Algebra word problems...?
I need to know how to solve the following problems. I need to know the process of how to solve them. I know this is a lot so maybe each person who reads this could just do one. Any help is greatly appreciated.
1. An investment of $2500 is made at an annual simple interest rate of 5.5%. How much additional money must be invested at an annual simple interest rate of 8% so that the total interest earned is 7% of the total investment?
2. A motorboat left a harbor an traveled to an island at an average rate of 15 knots. The average speed on the return trip was 10 knots. If the total trip took 7.5 hours, how many nautical miles is the harbor from the island?
3. 2 seconds after firing a rifle at a target, the shooter hears the impact of the bullet. Sound travels at 1100 feet per second and the bullet at 1865 feet per second. Determine the distance to to the target.
4. Ho many liters of water should be evaporated from 160 liters of a 12% saline solution so that the solution remains a 20% saline solution?
5. A grocery mixes some dried cranberries that cost $6 per pound with some granola that costs $3 per pound. How many pounds of each should be used to make a 25 pound mixture that costs $3.84 per pound?
6. A jeweler wants to make a silver alloy to be used to make in necklaces. How many ounces of a silver alloy that costs $6.50 per ounce should be mixed with one that costs $8.00 per ounce to make a new 20 ounce alloy that costs $7.40 per ounce?
Comments
1)
Let the amount needed be x.
We have:
(0.055)(2500) + (0.08)(x) = (0.07)(2500 + x)
137.5 + 0.08x = 175 + 0.07x
0.08x - 0.07x = 175 - 137.5
0.01x = 37.5
x = 37.5 * 100
x = 3750
Hence, $3750 must be invested at 8%
2)
Speed = distance / time
Let the time taken while travelling at 15 knots be x.
Thus, we have:
15 = distance / x and 10 = distance / (7.5 - x)
distance = 15x and distance = 10(7.5 - x)
Putting the two together, we will get:
15x = 10(7.5 - x)
15x = 75 - 10x
15x + 10x = 75
25x = 75
x = 75 / 25
x = 3
Distance from harbour to island
= 15x nautical miles
= 15(3) nautical miles
= 45 nautical miles
3)
Speed = Distance / time
Let the time taken by the bullet to reach the target be x.
Thus, time taken for sound to reach shooter will be 2 - x.
Thus, we have:
1100 = Distance / x and 1865 = Distance / (2 - x)
Distance = 1100x and Distance = 1865(2 - x)
Putting the two together, we have:
1100x = 1865(2 - x)
1100x = 3730 - 1865x
1865x + 1100x = 3730
2965x = 3730
x = 3730 / 2965
x = 746 / 593
Distance to the target
= 1000x
= 1000(746/593)
= 746000 / 593 feet
4)
12% saline solution means 12% is not water
0.12 * 160 liters = 19.2
Suppose 19.2 liters represent 20% now.
Then 100% will be 19.2 liters * 5 = 96 liters
Hence, water to be evaporated
= 160 - 96
= 64 liters
5)
Let the amount of cranberries used be x.
Thus, the amount of granola used is 25 - x
We have:
6x + 3(25 - x) = (3.84)(25)
6x + 75 - 3x = 96
3x = 96 - 75
3x = 21
x = 21 / 3
x = 7
Thus, we should use 7kg of cranberries and 25 - 7 = 18kg of granola.
6)
Let the amount $6.50 alloy used be x ounces.
Thus, the amount of $8 alloy used is (20 - x) ounces.
We have:
(6.5)(x) + (8)(20 - x) = (7.4)(20)
6.5x + 160 - 8x = 148
-1.5x = 148 - 160
-1.5x = -12
x = -12 / - 1.5
x = 8
Hence, 8 ounces of $6.50 alloy should be used and 20 - 8 = 12 ounces of $8 alloy should be used.