EXAMPLES FOR PRACTICE. Reduce 1. 3102 in the quaternary scale to the ternary scale. Ans. 21210. 2. 4052 in the senary scale to the quinary scale. Ans. 12041. 3. 1234 in the octary scale to the binary scale. Ans. 1656. 5. 7605 in the octary scale to the nonary scale. Art. 77. To add, subtract, multiply, or divide in a nondecimal scale. Rule. Proceed as in the decimal scale, using the given radix as 10 is used in similar cases in the decimal scale. NOTE.-Examples of these operations have already been presented in Articles 34, 39, 54, and 71. EXAMPLES FOR PRACTICE. 1. Add 53, 47, 253, 741, and 124 in octary notation. VARIATION. Art. 78. One quantity varies directly as another, when it increases proportionally as the other increases, and decreases proportionally as the other decreases. Thus, if the price of a unit of any article is fixed, the cost of any number of articles varies directly as their number; and in general, the quantity of effect varies directly as the quantity of cause. ILLUSTRATION.-At 10 cents per pound, 2 pounds of sugar cost 2 times 10 cents, 3 pounds cost 3 times 10 cents, &c. Again, at 25 miles per hour, a train moves, in 2 hours, 2 times 25 miles, in 3 hours 3 times 25 miles, &c.; that is the distance traveled varies directly as the time. One quantity varies inversely as another when it increases proportionally as the other decreases, and decreases proportionally as the other increases. Thus, the time occupied in traveling a certain distance varies inversely as the velocity; that is, with twice a given velocity, the distance would be traveled in half the time; with three times the given velocity, in one-third of the time. ILLUSTRATION.—At 2 miles per hour, it would require 6 hours to walk 12 miles; but, at twice 2 miles per hour, or 4 miles per hour, it would require only one-half of 6 hours, that is, 3 hours. Art. 79. In respect to their variation, the quantities of a problem are either variables or constants. A variable is a quantity which may have different values without altering the form of the problem. A constant is a quantity which does not vary with the other quantities with which it is connected. Art. 80. The sign of variation is the symbol ∞, placed between the symbols which represent the varying quantities. Thus, if C is a symbol for cost, and N for the number of articles, the fact that cost varies directly as quantity is expressed by C∞ N. If D represents distance traveled, and T the time of travel, the fact that distance varies directly as time is expressed by D ∞ T. Inverse variation is represented by writing the second member of the expression as a fraction having the numerator 1. Thus, if V represents velocity and T time, the fact that the time of traveling a certain distance varies inversely as the velocity is expressed by This, interpreted, means that, if the value of T is taken as twice what it was in a certain case, that of V must be half what it was in that case; if T is taken as 3 times what it was, V must be one-third of what it was, &c. RELATIONS BETWEEN FACTORS AND PRODUCT. Art. 81. The product varies directly as the factors. ILLUSTRATION. Since any factor may be considered the multiplier, doubling a factor doubles the number of times the other factor is taken, halving a factor halves the number of times the other factor is taken, &c. Art. 82. The product is constant when one factor varies inversely as the other. ILLUSTRATION. The product of a factor, taken a certain number of times, is the same as that of twice that factor, taken half as many times, or as one-third of that factor, taken three times as many times, &c. RELATIONS BETWEEN DIVIDEND, DIVISOR, AND QUOTIENT. Art. 83. When the divisor is constant, the quotient varies directly as the dividend. ILLUSTRATION.-The same divisor is contained in twice the dividend twice as many times; in three times the dividend three times as many times; in half the dividend half as many times, &c. Art. 84. When the dividend is constant, the quotient varies inversely as the divisor. ILLUSTRATION.-In the same dividend twice the divisor is contained half as many times; three times the divisor, one-third as many times, one-half of the divisor, twice as many times, &c. Art. 85. When the dividend and divisor vary proportionally, the quotient is constant. ILLUSTRATION.-Twice the divisor is contained in twice the dividend as many times as the divisor is contained in the dividend. The same is true of three times the divisor in three times the dividend, one-half of the divisor in one-half of the dividend, &c. PROBLEMS IN THE FUNDAMENTAL RULES. Art. 86. Addition, Subtraction, Multiplication, and Division are called the fundamental rules of Arithmetic, because every arithmetical process depends upon one or more of them. These four operations, are strictly speaking, reducible to two; namely, Addition, which has for its object the increase of a quantity, and Subtraction, which has for its object the decrease of a quantity. Multiplication may be considered as only a convenient method of obtaining the results of certain additions, and Division as only a convenient method of obtaining the results of certain subtractions. PROBLEMS IN ADDITION. Art. 87. The fundamental problem of Addition is Given the parts of an unknown quantity, required to find that quantity. The specific forms of this problem are 1. Given two or more numbers to find their sum. 2. Given the parts to find the whole. 3. Given the less of two numbers and their difference, to find the greater. ILLUSTRATIONS. 1. Find the sum of 483679, 8007346, 800, 800400, and 7960432. 2. Find the sum of 809, 4768407, 80004324, 7960434, and 7764383. 3. In a farm, A owned 640 acres, B 720, and C 872. How many acres in the whole farm? 4. The smaller of two numbers is 7963, their difference is 728. What is the greater number? PROBLEMS IN SUBTRACTION. Art. 88. The fundamental problem of Subtraction is Given the whole of a certain quantity, and one of its two component parts, required to find the other component part. The specific forms of this problem are 1. Given two numbers, to find their difference. 2. Given the sum of two numbers, and one of them, to find the other. 3. Given the whole of a quantity, and all the parts except one, to find that part. 4. Given the greater of two numbers, and their difference, to find the less. ILLUSTRATIONS. 1. From 78400038 take 46784. 2. The sum of two numbers is 8496384, one of the numbers is 79849; what is the other? 3. A, B, C, and D own a farm containing 7644 acres. A owns 763 acres, B 972, D 2464, and C the remainder. How many acres does Clown? 4. The greater of two numbers is 48796, and their difference is 40006. What is the other number? PROBLEMS IN MULTIPLICATION. Art. 89. The fundamental problem of Multiplication is Given a quantity, and the number of quantities of the same kind and value required to find the sum of that number of those quantities. The specific forms of this problem are 1. Given equal quantities, and their number, to find their sum. 2. Given one of the equal parts of a quantity, and their number, to find that quantity. 3. Given the number of causes, and the value of each, to find the effect. ILLUSTRATIONS. 1. B has 486 sheep in each of 13 fields. How many sheep has he? 2. An estate was divided among 43 heirs, each receiving $2748. What was the estate worth? 3. If 16 men build a wall in 35 days, how long will it take 1 man to build it? PROBLEMS IN DIVISION. Art. 90. The fundamental problems of Division are 1. Given a quantity, and one of its equal parts, required to find the number of such parts. 2. Given a quantity, and the number of its equal parts, required to find one of those parts. |