If I have a mixture of 15% one, 30% two, 15% three, and 40% other ingredients in one teaspoon, how many grams of two would I have to add to make the ratio 15/40/15/30?

To solve this problem, we need to adjust the proportions of the mixture so that the final mixture has the desired ratios. Let’s denote the weight of the original teaspoon mixture as \( W \) grams.

### Step 1: Express the original amounts of each component
- **Component One (15%)**: \( 0.15W \) grams
- **Component Two (30%)**: \( 0.30W \) grams
- **Component Three (15%)**: \( 0.15W \) grams
- **Other Ingredients (40%)**: \( 0.40W \) grams

### Step 2: Define the new amount of Component Two
Let \( x \) be the amount (in grams) of Component Two that we need to add.

### Step 3: Write the equation for the new total weight
The new total weight will be \( W + x \) grams.

### Step 4: Express the new ratios
- **New Component One**: \( \frac{0.15W}{W + x} \)
- **New Component Two**: \( \frac{0.30W + x}{W + x} \)
- **New Component Three**: \( \frac{0.15W}{W + x} \)
- **New Other Ingredients**: \( \frac{0.40W}{W + x} \)

### Step 5: Set up the equations for the desired ratios
We want the new mixture to have the following ratios: 15% for One, 40% for Two, 15% for Three, and 30% for Other Ingredients. Therefore, we want:

\[
\frac{0.15W}{W + x} = 0.15
\]

\[
\frac{0.30W + x}{W + x} = 0.40
\]

\[
\frac{0.15W}{W + x} = 0.15
\]

\[
\frac{0.40W}{W + x} = 0.30
\]

### Step 6: Simplify the equations

For Component Two:
\[
\frac{0.30W + x}{W + x} = 0.40
\]
\[
0.30W + x = 0.40(W + x)
\]
\[
0.30W + x = 0.40W + 0.40x
\]
\[
x - 0.40x = 0.40W - 0.30W
\]
\[
0.60x = 0.10W
\]
\[
x = \frac{0.10W}{0.60}
\]
\[
x = \frac{W}{6}
\]

### Step 7: Conclusion
To achieve the desired 15/40/15/30 ratio, you need to add \(\frac{W}{6}\) grams of Component Two to the original mixture.

### Example:
If the original teaspoon weighs \( W = 5 \) grams, then you would add:

\[
x = \frac{5}{6} \approx 0.833 \text{ grams of Component Two}
\]

Would you like more details on this, or do you have any other questions?

---

1. What happens to the percentages of other components when you add more of one component?
2. How would the calculations change if instead of adding Component Two, you removed some of the "Other Ingredients"?
3. How can you check if the final mixture matches the desired ratio after adding \( x \)?
4. What mathematical principles are used to solve this type of problem?
5. How would the solution differ if the original mixture was not in percentage form but in grams?

**Tip:** When dealing with mixtures and proportions, it’s essential to keep track of the total weight, as all percentages depend on this value.

Explore how to adjust ingredient ratios in mixtures using mathematical proportions. Learn step-by-step how to calculate the amount of Component Two needed to achieve a specific ratio in a mixture of 15% One, 30% Two, 15% Three, and 40% Other Ingredients. Suitable for high school students and anyone interested in understanding mixture problems and proportions.

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