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1)Danny invested some money at 3% simple interest and some money at 7% simple interest. The amount invested at the higher rate was $3000 more than the amount invested at the lower rate. If the total interest on the investments for one year was $810, then how much did he invest at each rate?

2)how many liters of a 10% alcohol solution should be mixed with 12 liters of a 20% alcohol solution to obtain a 14% alcohol solution?

")Danny invested some money at 3% simple interest and some money at 7% simple interest."

x = amount invested at 3%

y = amount invested at 7%

"The amount invested at the higher rate was $3000 more than the amount invested at the lower rate."

y = x + 3000

"the total interest on the investments for one year was $810"

0.03x + 0.07y = 810

substitution

0.03x + 0.07(x + 3000) = 810

x = 6000

y = x + 3000 = 9000

:::::

"how many liters of a 10% alcohol solution should be mixed with 12 liters of a 20% alcohol solution to obtain a 14% alcohol solution?"

x liters of 10% solution contain 0.10x liters of alcohol

12 liters of 20% solution contain 0.20·12 = 2.4 liters of alcohol

total volume = 12 + x liters

12 + x liters of 14% solution contain 0.14(12 + x) = 1.68 + 0.14x liters of alcohol

1.68 + 0.14x = 0.10x + 2.4

x = 18

18 liters of 10% solution mixed with 12 liters of 20% solution yields 30 liters of 14% solution.

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