multiple of 48 and 54, and we find 432, which is also the least common multiple of all the given numbers. Examples. 1. Required the least common multiple of 3, 5, 7, 4, and 11. Answer 4620. To multiply numbers with facility, much often depends on the order in which they are taken. Thus, in the present example, without writing a figure, we first say, 4 times 5 is 20; and, leaving 0 till the last, we say 3 times 2 is 6, and 7 times 6 is 42. Then, (126,) for 11 times 42, as the sum of 4 and 2 does not exceed 9, we place their sum 6 between those figures, and have 462. Lastly, we place 0 on the right, and have 4620 for the product. 2. Required the least common multiple of 2, 5, 10, 11, and 55. Answer, 110. 3. Required the least common multiple of 5, 8, 9, 11, and 13. Answer, 51480. 4. What is the least common multiple of the nine digits? Answer, 2520. 5. What is the least common multiple of 3, 17, 18, 51, and 54 ? Answer, 918. 6. What is the least common multiple of 6, 14, 39, 42, 78, and 91? Answer, 546. 7. What is the least common multiple of 18, 28, 33, 42, 48, 63, and 99? Answer, 11088. 8. What is the least common multiple of 6, 8, 12, 14, 16, 18, 21, and 22? Answer, 11088. 186. The number which divides the greater of two numbers and the less, will (172) divide their difference. Supposing the difference greater than the less number, the number which divides the less and the difference will also divide their difference. Again, supposing the difference still greater than the less, the number which divides the less number and this second difference will also divide their difference. Now, this continual subtraction of the less number is the same as the division of the greater number by the less. The final remainder must be less than the less number, and we might reason, as above, with respect to this remainder and the less number. Again, if there be still a remainder, the same reasoning may be pursued with respect to this remainder and the first remainder. Pursuing the same method, as a greater number cannot divide a less, the last remainder which succeeds in dividing the preceding will be the greatest common measure. Wherefore, to find the greatest common measure of two composite numbers, we have the following Rule. Divide the greater number by the less; if there is no remainder, the number divided by is the greatest common measure. If there is a remainder, divide the less number by this remainder, and if this division gives no remainder, the number last divided by is the greatest common measure. If there is still a remainder, divide the first remainder by the second; and thus continue, always dividing the preceding remainder by the last, till the division becomes exact. The last divisor is the greatest common measure. When the last divisor is a unit, the numbers are prime to each other. Examples. 1. Required the greatest common measure of 9388 and 16429. We proceed, according to the rule, thus: 9388) 16429 (1 7041) 9388 (1 2347) 7041 (3 **** and have 2347, the last divisor, for the required measure. Because 2347 divides 7041, it will evidently divide 7041+ 2347, which is 9388. Again, because 2347 divides 7041 and 9388, it will divide their sum, which is 16429. fore 2347 is a common measure of 9388 and 16429. There No number greater than 2347 can be their common measure: for, if possible, let x be greater than 2347, and, at the same time, their common measure. Then, because x measures 9388 and 16429, it measures their difference 7041 also, because it measures 9388 and 7041, it measures their difference 2347; that is, a number greater than 2347 measures it, which is absurd. Therefore 2347 is the greatest common measure. 2. Required the greatest common measure of 3675 and 5880. Answer, 735. 3. Find the greatest common measure of 4437 and 5899. Answer, 17. 4. Find the greatest common measure of 23205 and 31395. Answer, 1365. 5. Required the greatest common measure of 46503 and 57546. Answer, 9. 6. Required the greatest common measure of 698 and 5343. Answer, 7. Required the greatest common measure of 34177 and 1012924. Answer, 11. OF RATIO. 187. In division we say that the greater the dividend is, the greater will the quotient be, and the less the dividend, the less the quotient, (165.) This relation of the dividend and quotient is called direct ratio. Again, the greater the divisor is, the less will the quotient be; and the less the divisor, the greater the quotient. This relation of the divisor and quotient is called inverse ratio. 