SOLUTION BY ARITHMETIC. The less number, plus the greater number, equals 30. (a) Hence the less number, plus 4 times the less number, equals 30 (6) Therefore 5 times the less number equals 30. (c) Hence the less number is one-fifth of 30, or 6. (d) Then the greater number is 4 times 6, or 24. 2. A, B, and C together have $66. A has one-half as much as B, and C has as much as A and B together. How much has each? Let Then and the number of dollars A has. 2x = the number of dollars B has, x+2x, or 3x = the number of dollars C has. By the conditions, x+2x+3x = 66. PROBLEMS. 4. The greater of two numbers is 5 times the less, and their sum is 42. What are the numbers? 5. The sum of the ages of A and B is 68 years, and B is 6 years older than A. What are their ages? 6. Divide $1200 between A and B, so that A may receive $128 less than B. 7. A man had $3.72; after spending a certain sum, he found that he had left 3 times as much as he had spent. How much had he spent? 8. Divide $260 between A, B, and C, so that B may receive 3 times as much as A, and C 3 times as much as B. 9. Divide the number 125 into two parts, one of which is 21 less than the other. 10. The sum of three numbers is 98; the second is 3 times the first, and the third exceeds the second by 7 What are the numbers? 11. A, B, and C together have $127; C has twice as much as A, and $13 more than B. How much has each? 12. My horse, carriage, and harness together are worth $400. The horse is worth 11 times as much as the harness, and the carriage is worth $175 less than the horse. What is the value of each? 13. The sum of three numbers is 108. The first is onethird of the second, and 33 less than the third. What are the numbers? 14. Divide the number 210 into three parts, such that the first is one-half of the second, and one-third of the third. 15. A man bought a cow, a sheep, and a hog, for $75; the price of the sheep was $27 less than the price of the cow, and $6 more than the price of the hog. What was the price of each? NEGATIVE QUANTITIES. 44. The signs + and -, besides indicating the operations of addition and subtraction, are also used, in Algebra, to distinguish between quantities which are the exact reverse of each other in quality or condition. Thus, in the thermometer, we may speak of a temperature above zero as +, and of one below as For example, +25° means 25° above zero, and -10° means 10° below zero. 一。 In navigation, north latitude is considered +, and south latitude -; west longitude is considered +, and east longitude For example, a place in latitude -30°, longitude +95°, would be in latitude 30° south of the equator, and in longitude 95° west of Greenwich. Again, in financial transactions, we may consider assets as +, and debts or liabilities as For example, the statement that a man's property is - $100, means that he owes or is in debt $100. And in general, when we have to consider quantities the exact reverse of each other in quality or condition, we may regard quantities of either quality or condition as positive, and those of the opposite quality or condition as negative. A 45. The thermometer affords an excellent illustration of the relation between positive and negative quantities. +7 +6 +5 +4 • Let OA represent the scale for temperatures above zero, and OB the scale for temperatures +3 0 0 1 At 7 A.M. the temperature is - 6°; at noon it is 2 3 11° warmer, and at 6 P.M. it is 9o colder than at B 4 noon. Required the temperatures at noon and at 5 6 P.M. 6 -7 Beginning at the scale-mark 6. and counting 11 degree-spaces upwards, we reach the scale-mark +5; and counting from the latter 9 degree-spaces downwards, we reach the scale-mark -4. Hence, the temperature at noon is + 5°, and at 6 P.M. — 4°. EXERCISES. 46. 1. At 7 A.M. the temperature is 8°; at noon it is 7° warmer, and at 6 P.M. it is 3o colder than at noon. Required the temperatures at noon and at 6 P.M. 2. A certain city was founded in the year 151 B.c., and was destroyed 203 years later. In what year was it destroyed? Re 3. At 7 A.M. the temperature is +4°; at noon it is 10° colder, and at 6 P.M. it is 6° warmer than at noon. quired the temperatures at noon and at 6 P.M. 4. What is the difference in latitude between two places whose latitudes are + 56° and - 31°? 5. A man has bills receivable to the amount of $2000, and bills payable to the amount of $3000. How much is he worth? 6. At 7 A.M. the temperature is -3°, and at noon it is +11°. How many degrees warmer is it at noon than at 7 Α.Μ.? 7. What is the difference in longitude between two places whose longitudes are + 25° and 90°? 8. The temperature at 6 A.M. is - 7°, and during the morning it grows warmer at the rate of 3o an hour. Required the temperatures at 8 A.M., at 9 A.M., and at noon. 47. The absolute value of a quantity is the number represented by the quantity, taken independently of the sign affecting it. Thus, the absolute value of 5 is 5. II. ADDITION. 48. Addition, in Algebra, is the process of collecting two or more quantities into one equivalent expression, called the sum. Thus, the sum of a and b is a + b (Art. 8). 49. If either quantity is negative, or a polynomial, it should be enclosed in a parenthesis (Art. 20); thus, The sum of a and b is indicated by a + (-b). The sum of a - b and c -d is indicated by Using the interpretation of negative quantities as explained in Art. 44, if a man incurs a debt of $100, we may regard the transaction either as adding - $100 to his property, or as subtracting $100 from it That is, Adding a negative quantity is equivalent to subtracting a positive quantity of the same absolute value (Art. 47). Thus, the sum of a and b is obtained by subtracting b from a; or, a+(-b) = a - b. 51. It follows from Arts. 48 and 50 that the addition of monomials is effected by uniting the quantities with their respective signs. Thus, the sum of a, b, c, d, and - e, is a-b+c-d-e. It is immaterial in what order the terms are united, provided each has its proper sign. Thus, the above result may also be expressed c+a-e-d-b, |