2. Find the least common multiple of (1) 12, 8, and 9. (2) 8, 12, and 16. (3) 6, 10, and 15. (4) 8, 12, and 20. (17) 1, 2, 3, 4, 5, 6, 7, 8, and 9. Ex. IX. Miscellaneous Questions and Examples on the foregoing Articles. I. (1) Explain the principle of the common system of numerical notation. Multiply 603 by 48, and give the reasons for the several steps. (2) Write at length the meaning of 9090909, and of 90909. Find their sum and difference, and explain fully the processes employed. (3) Find the difference between the sum of 4715 added to itself 398 times, and the sum of 2017 added to itself 408 times. (4) A person, whose age is 73, was 37 years old at the birth of his eldest son; what is the son's age? (5) Explain the meaning of the terms 'vinculum', 'bracket'; and of the signs +, −, =,.., X. Find the value of the following expression: 15 × 37153-73474-67152÷4+40734 × 2. II. (1) Define 'a Unit', 'Number', 'Arithmetic'. What is the difference between Abstract and Concrete numbers? (2) The annual deaths in a town being 1 in 45, and in the country 1 in 50; in how many years will the number of deaths out of 18675 persons living in the town, and 79250 persons living in the country, amount together to 10000? (3) Define 'Notation', 'Numeration'; express in numbers seven hundred thousand four hundred and nine billions. (4) Find the value of 494871-94853+(45079-3177)-(54312-3987)-(1763+231)+379×379. (5) What number divided by 528 will give 36 for the quotient, and leave 44 as a remainder? III. (1) Define Multiplication, and Division. Shew that the product of two numbers is the same in whatever order the operation is performed. (2) The Iliad contains 15683 lines, and the Eneid contains 9892 lines; how many days will it take a boy to read through both of them, at the rate of eighty-five lines a day? (3) Explain what is meant by the greatest common measure, and by the least common multiple of two or more numbers; and shew that the product of two numbers is the product of their least common multiple into their greatest common measure. Find the least common multiple of 12, 16, 21, 52, and 70. (4) Explain the meaning of the sign÷, and find the value of (7854-4913) × 3-(20374-12530)÷53-6+(395456-2364)÷556. (5) At a game of cricket A, B, and C together score 108 runs; B and C together score 90 runs, and A and C together score 51 runs; find the number of runs scored by each of them. IV. (1) Define Addition, and Subtraction. What is meant by a prime number? When are numbers said to be prime to each other? Give examples. Explain the rule of carrying in the addition of numbers; exemplify it in the addition of 3864, 4768, and 15938. (2) There are two numbers of which the product is 373625; the greater number is 875; find the sum and difference of the numbers. (3) A father was 21 years old when his eldest son was born; how old will his son be when he is 50 years old, and what will be the father's age when the son is 50 years old? (4) Write in figures one hundred millions, one hundred thousand, one hundred and one; and in words 1010101010. Express in figures M.DCCC.XL. (5) When are numbers said to be 'composite'? Find the greatest number which can divide each of the two numbers 849 and 1132; also the least number which can be divided by each of them; explaining the process in each case. ས. (1) Multiply 478 by 146, and test the result by casting out the nines. In what cases does this method of proof fail? Divide 4843 by 99, and prove the correctness of the operation by any test you please. (2) What number multiplied by 86 will give the same product as 163 by 430? (3) In the city of Prague, for every two persons who speak German only, three speak Tschech only, and seven both German and Tschech; and the whole population is 120000. How many speak German only, Tschech only, and both German and Tschech? (4) A gentleman dies, and leaves his property thus: 10000 pounds to his widow; 15000 pounds to his eldest son, on the condition of his building a national school at a cost of 350 pounds; 5500 pounds to each of his four younger sons; 3750 pounds to each of his three daughters; 4563 pounds to different societies; and 599 pounds in legacies to his servants. What amount of property did he die possessed of? (5) The quotient arising from the division of 9281 by a certain number is 17, and the remainder is 373. Find the divisor. VI. (1) Explain briefly the Roman method of Notation. Express 1563 and 9000 in Roman characters. (2) Explain the terms 'factor', 'product', 'quotient'; shew by an example how the process