A certain car's cooling system has an 8-qt capacity and is filled with a mixture that is 30% alcohol. How much of this mixture must be drained off and replaced with pure alcohol if the solution is to be 50% alcohol?
the amount of alcohol in the parts must add up to the whole. So,
.3 * (8-x) + 1.0 * x = .5*8
x = 16/7
To solve this problem, we need to calculate the amount of the mixture that needs to be drained off and replaced with pure alcohol in order to achieve a 50% alcohol solution. Here's the step-by-step process:
1. Let's assume that x quarts of the mixture should be drained off and replaced with pure alcohol.
2. The amount of alcohol in the mixture before draining off is 30% of 8 quarts, which is 0.30 * 8 = 2.4 quarts.
3. The initial amount of alcohol is still present after draining off x quarts, so it becomes 2.4 - (0.30 * x) quarts.
4. After draining off x quarts, the total amount of liquid remaining in the coolant is 8 - x quarts.
5. The amount of pure alcohol added will be x quarts.
6. Thus, the final amount of alcohol is 2.4 - (0.30 * x) + x quarts.
7. The final alcohol concentration is desired to be 50%. Therefore, we can set up the following equation:
Final amount of alcohol / Total volume after draining = 50%.
(2.4 - 0.30x + x) / (8 - x) = 0.50.
8. To simplify the equation, let's multiply both sides by (8 - x) to get rid of the denominator:
(2.4 - 0.30x + x) = 0.50(8 - x).
9. Expand the right side and simplify:
2.4 - 0.30x + x = 4 - 0.50x.
10. Combine like terms:
-0.30x + x = 4 - 2.4 - 0.50x.
11. Simplify further:
0.70x = 1.60 - 0.50x.
12. Add 0.50x to both sides:
0.70x + 0.50x = 1.60.
13. Combine like terms:
1.20x = 1.60.
14. Divide both sides by 1.20:
x = 1.60 / 1.20.
15. Calculate the value of x:
x = 1.3333.
Therefore, approximately 1.3333 quarts of the mixture should be drained off and replaced with pure alcohol to achieve a 50% alcohol solution.
Note: Since we cannot have fractional quarts, we may need to round up or down based on practical considerations. In this case, you would most likely round down to 1 quart to maintain a whole number value.