John has $20,000 to invest. He invests part of his money at an annual interest rate of 6%, the rest at 9% annual rate. The return on these two investments over one year is $1,440. How much does he invest at each rate? Solution ? Paul made two investments totaling $15,000. The percentage return on the first investment was 7% annually, while the the percentage return on the second one was 10% annually. If the total return on the two investments over one year was $1,350, how much was invested at each rate? ? Ben invested $30,000, part of which at 5% annual interest rate, the rest at 9% annual interest rate.

The interest earned from the investments was $2,100 at the end of one year. How much did he invest at each rate? Solution ? Jason invested $20,000 for one year, Part of his money was invested at an annual interest rate of 6%, the rest at an annual interest rate of 10%. If his total income from the two investments over one year was $1,700, how much was invested at each rate? ? Jane had $20,000 to invest for one year. She deposited part of which into an account paying 5% annual interest. the rest into another account paying 8% annual interest.

If the total interest earned at the end of one year was $1,390, how much was invested at each account? ? A total of $18,000 was invested for 6 months, part at 4% annual interest rate and part at 7% annual interest rate. The total interest earned over the 6 month period was $450, how much was invested at each rate? Solution ? $12,000 was invested for three months. Part of which was invested at 6% annual interest rate and the rest at 10% annual interest rate. If the total income for three months from the investments was $240, how much was invested at each rate? Sue has $15,000 to invest for 5 months, part at 6% annual rate, the rest at 10% annual rate. If the total interest earned at the end of five months is $450, how much was invested at each rate? 1. Lisa requires additional income to meet her everyday expenses. She has $30,000 to invest. To generate the required additional income, the annual return rate has to be 6%. She deposits part of her capital into an account paying 4% per year, and invest the rest in stocks earning 10% per year. How much does she need to invest at each rate? 2.

To balance risk and return on his investment, Ben invests part of his money in an low risk, low return bank saving account paying 5% per year, the rest in high risk, high return stocks earning 15% per year. To achieve the goal of a 12% annual return on $24,000 investment, how much does he need to invest at each rate? 3. Sue has invested $12,000 at an annual interest rate of 6%. To realize an annual return of 8% on her investment, how much more funds must she invest at an annual rate of 12%? 4. Paul deposited $21,000 into his bank saving account paying 5% per year.

How much additional funds must he invest at 10% annually so that the annual return on his total investment is 7%? 5. Joe made two investments. She earned 8% annually on her first investment, but lost 12% annually on her second investment. If her total investment was $15,000, and the total income was $240 for one year, how much money was allocated to each investment? 6. Allan made two investments totaling $20,000. He earned 5% annually on his first investment, but lost 10% annually on his second investment. If the net loss was $140 for one year, how much money was allocated to each investment? 7.

Jan made two investments. She made a 7% profit on her first investment, but lost 10% annually on her second investment. If her total investment was $16,000, and the total income was $100, how much money was allocated to each investment? 8. Paul made two investments totaling $25,000. He made a profit of 6% on his first investment, but made a loss of 10% on his second investment. If the net loss was $260, how much money was allocated to each investment? 9. Ben invests 30% of his total funds at 5% annual rate, the rest at 8% annual rate. If his total income for one year is $625, how much does he invest at each rate? 10.

Ben invests 30% of his total funds at 5% annual rate, 40% at 6% annual rate, the rest at 8% annual rate. If his total income for one year is $560, how much does he invest at each rate? 11. ? Ben invests 30% of his total funds at 5% annual rate, the rest at 8% annual rate. If his total income for one year is $639, how much does he invest at each rate? Solution 12. ? Ben invests 30% of his total funds at 5% annual rate, 40% at 6% annual rate, the rest at 8% annual rate. If his total income for one year is $630, how much does he invest at each rate? Solution 13. ? A certain amount of money is to be invested for a period of one year.