of Division can be abridged, if the divisor terminate with cyphers. (3) The remainder of a division is 97, the quotient 665, and the divisor 91 more than the sum of both. What is the dividend? (4) Express in words the numbers 270130 and 26784; also write down in figures the number ten thousand, two hundred and thirty four; and find the least number which added to the last number will make it divisible by 8. (5) A gentleman, whose age is 60, has two sons and a daughter; his age equals the sum of the ages of his children; two years since his age was double that of his eldest son; the sum of the ages of the father and the eldest son is seven times as great as that of the youngest son; find the ages of the children. FRACTIONS. 58. If 1 represent any concrete quantity, as for instance 1 yard, it is divisible into parts: suppose the parts to be equal to each other, and the number of them 3; one of the parts would be denoted by (read one-third), two of them by (read two-thirds), three of them or the whole yard by or 1; if another equal portion of a second yard divided in the same manner as the first be added, the sum would be denoted by ; if two such portions were added, by §; and so on. Such expressions, representing any number of parts of a unit, that is, of the quantity which is denoted by 1, are termed BROKEN NUMBERS or FRACTIONS; we may therefore define a fraction thus: 59. DEF. A FRACTION denotes a part or parts of a unit; it is expressed by two numbers placed one above the other with a line drawn between them; the lower number is called the DENOMINATOR, and shews into how many equal parts the unit is divided; the upper is called the NUMERATOR, and shews how many of such parts are taken to form the fraction. Thus denotes that the unit is divided into 6 equal parts, and that 5 of these parts are taken to form the fraction: so, if a yard were divided into 6 equal parts, and 5 of them were taken, then denoting one yard by 1, we should denote the parts taken by the fraction . Again, 7 denotes that the unit is divided into 6 equal parts, and that 7 such parts are taken to form the fraction; for instance, in the example before us, one whole yard would be taken, and also one of the equal parts of another yard divided in the same manner as the first. 60. A Fraction also represents the quotient of the numerator by the denominator. Thus, represents 5÷6; for we should obtain the same result, whether we divide one unit into 6 equal parts, and take 5 of such parts (which would be represented by §); or divide five units into 6 equal parts, and take 1 of such parts, which would be equivalent to 1th part of 5 units, i. e. 5÷6: hence & and 5÷6 will have the same meaning. 61. When fractions are denoted in the manner above explained, they are called VULGAR FRACTIONS. Fractions, whose denominators are composed of 10, or 10 multiplied by itself, any number of times, are often denoted in a different manner; and when so denoted, they are called DECIMAL FRACTIONS. VULGAR FRACTIONS. 62. In treating of the subject of Vulgar Fractions, it is usual to make the following distinctions: (1) A PROPER FRACTION is one whose numerator is less than the denominator; thus, 2, 3, 4 are proper fractions. (2) AN IMPROPER FRACTION is one whose numerator is equal to or greater than the denominator; thus, %, %, are improper fractions. (3) A SIMPLE FRACTION is one whose numerator and denominator are simple integer numbers; thus, , are simple fractions. (4) A MIXED NUMBER is composed of a whole number and a fraction; thus, 5, 7 are mixed numbers, representing respectively 5 units, together with th of a unit; and 7 units, together with ths of a unit. (5) A COMPOUND FRACTION is a fraction of a fraction; thus, of 2, of of are compound fractions. (6) A COMPLEX FRACTION is one which has either a fraction or a mixed number in one or both terms of the fraction; thus, 2주 2 213 21 7' 3'43' 51' 63. It is clear from what has been said, that every integer may be considered as a fraction whose denominator is 1; thus, 5=5, for the unit is divided into 1 part, comprising the whole unit, and 5 of such parts, that is 5 units, are taken. 64. To multiply a fraction by a whole number, multiply the numerator of the fraction by it. Thus, & × 3=4. Reason for the above process. In the unit is divided into 7 equal parts, and 2 of those parts are taken : whereas in the unit is divided into 7 equal parts, and 6 of those parts are taken; i. e. 3 times as many parts are taken in as are taken in, the value of each part being the same in each case. |