EXERCISES 75. 1. If a man has a sons, b daughters, and 1 wife, how many persons are in the family? 2. A boy who had 15 cents found m cents and earned 4 ne How much had he then ? cents. 3. My house is d feet long and c feet wide. What is the distance around it? 4. Tom walked due east m hours, then due west n hours. If his rate was 3 miles an hour, at what distance from his starting point did he stop? 5. If m in the above problem is equal to n, where did Tom stop? = 3, how far, and in what direction, 6. If m = 5 and n = from his starting point did he stop? 7. Locate his stopping place if m = 4 and n = 6. 8. D earns a dollars each week and spends b dollars. How much will he have at the end of 8 weeks? 9. What will be his financial condition if a = $15 and b = $10? What will it be if a = $12 and b = $16? 10. A thermometer indicated 40°; it then rose first 20°, and then 30°. What temperature did it then indicate? 11. A balloon is pulling upward with a force of 400 lb.; its two occupants weigh 150 lb. each. What is the combined weight? How is it expressed algebraically, if weight is considered positive? 12. A man has a house worth $4000, and other property worth $3500. He owes $8000. Taking debt negatively, what represents his financial condition? 13. A man has $500 and has debts to pay as follows: To A, $100; to B, $ 150, and to C, $50. Show that he will have the same amount ($200) left, whatever be the order of payment. What law is illustrated? SUBTRACTION 76. 1. At 5 A.M. a thermometer registered 3° above zero (+3°); at 7 A.M. it registered 8° above zero (+8°). At 4 P.M. of the same day it registered 8° above zero, and at 6 P.M. 3° above zero. In the first case, the difference in temperature was +5°, and in the second case it was -5°; that is, +8°. +3° = +5°, and +3° +8=-5°. The positive result, +5°, indicates that the mercury rose 5°; that is, moved in the positive direction from +3° to +8°. The negative result, -5°, shows that the mercury fell 5°; that is, moved in the negative direction from +8° to +3°. 2. A thermometer at 7 A.M. registered -6° (i.e., 6° below 0); at 12 M. it registered +9° (i.e., 9° above 0); and at 5 P.M. it again registered -6°. What was the rise in the forenoon? The fall in the afternoon? From 6° below zero to 9° above zero is how many degrees in the positive direction? From 9° above to 6° below is 15° in which direction? Does adding a positive number give the same result as subtracting an equal negative number? 3. If to-day a thermometer registers -4°, and yesterday it registered -8°, how much warmer is it to-day than it was yesterday? That is, -4° -8° = ? How much colder was it yester day than to-day? -4° That is, -8° -4° = ? -4° ++8° = 4. What is the remainder after subtracting -3 from -10? After subtracting 10 from 3? What number added to -3 gives 10? What number added to -10 gives -3? 77. In algebra, as in arithmetic, the minuend is the sum of the subtrahend and the remainder (difference). As an operation Subtraction is the inverse of addition. +10 = −4, the remainder. Thus, 1046, the sum, which we may regard as a minuend. -11 - +2 = −13. -11 = -13+ +2. -4+10, the remainder. 1- 2=-1. 1+2=-1. 1=-1+ 2. Observe that subtracting +2 is equivalent to adding -2; also, + Observe that subtracting -4 is equivalent to adding 4; and again that the Minuend Remainder + Subtrahend. ILLUSTRATION I A man whose income is $100 a month spends $60, and saves $40. If his income is reduced $10 a month, he will save $30. Or, if his expenses are increased $10 a month, he will save Hence, to take away $10 income is equivalent to adding $10 expenses. Either reduces his savings to $30. Calling income and savings positive, and expenses negative, we have the following algebraic expression of this relation: If his income is increased $10 a month, he will save $50. $110 $60 = $50. Or, if his expenses are reduced $10 a month, he will save $50. Hence, to take away $10 expenses is equivalent to adding $10 Either increases his savings to $50. The relation is income. 78. These results are nicely illustrated by reference to the representation of positive and negative numbers as standing on opposite sides of zero as given in Art. 64. 1. The result of subtracting +2 from +3 is found by counting from +3 (whose distance from 0 is 3 positive units) 2 units to the left, or in the negative direction; it is +1, the number of units from 0 to +1. How may the result of subtracting +2 from -3 be found? 2. The result of subtracting 2 from +3 is found by counting from +3 two units to the right, or in the positive direction ; that is, in a direction opposite to that indicated by the sign belonging to 2; it is +5, which means that the distance from -2 to +3 is 5 units in the positive direction. From 3 to 2 is 5 units in the negative direction; hence -2 - +3 = −5. In 3. In arithmetic the difference between two numbers is the remainder found by subtracting the less from the greater. algebra, however, it is the remainder found by subtracting either from the other; it is the number which, added to the subtrahend, will produce the minuend. Hence, whether the difference is positive or negative depends upon which of the two numbers is regarded as the minuend. This is illus trated by the following examples. Verify them by reference to the diagram. Minuend -2 +7 +2 -7 Difference, or Remainder +5 -5 -5 +5 +9 -9 -9 +9 4. It appears from the preceding that the remainder is less than the minuend when the subtrahend is positive, and greater than the minuend when the subtrahend is negative. In arithmetic the remainder is never greater than the minuend. 79. From the preceding exercises and illustrations we derive the following: The subtraction of any algebraic number is equivalent to the addition of the equal opposite number; or, which is the same thing, the addition of the number with its sign of quality reversed. 80. The formal proof of the principle involved may be given as follows: 1. To subtract +2 from +5 is to find a number such that if +2 be added the sum will be +5. The result may be written +5 +2. Then, since by definition of subtraction, remainder plus subtrahend equals minuend, we have |