Elementary Algebra and Geometry for Schools

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 HOME : Mathematics - Geometry Introduction to coin problems and investment problems. If you have 7 nickels, what is the value of your nickels in cents? If you have 4 dimes, what is the value of your dimes in cents? These are simple questions that you can readily answer, but they involve the general principle that is used in most "coin" problems: If you know the number of a certain kind of coin, and if you know the value of that coin in cents, then the number of coins multiplied by the value (in cents) of each coin is the total value (in cents) of the coins. Thus the value of 7 nickels is (7) (5) cents, or 35 cents; the value of 4 dimes is (4) (10) cents, or 40 cents; the value of x quarters is (x) (25), or 25x, cents. If, instead of using cents as your unit of value, you used dollars, then 9 dimes would be worth (9)(.10) dollars; the value of 12 nickels would be worth (12)(.05) dollars; the value of y quarters would be .25y dollars. ORAL EXERCISES 1. A man has three dollars and seventy-five cents. What number represents the value of the man's money in cents? What number represents the value of the man's money in dollars? 2. A boy has \$1.37. What number represents the value of his money in cents? 3, A girl has 63 dimes. What number represents the value of the girl's money in cents? What number represents the value of the girl's money in dollars? 4. A man has x quarters. How would you represent the value of these quarters in cents? How would you represent the value of these quarters in dollars? 5. If x represents the number of nickels that a man has, what number represents the value of these nickels in cents? 6. If 3x represents the number of half dollars that a man .has, how would you represent the value of his half dollars in cents?. 7. If (3x -8) represents the number of dimes that a man has, how would you represent the value of these dimes in cents? 8. If 5(2x +7) represents the number of quarters that a man has, how would you represent the value of these quarters in cents? 9. If (a -b) represents the number of nickels that a man has, how would you represent the value of these nickels in cents? 10. A man has 18 coins consisting of dimes and quarters. If x represents the number of dimes, how would you represent the number of quarters that he has? How would you represent the value of these quarters in cents? EXERCISES: Arranging work in coin problems Copy the following boxes onto a paper and fill in the blank spaces with the correct data.  ILLUSTRATIVE EXAMPLE: Coin problem A man has 3 more quarters than nickels. If the value of these quarters and nickels is \$3.45, how many coins of each kind has he? Solution. Since we are given a relationship between the number of nickles and quarters that the man has, we can represent the number of nickels and the number of quarters in terms of a single unknown. Let the number of nickels be represented by x; then the number of quarters can be represented by x +3, and we can indicate this conveniently by using a box. Our equation will come from the fact that the man's coins are worth \$3.45; hence we are interested in the value of the man's nickels and the value of the man's quarters. Since the "value" is to be represented by a number, we must decide on the unit of value which we are going to use; that is, we must express the value either in cents or in dollars. If we decide to use cents as our unit of value, then we will not have expressions involving decimals. Completing the box by filling in the "value" column: But the value of the nickels and quarters is \$3.45, or, in cents, 345. Hence we can write the equation: From the box we proceed to answer the question. The number of nickels that the man has, represented by x, is 9. The number of quarters that the man has, represented by x + 3, is 9 + 3, or 12. EXERCISES: Coin problems 1. A man has twice as many dimes as he has nickels. If the value of his nickels and dimes together is \$1.75, how many coins of each kind has he? 2. A man has 12 coins, some of them nickels and the rest quarters. If the value of his coins is \$2.20, how many coins of each kind does he have? 3. A boy has 12 more dimes than he has quarters. If the value of his dimes and quarters is \$6.80, how many coins of each kind has he? 4. A man has 7 more dimes than nickels and twice as many quarters as nickels. If the value of these coins is \$5.90, how many coins of each kind has he? 5. A man has three times as many quarters as half dollars and 2 more dimes than half dollars. If the value of these coins is \$11, how many coins of each kind has he? 6. A boy has \$13.00 in nickels and dimes in a coin bank. He finds that the number of dimes is 2 less than five times the number of nickels. How many coins of each kind has he? 7. A man has 5 more half dollars than he has nickels, and the number of quarters is 3 less than twice the number of nickels. If the value of these coins is \$17.50, how many coins of each kind has he? 8. I have \$7.20 in nickels, dimes, and quarters, with twice as many dimes as quarters, and the number of nickels is three times the number of dimes and quarters together. How many coins of each kind have I? 9. A man has 71 coins consisting of nickels, dimes, and quarters, with the number of nickels 8 more than twice the number of dimes. If the, coins have a value of \$6.25, how many coins of each kind does he have? 10. A man has \$12.00 consisting of nickels, dimes, and quarters. There are 8 more nickels than quarters, and the number of dimes is 10 less than twice the number of nickels. How many coins of each kind are there? Answers : 1. 7 nickels, 14 dimes. 2. 4 nickels, 8 quarters. 3. 16 quarters, 28 dimes. 4. 8 nickels, 15 dimes, 16 quarters. 5. 8 half dollars, 10 dimes, 24 quarters. 6. 24 nickels, 118 dimes. 7. 