Question 1157116
<i>1. The sum of two numbers is 30. Five times the smaller exceeds twice the larger by 17. Find the numbers.</i>
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Smaller number = x
Larger number = 30 - x
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5x = 2(30 - x) + 17
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5x = 60 - 2x + 17
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5x = 77 - 2x
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7x = 77
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x = 11
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Smaller number = x = <b>11</b>
Larger number = 30 - x = 30 - 11 = <b>19</b>
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<i>2. Tim is 1/5 as old as Tom. In six years, Tim will be 1/3 as old. Find their present ages.</i>
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Tom's age = x
Tim's age = x/5
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(x + 6)/3 = x/5 + 6
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x/3 + 2 = x/5 + 6
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x/3 - x/5 = 4
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5x/15 - 3x/15 = 4
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2x/15 = 4
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2x = 60
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x = 30
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Tom's age = x = <b>30</b>
Tim's age = x/5 = 30/5 = <b>6</b>
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<i>3. A man invested part of $15,000 at 12% and the remainder at 8%. If his annual income from the two investments is $1456, how much does he have invested at each rate?</i>
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<u>12% rate</u>
Principal = p
Rate = 0.12
Interest = 0.12p
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<u>8% rate</u>
Principal = 15000 - p
Rate = 0.08
Interest = 0.08(15000 - p) = 1200 - 0.08p
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0.12p + (1200 - 0.08p) = 1456
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0.12p + 1200 - 0.08p = 1456
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0.04p = 256
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p = 6400
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<u>12% rate</u>
Principal = p = <b>$6400</b>
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<u>8% rate</u>
Principal = 15000 - p = 15000 - 6400 = <b>$8600</b>
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<i>4. How many gallons of cottonseed oil must be added to 45 gallons of a solution of olive oil, which is 90% olive oil, in order to make the resulting solution 80% olive oil?</i>
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{{{0.90(45)/(45 + x)}}} = 0.80
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{{{40.5/(45 + x)}}} = 0.80
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0.80(45 + x) = 40.5
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36 + 0.80x = 40.5
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0.80x = 4.5
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x = <b>5.625</b>
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<i>5. Tea worth $4.10 per pound is to be mixed with tea worth $4.90 per pound. How many pounds of each should be used to obtain 25 lb. of a blend worth $4.40 per pound?</i>
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Pounds of $4.10 tea = x
Pounds of $4.90 tea = y
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x + y = 25
4.10x + 4.90y = 25(4.40)
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x + y = 25
4.10x + 4.90y = 110
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Multiply first equation by -4.90:
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-4.90x - 4.90y = -122.50
4.10x + 4.90y = 110
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Add both equations together:
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-0.80x = -12.50
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x = 15.625
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We now know x = 15.625.  To find y, we simply plug that into:
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x + y = 25
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15.625 + y = 25
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y = 9.375
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Pounds of $4.10 tea = x = <b>15.625</b>
Pounds of $4.90 tea = y = <b>9.375</b>