To prepare 60 grams of a 3% cream using a 1% cream and a 10% cream, follow these steps:
### Step 1: Identify the amounts in grams
First, let's find out how many grams of each cream we need:
- Total cream required: 60 grams
- Percentage of 1% cream: 1%
- Percentage of 10% cream: 10%
- Target percentage: 3%
Let:
- `x` be the amount of 1% cream needed (in grams).
- `y` be the amount of 10% cream needed (in grams).
Since the total amount of cream required is 60 grams,
[tex]\[ x + y = 60 \][/tex]
### Step 2: Set up percentage equation
We will set up an equation based on the target percentage of the final mixture being 3%:
[tex]\[ 1\% \times x + 10\% \times y = 3\% \times 60 \][/tex]
Convert percentages to decimal form to simplify calculation:
[tex]\[ 0.01x + 0.10y = 0.03 \times 60 \][/tex]
[tex]\[ 0.01x + 0.10y = 1.8 \][/tex]
### Step 3: Solve the system of equations
We have the following system of equations:
1. [tex]\( x + y = 60 \)[/tex]
2. [tex]\( 0.01x + 0.10y = 1.8 \)[/tex]
From equation 1, express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[ y = 60 - x \][/tex]
Substitute [tex]\( y = 60 - x \)[/tex] into equation 2:
[tex]\[ 0.01x + 0.10(60 - x) = 1.8 \][/tex]
[tex]\[ 0.01x + 6 - 0.10x = 1.8 \][/tex]
[tex]\[ -0.09x + 6 = 1.8 \][/tex]
[tex]\[ -0.09x = -4.2 \][/tex]
[tex]\[ x = \frac{-4.2}{-0.09} \][/tex]
[tex]\[ x = 46.666666666666664 \][/tex]
Now substitute [tex]\( x = 46.666666666666664 \)[/tex] back into the equation [tex]\( y = 60 - x \)[/tex]:
[tex]\[ y = 60 - 46.666666666666664 \][/tex]
[tex]\[ y = 13.333333333333336 \][/tex]
### Step 4: Convert grams to ounces
Convert the amounts from grams to ounces (1 gram = 0.03527396195 ounces):
For the 1% cream:
[tex]\[ x_{\text{ounces}} = 46.666666666666664 \times 0.03527396195 \][/tex]
[tex]\[ x_{\text{ounces}} = 1.6461182243333334 \][/tex]
For the 10% cream:
[tex]\[ y_{\text{ounces}} = 13.333333333333336 \times 0.03527396195 \][/tex]
[tex]\[ y_{\text{ounces}} = 0.4703194926666668 \][/tex]
### Conclusion
To prepare 60 grams of a 3% cream from a 1% cream and a 10% cream, you need:
- 46.67 grams (approximately) of the 1% cream, which is 1.646 ounces (approximately).
- 13.33 grams (approximately) of the 10% cream, which is 0.470 ounces (approximately).
