In what proportion should 95% alcohol be mixed with 30% alcohol to make 70% alcohol?
Ah, the sweet science of mixology! Well, if you want to make a jazzy concoction of 70% alcohol, you'll need a bit of mathematical magic. Let's call the amount of 95% alcohol you need to mix 'x' (in any units you fancy), and the amount of 30% alcohol 'y'. Now, we have two equations to tickle our funny bones:
0.95x + 0.30y = 0.70(x + y) (from the alcohol content)
x + y = 1 (since we're talking proportions here)
To simplify things (and avoid making you dizzy), I'll spare you the gory details of solving these equations; trust me, they can be a real clown fiesta! The result is a mix of 2 parts 95% alcohol to 3 parts 30% alcohol. So, grab your lab coat and mix those spirits with precision! Cheers! 🍸
To calculate the proportion of 95% alcohol and 30% alcohol needed to create a mixture of 70% alcohol, we can use the method of alligation or algebraic equations.
Method 1: Alligation
Step 1: Draw a line and divide it into two parts representing the two different concentrations of alcohol (30% and 95%). Label the left side with "30%" and the right side with "95%."
Step 2: Place the desired concentration (70%) at the center of the line.
Step 3: Calculate the differences (in percentages) between the desired concentration and the two given concentrations.
- Difference between 70% and 30%: 70% - 30% = 40%
- Difference between 95% and 70%: 95% - 70% = 25%
Step 4: Place the differences below the corresponding concentrations on the line.
30% ------------ 40%
70%
95% ------------ 25%
Step 5: The proportion of the two alcohols can be determined by the distances of the differences from the desired concentration. In this case, the amount of 30% alcohol needed is in a 40:25 ratio compared to 95% alcohol.
Step 6: Calculate the proportion by dividing the distances: 40 / (40 + 25) = 40 / 65 = 4/13.
Therefore, the proportion of 95% alcohol to 30% alcohol is 4:13.
Method 2: Algebraic Equations
Let's assume a certain quantity (let's call it x) of the 30% alcohol is mixed with the remaining quantity (1 - x) of the 95% alcohol to get a 70% alcohol mixture.
The equation can be set up as follows:
30%(x) + 95%(1 - x) = 70%(1)
Simplify the equation:
0.3x + 0.95 - 0.95x = 0.7
Combining like terms:
-0.65x = -0.25
Solve for x:
x = -0.25 / -0.65
x ≈ 0.3846
Therefore, approximately 38.46% of the 30% alcohol should be mixed with approximately 61.54% of the 95% alcohol to create a 70% alcohol mixture.
To solve this problem, we'll need to use a concept called mixture or ratio.
Let's assume you want to mix x liters of the 95% alcohol with y liters of the 30% alcohol to get a total volume of z liters of 70% alcohol.
To determine the proportion, we need to consider the amount of pure alcohol in each solution.
Amount of pure alcohol in 95% alcohol = 95% * x
Amount of pure alcohol in 30% alcohol = 30% * y
Now, let's set up an equation based on the proportion of pure alcohol:
Amount of pure alcohol in 95% alcohol + Amount of pure alcohol in 30% alcohol = Amount of pure alcohol in the final mixture
(95% * x) + (30% * y) = 70% * z
To simplify the equation, convert the percentages to decimals:
(0.95x) + (0.30y) = 0.70z
Now we have one equation with three variables, so we need more information to solve it.
Do you have any specific values for x, y, or z?
0.......30%......70%......95%......100
.............40.......25..............
Look at the proportions above,
between 70% and 30%, the difference is 40%
and between 95% and 70%, the difference is 25%.
They should be mixed in the inverse ratio, i.e.
25 parts of 30% to 40 parts of 95%.
Converted to ratios,
25/(25+40)=25/65=5/13
40/(25+40)=40/65=9/13
So we mix (5/13) parts of 30% with (8/13) parts of 95% alcohol.
You can also solve it with algebra:
Let x be the fraction of 30% alcohol, then
30%*x+95%*(1-x) = 70%
Solve for x to get 5/13