words multiple and measure are thus connected: Since 4 is a measure of 24, 24 is a multiple of 4. The number 96 is a multiple of 8, 12, 24,` 48, and several others. It is therefore called a common multiple of 8, 12, 24, 48, &c. The product of any two numbers is evidently a common multiple of both. Thus, 36 × 8, or 288, is a common multiple of 36 and 8. But there are common multiples of 36 and 8 less than 288; and because it is convenient, when a common multiple of two quantities is wanted, to use the least of them, I now shew how to find the least common multiple of two numbers. 103. Take, for example, 36 and 8. Find their greatest common measure, which is 4, and observe that 36 is 9 × 4, and 8 is 2 × 4. The quotients of 36 and 8, when divided by their greatest common measure, are therefore and 2. Multiply these quotients together, and multiply the product by the greatest common measure, 4, which gives 9 × 2 × 4, or 72. This is a multiple of 8, or of 4 × 2, by (55); and also of 36, or of 4×9. It is also the least common multiple; but this cannot be proved to you, because the demonstration cannot be thoroughly understood without some knowledge of algebra. But you may satisfy yourself that it is the least in this case, and that the same process will give the least common multiple in any other case which you may take. It is not even necessary that you should know it is the least. Whenever a common multiple is to be used, any one will do as well as the least. It is only to avoid large numbers that the least is used in preference to any other. When the greatest common measure is 1, the least common multiple of the two numbers is their product. The rule then is: To find the least common multiple of two numbers, find their greatest common measure, and multiply one of the numbers by the quotient which the other gives when divided by the greatest common measure. To find the least common multiple of three numbers, find the least common multiple of the first two, and find the least common multiple of that multiple, and the third, and so on. 104. Suppose it required to divide 49 yards into five equal parts, or, as it is called, to find the fifth part of 49 yards. If we divide 49 by 5, the quotient is 9, and the remainder is 4; that is (2), 49 is made up of 5 times 9 and 4. Let the line A B represent 49 yards: take 5 lines, C, D, E, F, and G, each 9 yards in length, and the line н, 4 yards in length. Then, since and H, are together equal to a B. equal parts, 1, K, L, M, and N, and each of the lines C, D, E, F, and G. It follows that the ten lines, C, D', E, F, G, I, K, L, M, N, are together equal to a B, or 49 yards. Now D length as c and I together, and so are E Therefore c and I together, repeated 5 49 is 5 nines and 4, C, D, E, F, G, Divide H, which is 4 yards, into five place one of these parts opposite to and K together are of the same and L, F and м, and G and N. times, will be 49 yards; that is, c and I together make up the fifth part of 49 yards. 105. c is a certain number of yards, viz. 9; but 1 is a new sort of quantity, to which hitherto we have never come. It is not an exact number of yards, for it arises from dividing 4 yards into 5 parts, and taking one of those parts. It is the fifth part of 4 yards, and is called 4 a FRACTION of a yard. It is written thus, (23), and is what we 5 must add to 9 yards in order to make up the fifth part of 49 yards. The same reasoning would apply to dividing 49 bushels of corn, or 49 acres of land, into 5 equal parts. We should find for the fifth part of the first 9 bushels, and the fifth part of 4 bushels; and for the second 9 acres, and the fifth part of 4 acres. We say then, once for all, that the fifth part of 49 is 9 49 ; which is usually written 91 or if we use signs, = 5 and or EXERCISES. 13 What is the seventeenth part of 1237? Answer, 72 17 106. By the term fraction is understood a part of any number, or the sum of any of the equal parts into which a number is divided. 49 4 20 Thus, 5.5' 7' are fractions. The term fraction even includes whole numbers: for example, 17 is 17 The upper number is called the numerator, the lower number is called the denominator, and both of these are called terms of the fraction. As long as the numerator is less than the denominator, the fraction is less than a unit: thus, is less than a unit; for 6 divided into 6 parts gives 1 for each part, and must give less when divided into 17 parts. Similarly, the fraction is equal to a unit when the numerator and denominator are equal, and greater than a unit when the numerator is greater than the denominator. Numbers which contain an exact number of units, such as 5, 7, 100, &c., are called whole numbers or integers, when we wish to distinguish them from fractions. To prove this, let A B be two yards, and divide each of the yards a c Then, because A E, EF, and F B, are all equal to one another, A E is the third part of 2. 3 2 I 3 order to get the length it makes no difference whether we divide two yards at once into three parts, and take one of them, or whether we divide one yard into three parts, and take two of them. By the same reasoning, may be found either by dividing 5 into 8 parts, and taking one of them, or by dividing 1 into 8 parts, and taking 5 of them. In future, of these two meanings, I shall use that which is most convenient at the time, as it is proved that they are the same thing. This principle is the same as the following: The third part of any number may be obtained by adding together the thirds of all the units of which it consists. Thus, the third part of 2, or of two units, is made by taking one-third out of each of the units, that is, 2 3 7 This meaning appears ambiguous, when the numerator is greater than the denominator: thus, 15 would mean that I is to be divided into 7 parts, and 15 of them are to be taken. We should here let as many units be each divided into 7 parts as will give more than 15 of those parts, and take 15 of them. 4' 108. The value of a fraction is not altered by multiplying the numerator and denominator by the same quantity. Take the fraction 3, mul15 tiply its numerator and denominator by 5, and it becomes which is 3 4 20' the same thing as ; that is, one-twentieth part of 15 yards is the same thing as one-fourth of 3 yards: or, if our second meaning of the word fraction be used, you get the same length by dividing a yard into 20 parts and taking 15 of them, as you get by dividing it into 4 parts and taking 3 of them. To prove this, let A B represent a yard; divide it into 4 equal parts, a C, C D, d E, and 3 E B, and divide each of these parts into 5 equal parts. Then A E is 4 But the second division cuts the line into 20 equal parts, of which a E contains 15. It is therefore are the same thing. 3 15 20 15 20 15 Therefore and 3 20 This Again, since is made from by dividing both the numerator and denominator by 5, the value of a fraction is not altered by dividing both its numerator and denominator by the same quantity. principle, which is of so much importance in every part of arithmetic, is often used in common language, as when we say that 14 out of 21 is 2 out of 3, &c. 15 20 109. Though the two fractions 3 and are the same in value, and either of them may be used for the other without error, yet the first is more convenient than the second, not only because you have a clearer idea of the fourth of three yards than of the twentieth part of fifteen yards, but because the numbers in the first, being smaller, are more convenient for multiplication and division. It is therefore useful, when a fraction is given, to find out whether its numerator and denominator have any common divisors or common measures. In (98) was given a rule for finding the greatest common measure of any two numbers; and it was shewn, that when the two numbers are divided by their greatest common measure, the quotients have no common measure except 1. Find the greatest common measure of the terms of the fraction, and divide them by that number. The fraction is then said to be reduced to its lowest terms, and is in the state in which the best notion can be formed of its magnitude. EXERCISES. With each fraction is written the same reduced to its lowest terms. |