15 nickels, 20 half dollars, 27 quarters. 8. 8 quarters, 16 dimes, 72 nickels. 9. 18 dimes, 44 nickels, 9 quarters. 10. 22 quarters, 30 nickels, 50 dimes. Introduction to investment problems. If you deposit \$500 in a savings account at a bank that pays 2% interest annually, how much interest will you receive each year on this investment? In arithmetic you learned to solve such problems as this by multiplying 500 (the principal) by .02 (the rate) to find the interest (or income). In algebra we state this fact by the formula I= prt where I represents the number of dollars of interest, p represents the number of dollars of principal, r represents the annual rate of interest, and t represents the number of years the principal is invested. Applying this formula to the above problem, we can write: I = (500) (.02) (1) Hence I=10 Since I is expressed in dollars, the interest is \$10. EXERCISES: Investment problems 1. A man invests \$200 at 5% interest. How would you represent the number of dollars of interest that he would receive annually? You are not required to perform the indicated multiplication. I= (200)(.05) Answer 2. A man invests x dollars at 4% interest. How would you represent the number of dollars of interest that he would receive annually? 3. A boy has a \$1000 bond that pays 3½ % interest annually. How would you represent the number of dollars of interest that he would receive each year? 4. A man invests y dollars at 4½ % interest. How would you represent the number of dollars of interest that he would receive each year? 5. A girl invests (2x -100) dollars in stock that pays an interest of 6%. How would you represent the number of dollars of income that she receives each year from this investment? 6. How would you represent the annual interest received from an investment of (10,000 -x) dollars at 3% interest? 7. A man invests x dollars in stocks that pay an interest of 6½% and he invests 2x dollars in bonds that pay interest at 3%. How would you represent the number of dollars that he receives each year from the bonds? How would you represent the number of dollars that he receives each year from the stocks? How would you represent the number of dollars that he receives each year from the two investments? 8. A man invests x dollars in real estate and (15,000 -x) dollars in a savings account. If the real estate pays 8% interest and the savings account pays 3% interest, how would you represent the number of dollars in the man's annual income from these two investments? Answers : 2. .04x. 3. (.035) (1000). 4. .045 y. 5. .06(2x -100). 6. .03(10,000 -x). 7. .06x, .065x, .125x. 8. .08x + .03(15,000 -x). EXERCISES : Arranging work in investment problems Copy the following boxes onto a paper and fill in the blank spaces with the correet data.  Answers : 1. 250. 2. .04x. 3. .06(7000 -x). 4. .06(1000 -x). 5. .05x, .06(10,000 -x). 6. .045x, .07(25,000 -x). 7. .025x, .06(2x + 500). 8. .09x, .05(x + 8000), .18x.   ILLUSTRATIVE EXAMPLE: Investment problem A man invests \$15,000, part in bonds which pay 4% interest and the rest in stocks which pay 5% interest. If his annual income from these two investments is \$682.50, find the amount invested in bonds and the amount invested in stocks. Solution. We can represent the number of dollars invested in bonds by x and the number of dollars invested in stocks by 15,000 -x. Filling these values in the box, and also filling in the given rates, we have: We now answer the original question. The amount invested in bonds, x, is \$6750; the amount invested in stocks, 15,000 -x, is 15,000 -6750, or \$8250. EXERCISES: Investment problems 1. A man invests some money at 4% and twice as much money at 5%. If is annual income from these two investments is \$280, how much has he invested at each rate? 2. A man invests a certain amount of money at 6%, and he invests \$1000 more than this amount at 4%. If his annual income from these two investments is \$315, how much has he invested at each rate? 3. A man invests \$5000, part at 4%, and the rest at 5%. If his annual income from these two investments is \$215, how much has he invested at each rate? 4. A man invests \$12,000, part at 3½% and the rest at 5%. If the annual income on the 5% investment exceeds the annual income on the 3½% investment by \$260, how much has he invested at each rate? 5. A man invests a certain amount of money at 6% and \$1800 more than this amount at 4%. If his annual income from these two investments is \$492, how much is invested at each rate? 6. A man invests a certain amount of money at 4½% and twice as much at 5%. If his annual income from the two investments is \$522, how much has he invested at each rate? 7. A man invests a certain amount of money at 4% and a sum \$2000 less than this amount at 6%. If his annual income from these two investments is \$380, how much has he invested at each rate? 8. A man invests \$15,000, part at 5½% and the rest at 4%. If the interest for one year on the 4% investment is \$75 less than the interest for two years on the 5½%  investment, how much has he invested at each rate? 9. A man invests a certain sum of money at 6%, twice that sum at 3½% , and twice the second sum at 4%. If the total annual income from the three investments is \$1189, how much does he invest at each rate? 10. A man invests a certain amount of money at 4%, a second amount \$1500 greater than the first at 4½%, and a third amount \$1250 less than the first at 3½%. If the yearly income from the three investments is \$923.75, how much has he invested at each rate? Answers : 1. \$2000 at 4%, \$4000 at 5%. 2. \$2750 at 6%, \$3750 at 4%. 3. \$3500 at 4%, \$1500 at 5%. 4. \$4000 at 3½%, \$8000 at 5%. 5. \$4200 at 6%, \$6000 at 4%. 6. \$3600 at 4½%, \$7200 at 5%. 7. \$5000 at 4%, \$3000 at 6%. 8. \$10,500 at 4%, \$4500 at 5½%. 9. \$4100 at 6%, \$8200 at 3½%, \$16,400 at 4%. 10. \$7500 at 4%, \$9000 at 4½%, \$6250 at 3½%.