How much pure alcohol be added to 400 ml of a 15% solution to make its strength 32%?
Understand the Problem
The question is asking how much pure alcohol should be added to a 400 ml solution that currently has a concentration of 15%, in order to achieve a final concentration of 32%. This entails using the concept of concentration and dilution to set up an equation for the solution.
Answer
Add $100 \, \text{ml}$ of pure alcohol.
Answer for screen readers
The amount of pure alcohol that should be added is ( 100 , \text{ml} ).
Steps to Solve
- Calculate the amount of alcohol in the current solution
To find how much alcohol is in the existing solution, multiply the volume by the percentage concentration.
$$ \text{Current alcohol} = 400 , \text{ml} \times 0.15 = 60 , \text{ml} $$
- Set up the equation for the final concentration
Let ( x ) be the amount of pure alcohol to add. The total volume of the new solution will be ( 400 + x ) ml, and the total amount of alcohol will be ( 60 + x ) ml. We can set up the equation for the final concentration:
$$ \frac{60 + x}{400 + x} = 0.32 $$
- Solve for ( x )
Multiply both sides by ( 400 + x ) to eliminate the fraction:
$$ 60 + x = 0.32(400 + x) $$
Now, distribute ( 0.32 ):
$$ 60 + x = 128 + 0.32x $$
- Rearrange the equation
Rearranging gives us:
$$ x - 0.32x = 128 - 60 $$
Combining like terms:
$$ 0.68x = 68 $$
- Isolate ( x )
Now, divide both sides by ( 0.68 ):
$$ x = \frac{68}{0.68} = 100 , \text{ml} $$
The amount of pure alcohol that should be added is ( 100 , \text{ml} ).
More Information
To create a 32% alcohol solution, you need to add 100 ml of pure alcohol to the existing 400 ml solution that is 15% alcohol. This calculation relies on understanding dilution and concentration principles.
Tips
- Incorrectly calculating alcohol content: Make sure to use the correct percentage in calculations.
- Not rearranging the equation properly: Always check steps when isolating variables to avoid mistakes.
