Solve each problem. See Example 3. Aryan wishes to strengthen ...

Question

Solve each problem. See Example 3. Aryan wishes to strengthen a mixture from 10% alcohol to 30% alcohol. How much pure alcohol should be added to 7 L of the 10% mixture?

Accepted Answer

Hello everybody. I have everything right today. So we're going to be looking at this question that states a 5% ethanol solution is mixed with pure ethanol to get a 35% solution. And the volume of a of a 5% solution is 14 gallons. How much pure ethanol should be added to get this desired mixture? Now, answer A is 11.67 gallons. B is 12.35 gallons. C is 6.46 gallons and D is 35.11 gallons. Now for a question like this where they give me different percentages of different amounts and I'm trying to figure out what the correct mixture will be the right way to go is to try to come up with an equation that includes all of our values and has a variable which we are trying to solve. So in this case, we're trying to solve for the amount of pure ethanol. So our equation will be 14 or the gallons multiplied by the 5% solution because that's, we have 14 gallons of 5% solution. So we'll have 14, multiplied by 5% plus X. This is the variable which is the amount of pure ethanol so will have x multiplied by 100% since pure ethanol is 100% ethanol and that'll be equal to open parentheses X plus 14. So this is both amounts added together. That is the amount of the pure ethanol and the 14 is the 14 gallons of the 5% ethanol and that'll be multiplied by the 35% which is what we want. The total percentage of our solution to be. So, on the left hand side, we have kind of the parts divided so amount and percent plus amount and percent equal to. And on the right hand side, we have total so total amount and total percent. So we can turn the percentages into decimals and we'll have multiplying 0.5 because to go from percentage to decimal, we just move the decimal point to the left twice and we'll have that plus 100% is one. And so we'll have plus X equals X plus 14 in one parentheses, multiplying and open parentheses and 35% is 0.35 as a decimal. So now we can distribute m 14 multiplied by 0.5. We'll give us 0.7. So we'll have 0.7 plus X equals X multiplied by 0.35 is 0.35 X and then we'll have 14 multiplied by the 0.35 which will be plus 4.9. So all we did there was distribute. So now we can have all of our variables on one side and our numbers on the other. So on the left hand side, I'm gonna keep the variables and I'll have X minus 0. X. So I moved the 0.35 X from the right hand side to the left hand side by subtracting. And that will be equal to 4.9, which stays on the right hand side minus 0.7, which we bring over from the left hand side. And now if we simplify this, we'll have 0.65 X equals 4.2. Because when we have X minus 0.35 X, that's the same as saying one X. So I imagine there's a one in front of that X minus 0.35 X and 4.9 minus 0.7 is 4.2. And then we can divide both sides by 0.65 to have that X is equal to 4.2, divided by 0.65 which means that X is around 6. gallons and this will be the amount of gallons that we need from the pure ethanol solution or answer choice C to obtain the desired mixture. So I hope that was helpful. And until next time

Solve each problem. See Example 3. Aryan wishes to strengthen a mixture from 10% alcohol to 30% alcohol. How much pure alcohol should be added to 7 L of the 10% mixture?

Key Concepts

Concentration and Percentage Solutions

Setting Up and Solving Linear Equations

Conservation of Quantity in Mixtures

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