17. If a debt of $144 is paid by using the same number each of $1, $2, $5, and $10 bills, how many of each kind of bills is used? 18. At an election there were two candidates for the office of mayor. They together received 2360 votes. If one candidate was defeated by 328 votes, how many votes did each receive? 19. At an election there were three candidates A, B, and C for a certain office. They together received 3447 votes. If A received twice as many as B, and C 195 more than B, many votes did each receive? how 20. If a field requires 36 pounds of nitrogen for fertilization, how much nitrate of soda containing 18 % of nitrogen will be needed? 21. In an algebra class there are 24 pupils. If there are 6 more girls than boys in the class, how many boys are there? 22. A man buys twice as much hard coal as soft coal and pays $108. If hard coal is $7.50 a ton and soft coal is $3, how many tons of each does he buy? 23. Two trains leave Buffalo at the same time going in opposite directions. One travels 50 miles an hour and the other 40 miles an hour. In how many hours will they be 630 miles apart? 24. Two trains leave Buffalo at the same time going in the same direction. One travels 45 miles an hour and the other 38 miles. In how many hours will they be 35 miles apart? 25. A merchant's profits for the second year increased 25% over the first year's profits. If the total profits for the two years are $7623, how much are the profits for each year? SOLUTION. Let x number of dollars profit the first year. Then x + x + Hence x + or 7623, x = 7623. (Why?) .'.x= 3388, the number of dollars profit the first year. 26. A workman's weekly expenses are of his wages. How much does he earn each week if he has $5 left? 27. Two pupils together solve 28 algebra problems. One of them solves & as many as the other. How many problems does each one solve? 28. A rectangular field is as wide as it is long and its perimeter is 40 rods. Find the length and the width. 29. Divide 90 into two such parts that one part equals twice the other. 30. A farmer raised 3000 bushels of corn, wheat, and oats. If he raised 3 times as much corn as wheat and twice as much oats as wheat, how many bushels of each did he raise? 31. A farmer has 4 times as many hogs as cattle and twice as many sheep as hogs and cattle together. If he has 210 animals in all, how many of each kind has he? 32. Three newsboys sold 140 papers. If the first sold as many as the second and the third twice as many as the second, how many did each boy sell? 33. A mason and his helper together earn $6 a day. If the helper earns as much as the mason, how much does each receive? 34. A baseball team won 12 games, which was of the number of games played. How many games were played? 35. A boy bought a ball, a bat, and a glove for $2.50. The ball cost as much as the glove and the bat as much as the How much did each cost? ball. II. POSITIVE AND NEGATIVE NUMBERS 21. The first numbers with which we became acquainted were the whole numbers used in counting, such as 1, 2, 3. Later it was found necessary to enlarge our idea of numbers and include fractions, as, t, †, ů. 3 Still later it became necessary to express the value of the square roots and cube roots of numbers, as √2, 5. A still further extension of our number system will now be made, introducing negative numbers. · 10°. 22. A thermometer scale is marked as in the figure. To indicate that the temperature is 10° below zero we write To indicate that the temperature is 10° above zero we write +10°, or simply 10°. 1. At noon on a certain day the temperature was 8° above zero. At night it had fallen 6°. What was the temperature at night? Will the equation 8° -6° = 2°, indicate the method of finding the answer? = 2. Suppose the temperature is 8° above zero at noon and falls 12° in the next six hours. What is the temperature at 6 o'clock ? The equation, 8° — 12° — — - 4°, indicates the method of finding the answer. 3. If the temperature is 10° above zero in the morning and rises 15° during the forenoon, what is the temperature at noon? 100 90 70 50 F80 60 4. If the temperature is 10° below zero in the morning and rises 15° in the forenoon, what is the temperature at noon? ORAL EXERCISE 23. Explain and give the answers to the following: hend must not be larger than the minuend. Such an operation as 8-12 has no arithmetical meaning, for we cannot subtract from a number more units than the number contains. In algebra, however, we do subtract a larger number from a smaller number, and such subtractions give rise to negative numbers. = Thus, 8 12 8 −8 − 4 = 0 — 4, which we write - 4. 25. Positive and Negative Numbers. There are many pairs of opposite numbers similar to the numbers of the thermometer scale. The fact that numbers are so related to each other can be conveniently represented by the use of the signs + and When thus used to represent the quality of a number, these signs are read positive and negative respectively. Thus, +5 is read positive five and 7 is read negative seven. Numbers preceded by the sign + to indicate the quality of the number are positive numbers; numbers preceded by the sign to indicate the quality of the number are negative numbers. The student will note that each of the signs + and may have two distinct uses; they may indicate the operations of addition and subtraction, or they may indicate the quality of a number. 26. We usually omit the positive sign before positive numbers, writing and reading them exactly as in arithmetic. Sometimes, however, for emphasis or for contrast, we write the sign before a positive number, as (+5) or +5. The negative sign before a negative number is never omitted. To show that these signs are quality signs, and not operation signs, we often write such numbers within a parenthesis, thus (-3)+(+5), read negative 3 plus positive 5. ORAL EXERCISE 27. Read the following, using "positive" and "negative" as the names of these signs when they indicate quality. 1. (−3)+2+(−3); −3+2+(− 3). 2. 3+5; (− 3)+5; 5 +(− 3). 13. If we consider north positive, what should we consider south? If rising temperature is positive, what kind of temperature is negative? 14. What signs would you associate with each of the following: (1) Money earned and money spent? (2) A man's property and his debts? (3) Distance up and distance down? (4) Distance to the right and distance to the left? 28. The Algebraic Number Scale. Draw a straight line and divide it into spaces of equal length. Select some point as zero near the center and name the other points of division as indicated. This arrangement of numbers on a line is the algebraic number scale. (See figure, page 18.) Just as the arithmetical number scale (that part of the algebraic scale that extends from 0 to the right) is conceived as extending indefinitely to the right, so the negative numbers of the algebraic scale extend from 0 indefinitely to the left. |