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How much fresh water should be mixed to a 30% salt water solution, to obtain 90 gallons of a 23% salt water solution?
.30(90-x) + 0(x) = .23(90)
90 gallons of a 23% salt water =
90 * 23 / 100 = 2070 /100 = 20.7 gallons of salt.
x = quantity of salt wather
y = quantity of fresh wather
Fresh wather have 0 % salt.
This mean all salt come on 30 % salt wather.
Quantity of salt in 30 % salt wather = 0.3 x
0.3 x = 20.7 Divide both sides by 0.3
x = 20.7 / 0.3 = 69 gallons
x + y = 90
69 + y = 90 Subtract 69 to both sides
69 + y - 69 = 90 - 69
y = 21 gallons
69 gallons of salt water and 21 gallons of fresh water = 90 gallons
20.7 gallons of salt / 90 gallons = 0.23 = 23 %
To find the amount of fresh water that needs to be mixed with a 30% salt water solution to obtain 90 gallons of a 23% salt water solution, we can use a method called the "alligation method." This method allows us to determine the ratio of the two solutions needed to achieve the desired concentration.
Step 1: Set up a table with two rows. Label the top row with the two given concentrations (30% and 0%), and label the bottom row with the desired concentration (23%).
| 30% | 0% |
|----|---|
| | 23% |
Step 2: In the top row, write the difference between the desired concentration and each given concentration in the corresponding cell.
| 30% | 0% |
|----|----|
| 7% | 23% |
Step 3: Subtract the values in the top row to get the difference between the two given concentrations.
| 30% | 0% |
|----|----|
| 7% | 23% |
| 23% |
Step 4: Add up the values in the difference column (30% + 7% = 37%).
| 30% | 0% |
|----|----|
| 7% | 23% |
| 23% |
| 37% |
Step 5: Set up a proportion to find the ratio of the two solutions needed. Divide the difference between the two given concentrations (37%) by the difference between the desired concentration and the lower given concentration (7%).
37% / 7% = 37/7 = 5.28
This means that for every 5.28 parts of the 30% salt water solution, you need to mix it with 1 part of fresh water to obtain the desired 23% salt water solution.
Step 6: Calculate the total volume of the mixture required. Since we only have the ratio, we need to determine the total number of parts in the mixture.
5.28 + 1 = 6.28
Step 7: Divide the desired volume of the mixture (90 gallons) by the total number of parts (6.28) to find the volume of each part.
90 gallons / 6.28 = 14.33 gallons
Step 8: Multiply the volume of each part by the corresponding part of the ratio (5.28 parts) to find the volume of the 30% salt water solution.
14.33 gallons * 5.28 = 75.70 gallons
Therefore, you would need to mix approximately 75.70 gallons of the 30% salt water solution with approximately 14.33 gallons of fresh water to obtain 90 gallons of a 23% salt water solution.