188. A simple ratio is the relation which one number bears to another, with respect to how often the one contains the other, or to what part the one is of the other; and is, in either case, expressed by that one divided by that other. Thus, if we would have the ratio of 3 to 5, or find what part 3 is of 5: as 1 unit is of 5, it is plain that 3 units must be ; say, the ratio of 3 to 5 is 3 divided by 5. Again, have the ratio of 5 to 3, or find how often 5 contains 3, this is evidently §. that is to if we would Let A and B be any two homogeneous quantities—that is, quantities, the units of which are alike; then, if A be greater than B, shows how often A contains B: and, if A be less than B, shows what part A is of B; for, B being a whole or unit divided into equal parts, determines the value of these parts, and A is the number of the same parts which constitute the fraction. Wherefore, the ratio of A to B is , and that of B to A is and the ratio of 4 to 8 is B . The ratio of 8 to 4 is &=2, 1. Hence, every fraction expresses a ratio; namely, the ratio of the numerator to the denominator. B The ratio is called the reciprocal of the ratio; that is either ratio (or either fraction) is the reciprocal of the other. 189. A compound ratio is that which involves two or more simple ratios. If two fractions have each the same denominator, the ratio of the one to the other is the simple direct ratio of their numerators. Thus, the ratio of to is the same as that of 3 to 5; namely, ; being (188) no more than the ratio of 3 units of a certain order to 5 units of the same order. Again, if two fractions have each the same numerator, they have to one another the inverse ratio of their denominators. Thus, the ratio of to is 7. For, the greater the number of parts in the unit, the less they are in value: therefore, whatever number of times 7 is greater than 5, so many times are the units of the fraction less than those of ; or, so many times are the units of greater than those of 3. 5 But is this number of times: therefore, is the ratio of to 3. The ratio of to is 10-2; which shows that is twice Again, the ratio of to is; which shows that is one-half of 3. 10 Wherefore, any two fractions have to one another the direct ratio of their numerators, and the inverse ratio of their denominators. The ratio of one fraction to another, therefore, being compounded of both these ratios, is a compound ratio. Of this we shall treat more fully hereafter. 190. As a fraction is composed of one or more units, generated by the division of some standard unit into a certain number of equal parts, the number of which is signified by its denominator, we call one of these equal parts the generic unit of the fraction. Thus, the generic unit of is, that ofis, &c. Now, as fractions, having like numerators, are to one another inversely as their denominators, the generic unit of one fraction is to that of another in this same ratio, which is found by dividing the denominator of the second by that of the first. Thus, the ratio of to is g; the ratio of tois, &c. Hence we say, that fifths are to sixths as 6 to 5; fifths to ninths as 9 to 5, &c. How many times greater are thirds than tenths? 191. A whole number is expressed in the form of a fraction by giving it a unit for denominator; this unit, being the generic unit of the number, both before and (190) after it is so expressed. Thus, 3. Now, as both terms of may be multiplied by any number in the scale 2, 3, 4, 5, 6, &c., ad inf., without (165) altering the value of the number 3, it is evident that we can give this, or any other whole number, any denominator that we please, without at all altering its valuenamely, by first multiplying the whole number by that denominator. Thus, if we would express 6 as a fraction, having 7 6×7 42 for denominator, we say 6= = 7 7 We have already defined a fraction to be that which is less than a unit: consequently, that which is equal to, or greater than, a unit, though in fractional form, is not really a fraction. Therefore, when the numerator is less than the denominator, we call the fraction a proper fraction. But, when the numerator is equal to, or greater than the denominator, we call the fraction an improper fraction. The expression 42 is, therefore, an improper fraction. Express each of the numbers 3, 4, 5, 6, 7 as an improper fraction, having 8 for denominator. To reduce a mixed number to an improper fraction : 192. Express its integral part (191) as a fraction, having the same denominator as its fractional part: then, as the generic unit is the same in both parts, add their numerators together, and place the sum over the common denominator. 4X5 20 5 then Thus, 203 23 if we have 4g, we first put 4 5+5=5; that is, 4323. In the same manner reduce the following numbers: 9, 119, 154, 14,7%, and 34,53. 193. Having well understood the preceding article, we may proceed more expeditiously, thus: multiply the integral part by the denominator of the fractional part; and, to the product, add the numerator of the fractional part, for a numerator, under which place the denominator of the fractional part. 5X6+1 31 13×5+2 67 Thus, 51 6 = So also the following: 6' and 13% = 194. A fraction is in its lowest terms, when those terms are prime to each other. Let be a fraction, of which the |