The amount of money invested at 6% per year is twice as much as the amount invested at 9% per year. If the income for one year is $1680, how much is invested at each rate? Solution 14. ? Jeff has some money to invest for one year. The amount of money invested at 5% per year is $5,000 more than that invested at 8% per year. The interest earned is $1,160. How much does he invest at each rate? Solution 15. ? One third of the funds is invested at an annual interest rate of 4%, the rest at 6% annual rate. If the income for one year is $480, how much is invested at each rate? 16. The ratio of the the amount of money invested at 6% annual rate to the amount invested at 9% is 2 : 5. If the total income for one year is $570, how much is invested at each rate? 17. ? A certain amount of money is to be invested for one year. The amount of money invested at 6% per year is $2,000 more than twice the amount invested at 9% per year. If the income for one year is $1,800, how much is invested at each rate? 18. ? John has some money to invest, the amount of money invested at 5% per year is $3,000 less than three times the amount invested at 9% per year. If the income for one year is $1,050, how much is invested at each rate? Jason invested a total of $20,000 for one year, Part of which was invested at an annual interest rate of 6%, the rest at an annual interest rate of 10%. If the return on the 10% investment is $400 more than the return on the 6% investment, how much was invested at each rate? ? Ben made two investments totaling $17,000. The percentage return on the first investment was 7% annually, while the the percentage return on the second one was 10% annually. If the return on the 7% investment was $340 less than the return on the 10% investment, how much was invested at each rate? ? John has $14,000 to invest.

He invests part of his money at an annual interest rate of 6%, the rest at 9% annual rate. The return on the 6% investment is twice as much as the return on the 9% investment,. How much does he invest at each rate? ? Jane had $30,000 to invest for one year. She deposited part of the money into an account paying 5% interest annually. the rest into another account paying 8% interest annually. If return on the 5% investment was $1,400 less than three times the return on the 8% investment, how much was invested at each account? ? Ben invests 30% of his total funds at 5% annual rate, the rest at 8% nnual rate. If the return on the 8% investment is $82 more than the return on the 5% investment, how much does he invest at each rate? ? A certain amount of money is to be invested for a period of one year. The amount of money invested at 6% per year is twice as much as the amount invested at 9% per year. If the return on the 6% investment is $150 more than the return on the 9% investment,, how much is invested at each rate? ? Jeff has some money to invest for one year. The amount of money invested at 5% per year is $5,000 more than that invested at 8% per year.

If the return on the 5% investment is $200 less than the return on the 8% investment, how much does he invest at each rate? 1. ? The ratio of the the amount of money invested at 6% annual rate to the amount invested at 9% is 3 : 5. The return on the 9% investment is $216 more than twice the return on the 6% investment. How much is invested at each rate? 2. HERE ARE SOME EXAMPLES of problems that lead to simultaneous equations. 3. Example 1. 1000 tickets were sold. Adult tickets cost $8. 50, children’s cost $4. 50, and a total of $7300 was collected. How many tickets of each kind were sold? 4. Solution.

Let x be the number of adult tickets. Let y be the number of children’s tickets. 5. Always let x and y answer the question — and be perfectly clear about what they represent! 6. Now there are two unknowns. Therefore there must be two equations. (In general, the number of equations must equal the number of unknowns. ) How can we get two equations out of the given information? 1) | Total number of tickets:| x | +| y| =| 1000| | 2) | Total money collected:| 8. 5x | +| 4. 5y| =| 7300| 7. In equation 2), let us make the coefficients into whole numbers by multiplying both sides of the equation by 10: 1) | x | +| y| =| 1000| 2′) | 85x | +| 45y| =| 73,000| 8. We call the second equation 2′ (“2 prime”) to show that we obtained it from equation 2). 9. These simultaneous equations are solved in the usual way. 10. The solutions are: x = 700, y = 300. 11. To see the answer, pass your mouse over the colored area. To cover the answer again, click “Refresh” (“Reload”). Do the problem yourself first! 12. Example 2. Mrs. B. invested $30,000; part at 5%, and part at 8%. The total interest on the investment was $2,100. How much did she invest at each rate? 13. Solution. 1) | Total investment:| x | +| y| =| 30,000| | ) | Total interest| . 05x | +| . 08y| =| 2,100| 14. (To change a percent to a decimal, see Skill in Arithmetic, Lesson 3. ) 15. Again, in equation 2) let us make the coefficients whole numbers by multiplying both sides of the equation by 100: 1) | x | +| y| =| 30,000| | 2′) | 5x | +| 8y| =| 210,000| 16. These are the simultaneous equations to solve. 17. The solutions are: x = $10,000, y = $20,000. 18. Problem 1. Samantha has 30 coins, quarters and dimes, which total $5. 70. How many of each does she have? 19. To see the answer, pass your mouse from left to right over the colored area.

