Please help me solve the problem: A mixture of 12 ounces of vinegar and oil is 40 percent vinegar by weight. How many ounces oil must be added to the mixture to produce a new mixture that is only 25 percent vinegar?

12 oz @ 0.4 = 4.8 oz vinegar.

12 oz @ 0.6 = 7.2 oz oil.

You want 25% of some volume to be 4.8 oz vinegar.
0.25 X = 4.8
X = 19.2 total volume.
You had 12 oz; therefore, you must add 19.2 - 12.0 oz oil or you must add 7.2 oz oil to make a final volume of 19.2 oz.

You can check it to see if it is right.
4.8 oz vinegar + 7.2 oz oil (initially) + 7.2 oz oil added to make 19.2 oz total.
vinegar is 4.8/19.2 = 0.25
oil is 14.4/19.2 = 0.75
Check my thinking. Check my work.

To solve this problem, we'll use the concept of calculating the amount of vinegar in a mixture and then determine the amount of oil needed to achieve the desired vinegar concentration.

Let's break down the given information:

1. The original mixture contains 12 ounces of vinegar and oil.
2. The original mixture is 40% vinegar by weight.
3. We need to find out how many ounces of oil must be added to the mixture to achieve a new mixture that is 25% vinegar.

To begin, let's determine the amount of vinegar in the original mixture:

12 ounces * 0.40 (40%) = 4.8 ounces of vinegar

Next, let's determine the amount of oil in the original mixture:

12 ounces - 4.8 ounces = 7.2 ounces of oil

Now, let's understand the mixing process:

1. We have 4.8 ounces of vinegar and 7.2 ounces of oil in the original mixture.
2. We need to add some amount of oil to dilute the vinegar concentration to 25%.

Now, let's solve for the amount of oil needed in the new mixture:

Let x be the amount of oil to be added.

(4.8 ounces of vinegar) / (12 ounces + x ounces) = 25%

To convert 25% to decimal form, divide by 100:

25% = 0.25

0.25 = 4.8 ounces / (12 ounces + x ounces)

0.25 * (12 ounces + x ounces) = 4.8 ounces

3 ounces + 0.25x ounces = 4.8 ounces

0.25x ounces = 4.8 ounces - 3 ounces

0.25x ounces = 1.8 ounces

To isolate x, divide both sides by 0.25:

x = 1.8 ounces / 0.25

x = 7.2 ounces

Therefore, 7.2 ounces of oil must be added to the mixture to achieve a new mixture that is only 25% vinegar.

To solve this problem, we need to set up an equation based on the given information and then solve for the unknown variable.

Let's start by analyzing the given information. We have a mixture of 12 ounces of vinegar and oil, which is 40% vinegar by weight. This means that in the original mixture, 40% of the total weight (12 ounces) is vinegar. The rest, 60%, is oil.

To set up the equation, we can define the following variables:
- Let "x" represent the number of ounces of oil to be added to the mixture.
- The total weight of the new mixture will be (12 + x) ounces.

We need to find how many ounces of oil must be added to the mixture to produce a new mixture that is 25% vinegar by weight. This means that in the new mixture, 25% of the total weight (12 + x) ounces should be vinegar. The remaining 75% should be oil.

Now we can set up the equation based on the percentage of vinegar and oil in the mixtures:
(40% of 12) + (0% of x) = 25% of (12 + x)

To solve this equation, we can simplify it:
(0.4 * 12) + 0 = 0.25 * (12 + x)

4.8 + 0 = 3 + 0.25x

4.8 = 3 + 0.25x

Now we can isolate the variable "x" by subtracting 3 from both sides of the equation:
4.8 - 3 = 0.25x

1.8 = 0.25x

Lastly, we can solve for "x" by dividing both sides of the equation by 0.25:
1.8 / 0.25 = x

x = 7.2

Therefore, 7.2 ounces of oil must be added to the mixture to produce a new mixture that is 25% vinegar.