20. The top of my desk contains 6b square inches. If yours is one-fourth as large, what represents the number of square inches in yours? 21. In two quarters there are 2 x 25 cents. How many cents in d quarters ? in r quarters ? 22. If a train runs m miles in 1 hour, how many miles will it run in 3 hours ? in y hours? in 3 y hours ? 23. What is the next whole number after 12? If x is a whole number, what represents the next whole number after x? : 24. If y is an even number, what is the next even number after y? 25. If z is an odd number, what is the next odd number after 2? 26. If z is an odd number, what is the next even number after z? 27. Find the value of each of the following when a=1 and b=2: (a) a+b (c) b-a . (e) 2b-a (b) 2 a+b (d) ab (f) 3 ab For further exercises on this topic, see the review list, p. 20, and Appendix, p. 289. 4. Factor. In arithmetic when two or more numbers are multiplied together each is called a factor of the product. Thus, in the product 5X3X2 the factors are 5, 3, and 2. The same is true in algebra. Thus, the factors of 3 a are 3 and a; the factors of 2 xy are 2, x, and y. 5. Coefficient. In algebra whenever a number is separated into two factors either is called the coefficient of the other. Thus, in the expression 3 a, 3 is the coefficient of a, or it may be said also that a is here the coefficient of 3. Likewise, in 15 ab the coefficient of ab is 15. ORAL EXERCISES Name the factors in each of the following expressions. 1. 5x. 2. 7b. 3. 11 bx. 4. 19 xyz. 5. 29 mn. 6. 31 pqrs. State the coefficient of x in each of the following expressions. 7. 5x. 8. 13x. 9. 19 ax. 10. 7 abr. 11. 19 pqxr. 12. Explain the difference between 2.3 and 2.3; between 2.13 and 2. 13. 13. State the simplest form for each of the following expressions. (a) 3X16. (e) 10X3 a. (i) 10X5z. (m) 1 of 12 a. (6) 2X3X5. (f) 3 ax10. () 10Xk. (n) of 24 x. (c) 8Xa. (g) 6X4r. (k) i xk X 10. (o) of 6 rs. (d) 8X4 a. (h) 4rX6. (1) 6X}xk. (p) of 10 abc. 14. State the simplest form for each of the following expressions. (a) 12 ar · 5 (d) 24 b 7c (g) 6.5 rs · 3 (1) 2 · xyz (b) 16 abc · 2 (e) 9 rs · 6 (h) { bc:6 (k) 6 mn · 3 p (c) 5 axy 9 (f) 14 a · () 1 a·b·20 (1) 5 · 10 yz 15. Sometimes the product of two factors is given together with one of the factors, and it is required to find the other factor. This leads, as in arithmetic, to division. Thus, if the given product is 27 and the given factor is 9, the other factor is 27 -9, or 3. In each of the following expressions, state what the given product is, what the given factor is, and what the other factor would be. (a) 81-9. (c) 21 x +3. (e) 6 m :3. (g) 10 ab +5. (b) 16:4. (d) 18 6:18. (f) 75 q=25. (h) 125 xyz = 25. 6. The Equation. We are now ready to see how the literal number may be used in solving problems. Here the advantage of using letters, mentioned in § 1, will become clearer. EXAMPLE 1. If twice a certain number is increased by 5, the result is 29. What is the number? SOLUTION. Let x stand for the number sought. Then, from what the problem says, we must have 2 x+5=29. This statement is called an equation, since it is an equality between two numbers. It may be compared to a balance (see Fig. 1), on one side of which is 2x+5 and on the other side is 29. Now, the balance will, of course, remain undisturbed if we subtract 5 from each side. The result (see Fig. 2) when stated in the form of a new equation is 2 x=24. Let us now divide both sides by 2. This gives (see Fig. 3) the new equation r=12. Therefore, the number sought is 12. Ans. CONDENSED SOLUTION. The work which we have just done in finding x may be greatly condensed. All we really need to do is to write down three steps as follows: From the problem, we know, as before, that 2 x+5=29. We shall now solve another example, but we shall condense the steps in this way from the beginning. EXAMPLE 2. A person wishes to find the weight of a tennis ball. He puts three such balls in the left scale pan of the balance and six ounces in the right pan. He finds this too much, but by adding 1 ounce to the left pan he se cures a good balance. How much Fig. 4. does one ball weigh? SOLUTION. Let x stand for the number of ounces in one ball. Then, from what the problem says, we must have 3 x+1=6. Fig. 5. Subtracting 1 from both sides, we find 3 x=5 Dividing both sides by 3, we have x=ls, the number of ounces each ball weighs. Ans. CHECK. Whenever an answer does what the problem demands of it, it is said to check. In this problem, the answer 1s checks because 3x1}+1=6, just as the first equation (or 3 x+1=6) demands. WRITTEN EXERCISES Solve the following exercises, letting x stand for the unknown number. Condense the work as in § 6 and check each answer.† 1. If three times a certain number is increased by 2 the result is 20. What is the number? Ans. 6. † It is true that these exercises, like those in g 6, may be worked by arithmetic, but you should work them by algebra in order to be prepared for the more difficult problems that are coming later, which cannot be worked easily by arithmetic. 2. A boy paid $1.25 for a base ball bat and three balls. The price of the bat was 50 cents. What was the price of a ball ? (Hint. Work in cents.] 3. A dozen eggs are placed in a basket that weighs 6 oz. and the whole is then found to weigh 30 oz. What is the weight of one egg ? 4. A wagon loaded with 500 paving bricks weighs 1250 lb., and the empty wagon weighs 500 lb.. Find the weight of a single brick. 5. A jar containing 40 cakes of chocolate weighs 44 lb. The jar alone weighs 6 lb. Find the weight of one of the cakes of chocolate. 6. A wall twelve feet high has 64 layers of mortar and 64 courses of brick. If each brick is 2 in. thick, find the thickness of each layer of mortar. 7. A box containing a ream (20 quires) of paper weighs 3 lb. The box alone weighs 8 oz. Find the weight of one quire of paper. Find the weight of a single sheet of this paper, if 1 quire = 24 sheets. 8. A stick ten feet long is cut into two pieces, one of which is 2 feet longer than the other. Find the length of each piece. 9. Two hunters together shot thirty quail, but one of them shot four more than the other. How many quail did each shoot? 10. Three basket ball teams played fifteen games. One team won three games and the other two teams divided the remaining victories equally between them. How many games did each of the other teams win? |