To cover the answer again, click “Refresh” (“Reload”). Do the problem yourself first! 20. Let x be the number of quarters. Let y be the number of dimes. 21. The equations are: 1) | Total number of coins:| x| +| y| =| 30| | 2) | Total value:| . 25x| +| . 10y| =| 5. 70| 22. To eliminate y: 23. Multiply equation 1) by ? 10 and equation 2) by 100: 1′) | ? 10x| ?| 10y| =| ? 300| | 2′) | 25x| +| 10y| =| 570| | Add:| | | 15x| | | =| 270| | | | | x| =| 270 15| | | | | x| =| 18| 24. Therefore, y = 30 ? 18 = 12. 25. Example 3. Mixture problem 1. First: 26. “36 gallons of a 25% alcohol solution” 27. eans: 25%, or one quarter, of the solution is pure alcohol. 28. One quarter of 36 is 9. That solution contains 9 gallons of pure alcohol. 29. Here is the problem: 30. How many gallons of 30% alcohol solution and how many of 60% alcohol solution must be mixed to produce 18 gallons of 50% solution? 31. “18 gallons of 50% solution” means: 50%, or half, is pure alcohol. The final solution, then, will have 9 gallons of pure alcohol. 32. Let x be the number of gallons of 30% solution. Let y be the number of gallons of 60% solution. 1) | Total number of gallons| x | +| y| =| 18| | 2) | Gallons of pure alcohol| . x | +| . 6y| =| 9| | 2′) | | 3x | +| 6y| =| 90| 33. Equations 1) and 2′) are the two equations in the two unknowns. 34. The solutions are: x = 6 gallons, y = 12 gallons. 35. Example 4. Mixture problem 2. A saline solution is 20% salt. How much water must you add to how much saline solution, in order to dilute it to 8 gallons of 15% solution? 36. (This is more an arithmetic problem than an algebra problem. ) 37. Solution. Let s be the number of gallons of saline solution. Now all the salt will come from those s gallons. So the question is, What is s so that 20% of s — the salt — will be 15% of 8 gallons? 8. .2s = . 15 ? 8 = 1. 2 39. That is, 40. 2s = 12. 41. s = 6. 42. Therefore, to 6 gallons of saline solution you must add 2 gallons of water. 43. Example 5. Upstream/Downstream problem. It takes 3 hours for a boat to travel 27 miles upstream. The same boat can travel 30 miles downstream in 2 hours. Find the speeds of the boat and the current. 44. Solution. Let x be the speed of the boat (without a current). Let y be the speed of the current. 45. The student might review the meanings of “upstream” and “downstream,” Lesson 25. We saw there that speed, or velocity, is distance divided by time: v| =| d | 46. Therefore, according to the problem: Upstream speed| =| Upstream distance Upstream time| =| 27 3| =| 9| | Downstream speed| =| Downstream distance Downstream time| =| 30 2| =| 15| 47. Here are the equations: 1) | Upstream speed:| x | ? | y| =| 9| | | 2) | Downstream speed: | x | +| y| =| 15| 48. Enjoy! 49. (The solutions are: x = 12 mph, y = 3 mph. ) 50. Word problems that lead to simultaneous equations 51. Section 2 52. Back to Section 1 53. Problem 2. A total of 925 tickets were sold for $5,925. If adult tickets cost $7. 50, and children’s tickets cost $3. 0, how many tickets of each kind were sold? (Compare Example 1. ) 54. To see the answer, pass your mouse over the colored area. To cover the answer again, click “Refresh” (“Reload”). Do the problem yourself first! 55. Let x be the number of adult tickets. Let y be the number of childeren’s tickets. 56. Here are the equations: 1) | Total number of tickets:| x | +| y| =| 925| 2) | Total money collected:| 7. 5x | +| 3y| =| 5,925| 57. In equation 2), make the coefficients into whole numbers by multiplying both sides of the equation by 10: 1) | x | +| y| =| 925| 2′) | 75x | +| 30y| =| 59,250| 8. To eliminate y, for example: 59. Multiply equation 1) by ? 30, and add. 60. The solution is: x = 700, y = 225. 61. Problem 3. Mr. B. has $20,000 to invest. He invests part at 6%, the rest at 7%, and he earns $1,280 interest. How much did he invest at each rate? (Compare Example 2. ) 62. Let x be how much he inveted at 6%. Let y be how much he inveted at 7%. 63. Here are the equations: 1) | Total investment:| x | +| y| =| 20,000| 2) | Total interest:| . 06x | +| . 07y| =| 1,280| 2′) | | 6x | +| 7y| =| 128,000| 64. To eliminate x, for example, from equations 1) and 2′): 65. Multiply equation 1) by ? 6, and add. 66. The solution is: x = $12,000. y = $8,000. 67. Problem 4. Edgar has 20 dimes and nickels, which together total $1. 40. How many of each does he have? (Compare Problem 1. ) 68. Let x be the number of dimes. Let y be be the number of nickels. 69. Here are the equations: 1) | Total number of coins:| x | +| y| =| 20| 2) | Total value:| . 10x | +| . 05y| =| 1. 40| 2′) | | 10x | +| 5y| =| 140| 70. To eliminate x, for example, from equations 1) and 2′), multiply equation 1) by ? 10, and add. 71. The solution is: x = 8 dimes. y = 12 nickels. 72. Problem 5.

