: 3 :: 4 : 6 For 14, and 22 NOTE. For a more particular account of Proportion, see the Author's Algebra, or Dr. Wood's, by Lund. GEOMETRICAL MEAN PROPORTION. A geometrical mean proportioual between any two numbers is the square root of their product. If it be required to find a geometric mean between 4 and 16: Here 4 X 1664; then /648, the geometric mean required. .. 48: 8:16. To find two geometrical mean proportionals between two given numbers. Divide the greater of the two given terms by the less, and the cube root of the quotient will be the common ratio of the terms. Multiply the first term by this ratio for the first mean, and the product of this mean and the ratio will be the second mean. = Find two mean proportionals between 4 and 32. Here 324 = 8; then 3/8 = 2 the ratio. ..4 x 2 = 8 the first mean, and 8 x 2 = 16 the second mean. Hence 4, 8, 16, 32, T To find any number of geometrical means. Divide the greater term by the less, and take such a root of the quotient as is denoted by the number of terms plus 1; i. e. if 3 means be required, the fourth root of the quotient must be taken; if 4 means be required, the fifth root of the quotient must be taken; and so on for the ratio. Multiply the 1st term by the ratio for the 1st mean, this mean by the ratio for the 2nd mean. and so on. Required 3 geometrical means between 2 and 32. .. 2 × 2 = 4 the first mean; 4 × 2 = 8 the second; Hence 2, 4, 8, 16, 32. PERMUTATION AND COMBINATION. INTRODUCTION. PERMUTATION is a changing of the relative situation or position of any given number of things, so that no two parcels or quantities of them shall have the quantities of which they are composed placed in the same order. Thus, if we take the letters a and b, they will permute, or change their position two different ways, as ab and ba, or 1 × 2 = 2. Again, if we take three letters a, b and c, they may be permuted or changed six different ways, as abc, acb, bca, cab, and cba; or 1 × 2 × 3 = 6 different ways. If four letters be taken, as a, b, c and d, then will 1 × 2 × 3 X 4 = 24 different ways, and so on for any number of quantities. Or, how many changes may be rung on four bells? Thus, 1 × 2 × 3 × 4 = 24, Ans. Permutation is sometimes called Changes or Variation. COMBINATION is the several ways that a less number of things can be taken out of a greater. Thus, if we take two letters, a and b, only one combination can be formed, namely, ab or ba. If we take the quantities or letters a, b and c, we shall have only three combinations, by taking two and two, namely, ab, ac, and be; but they will form six permutations, ab, ba, ac, ca, bc, and cb. If we take the four letters, a, b, c, d, and combine them two and two; a may be combined as follows, ab, ac ad, bc, bd, and cd, thus we have six combinations of the four quantities, two together. But, for their permutation, we must write each combination twice, viz. ab and ba, &c., which will make twelve permu tations, when two are taken together: as how many changes may be rung with 2 bells out of 4? 4 × (4 −1) = 4 × 3 = 12. PROBLEM I.-To find the number of permutations, or changes that can be made of any given number of different things. RULE* Multiply all the terms of the given series continually together, and the last product will be the number of permutations or changes required. 1. How many changes may be rung on 6 bells? Here 1 x 2 x 3 x 4 × 5 × 6 = 720 Ans. 2. How many days can 8 persons be placed in a different position at dinner? Ans. 40320 days. 3. There are 13 bells in the Parish Church, Leeds. How many changes can be rung on the same; and what time will it require, allowing 10 changes per minute, and the year to consist of 365 days, 5 hours and 49 minutes? Ans. 6227020800 changes, and 1183 years, 350 days, 6 hours, 53 min. 11 sec. PROBLEM II.-Given any number of different things, to find how many permutations or changes can be made out of them. by taking a given number of them at a time. RULE.-Take a series of numbers, beginning at the given number of things, and decreasing by 1 to the number of quantities to be taken at a time, and the continued product of all the terms will be the answer required. 1. How many changes may be rung upon 2 bells out of 8; that is taking 2 at a time? = Ans. Here 8 (81) — 8 × 7 — 56 2. How many changes may be rung with 3 bells out of 8 ? Here 3 at a time are taken, Then 8x (8-1)x(7-1)=8X7X6-336 Ans. PROBLEM III.-To find how many combinations may be made out of a given number of different things, by taking a given number at a time, so that no two sets may be alike. RULE -Take the numbers 1, 2, 3, &c. up to the number of things to be taken at a time, and find their product. *The reason of this rule is obvious from the introduction. Take a series of as many terms, as the number of things to be taken at a time, beginning with the given number out of which the election is to be made, decreasing each succeeding term by 1, and find their product; divide this product by the former, and the quotient will be the number of combinations required. 1. How many combinations can be made of 4 letters out of 9? Here, 4 are to be taken at a time; then, 1 X2 X3 X 4 24, for the divisor. Again, 9 is the given number, and as 4 is to be taken out of them at a time, we must have a series of four numbers beginning with 9 as follows: 9 × (9-1) x (8-1) x (7-1) Or, 9 x 8 x 7 x 6 3024, the dividend. 24) 3024 (126 the number of combinations. 2. How many days may 10 men dine, taking 6 at a time, without the same company coming a second time? Ans. 210 days. PROBLEM IV.-Given a number of things, in which there are several of one sort and several of another, to find how many permutations or changes can be made out of then. RULE.-Take the series 1, 2, 3, 4, &c. up to the given number of things, and find their product. Take the series 1, 2, 3, 4, &c. up to the given number of things of the first sort, and the series 1, 2, 3, 4, &c. up to the given number of things of the second sort, and so on. Divide the product of all the terms of the first series by the joint product of all the terms of the remaining series, and the quotient will be the number of permutations or changes required. 1. How many variations can be made of the letters in the word Bacchanalia ? Here are 11 letters in the word, Then I×2×3×4×5×6×7×8×9×10×11=39916800; Again, we have 4 a's and 2 c's; Then 1×2×3× 4-24 for the a's, and 1X2 2 for the c's. Then 24X2=48; Therefore 39916800-48-831600 Ans. 2. How many variations can be made of the letters in the word Teetotalism? Ans. 3326400. |