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III. THE EQUATION Solve each of the following exercises by algebra; that is, let x represent the unknown number, form an equation, and solve it. Check each answer.
33. A farmer sold a horse and cow for $210. He sold the horse for four times as much as the cow. How much did he get for each?
$42, $168. Ans. [Hint. Let x=the selling price of the cow.]
34. A plumber and two helpers get $7.50 per day. How much does each earn per day if the plumber earns four times as much as each helper?
35. In a business enterprise the combined capital of A, B, and C was $8400.00. If A's capital was twice B's, and B's was twice C's, what was the capital of each ?
36. Three newsboys sold a total of 60 papers. If the first sold twice as many as the second, and the third sold three times as many as the second, how many did each sell?
37. A and B began business with a capital of $7500. If A furnished half as much as B, how much did each furnish?
[Hint. Let x=the number of dollars A furnished.]
38. Separate 72 into two parts one of which shall be one third the other.
39. A base ball nine won 12 games, which was three fourths of all the games it played. How many games did it play?
40. If one fifth of a certain number is added to the number the result is 12. What is the number?
41. The difference between two thirds of a certain number and two fifths of the same number is 16. What is the number?
42. A man pays a debt of $91 with ten-dollar bills and onedollar bills, paying three times as many one-dollar bills as tendollar bills. How many bills of each kind does he pay out?
43. A cablegram from New York to London costs 25 cents per word; one to Rome costs 31 cents per word; while one to Tokio costs $1.33 per word. If a business man wishes to send the same message to each of the three cities and keep his total expense within $10, how many words may be in the message? Just how much will he spend ?
44. A man has $100 in one bank and $25 in another. If he has $125 more to deposit, how should he divide it between the two banks so that the first account may become equal to the second ?
[HINT. Let x=the amount to be deposited in the first bank. Then, 125—x will equal the amount to be deposited in the second bank.]
For further review exercises on this chapter, see Appendix, pp. 289–291.
POSITIVE AND NEGATIVE NUMBERS
16. Introduction. In Figure 16 are two boys having a “tug of war” at the ends of a rope. One boy is pulling with a force of 50 lb., while the other boy is pulling with a force of 45 lb. This can be described very briefly by saying that one boy is pulling with a force of +50 lb. (read plus 50 lb.) while the other boy is pulling with a force of - 45 lb. (read minus 45 lb.). The two signs + and – as thus used simply indicate that the two pulls are opposite in
110 In just the same way the + and signs may be used in reading a thermometer. Thus, + 10° indicates 10° above zero, while – 10° represents 10° below zero.
10 Mercury 40° M.
Other familiar illustrations are the following: +26° of latitude means 26° north latitude, while – 26° latitude
means 26° south latitude. Again, a business man may use
+$15 to represent an asset (amount due) of $15, and he may use – $15 to represent a liability (amount owed) of $15. Thus, to say that you have – $10 means that you owe that amount.
Many other illustrations of this idea might be given, but these are enough to make clear how we often meet with quantities which seem to have precisely opposite senses. Numbers dealing with such quantities are called opposite numbers. The one given the + sign is called positive, while the one given the – sign is called negative.
ORAL EXERCISES 1. Using Fig. 17, give with proper sign the number which represents the boiling point. Do the same for the freezing point of water, for the freezing point of mercury, and for the temperature called blood heat.
2. What sign should be used to indicate the latitude of New York City ? Of St. Louis? Of Buenos Ayres? Of Calcutta ?
3. If we use the + sign to denote a man's income, what will the – sign denote?
4. Express by means of suitable signs the following: $18 gain ; $20 loss; a $40 debt; a liability of $100; an asset of $150; $8 profit.
5. Augustus Cæsar was born in the year 63 B.C. and died in the year 14 A.D. What do these dates become if we use the + sign for years that are A.D. ?
6. What is meant by saying that the Pyramids of Egypt were built about the year – 2500 ?
7. State what each of the following means when the number in it has its sign changed to – .
(a) John has $2.
SOLUTION. If we change the 2 to – 2, the statement becomes “John has – $2,” which means “ John owes $2.”
(6) My home is 25 miles northwest.
(g) The rabbit ran away from the hunter at the rate of 15 miles an hour.
(h) John owes $2.
(1) The wind carried us 8 miles toward shore. (m) He took 4 oranges out of the basket. (n) She added 100 words to her essay. (0) He turned on 10 lights. (p) After the bell rang, 5 people entered the room. (9) Please open the door about a foot more. (r) School begins at 8 o'clock. For further exercises see the review list, p. 47, and Appendix,
17. Addition of Positive and Negative Numbers. If you have $10 and receive $6 more, you have $16.. This may be expressed by writing
(+10)+(+6)= +16. Again, if you have $10 but owe $8, what you really have is $2. This may be expressed by writing