How many gallons of 20% alcohol solution and how many of 50% alcohol solution must be mixed to produce 9 gallons of 30% alcohol solution? (Compare Example 3. ) 73. (9 gallons of 30% alcohol solution = . 3 ? 9 = 2. 7 gallons of pure alcohol. ) 74. Let x be the number of gallons of 20% solution. Let y be the number of gallons of 50% solution. 75. Here are the equations: 1) | Total number of gallons:| x | +| y| =| 9| 2) | Total gallons of pure alcohol:| . 2x | +| . 5y| =| 2. 7| 2′) | | 2x | +| 5y| =| 27| 76. To eliminate x, for example, from equations 1) and 2′), multiply equation 1) by ? , and add. 77. The solution is: x = 6 gallons. y = 3 gallons. 78. Problem 6. 15 gallons of 16% disenfectant solution is to be made from 20% and 14% solutions. How much of those solutions should be used? 79. (15 gallons of 16% solution = . 16 ? 15 = 2. 4 gallons of pure disenfectant. ) 80. Let x be the number of gallons of 20% solution. Let y be the number of gallons of 14% solution. 81. Here are the equations: 1)| Total number of gallons:| x | +| y| =| 15| 2)| Total gallons of pure disenfectant: | . 20x | +| . 14y| =| 2. 4| 2′)| | 20x | +| 14y| =| 240| 82.

To eliminate x, for example, from equations 1) and 2′), multiply equation 1) by ? 20, and add. 83. The solution is: x = 5 gallons. y = 10 gallons. 84. Problem 7. It takes a boat 2 hours to travel 24 miles downstream and 3 hours to travel 18 miles upstream. What is the speed of the boat in still water, and how fast is the current? (Compare Example 4. ) 85. Let x be the speed of the boat in still water. Let y be the speed of the current. 86. Here are the equations: 1) | Downstream speed: | x | +| y| =| 24 2| =| 12| 2) | Upstream speed:| x | ? | y| =| 18 3| =| 6| 87.

To eliminate y, simply add the equations. 88. The solution is: x = 9 mph. y = 3 mph. 89. Problem 8. An airplane covers a distance of 1500 miles in 3 hours when it flies with the wind, and in 3| 1 3| hours when it flies against the| 90. wind. What is the speed of the plane in still air? (Compare Example 4. ) 91. Let x be the speed of the plane in still air. Let y be the speed of the wind. 92. Here are the equations: 1) | Speed with the wind:| x | +| y| =| 1500 3| = 500| 2) | Speed against the wind 😐 x | ? | y| =| | 93. To eliminate y, simply add the equations. 94. The solution is: x = 475 mph.