A boutique owner is mixing an 18-ounce batch of her custom salt scrub, which costs $3.40 per ounce to make. It is made up of a tropical oil blend with a cost of $7.95 per ounce and sea salt with a cost of $1.45 per ounce. How many ounces of each should be used to make this batch?

Question

Write your answers as whole numbers or as decimals rounded to the nearest tenth.
ounces of tropical oil blend
ounces of sea salt

Answer 1

Let \( x \) be the ounces of tropical oil blend and \( y \) be the ounces of sea salt in the batch. We know the following:

1. The total batch size is 18 ounces: \[ x + y = 18 \]

2. The total cost of the mixture is $3.40 per ounce, which means the total cost for 18 ounces is: \[ 18 \times 3.40 = 61.20 \text{ dollars} \]

3. The cost equations based on the costs per ounce: \[ 7.95x + 1.45y = 61.20 \]

Now we have a system of equations to solve:

1. \( x + y = 18 \)
2. \( 7.95x + 1.45y = 61.20 \)

From the first equation, we can express \( y \) in terms of \( x \): \[ y = 18 - x \]

Now substituting \( y \) into the second equation: \[ 7.95x + 1.45(18 - x) = 61.20 \]

Expanding the second equation: \[ 7.95x + 26.1 - 1.45x = 61.20 \]

Combining like terms: \[ (7.95 - 1.45)x + 26.1 = 61.20 \] \[ 6.50x + 26.1 = 61.20 \]

Now, subtract 26.1 from both sides: \[ 6.50x = 61.20 - 26.1 \] \[ 6.50x = 35.10 \]

Now, divide by 6.50: \[ x = \frac{35.10}{6.50} \approx 5.4 \]

Now, substitute \( x \) back into the equation for \( y \): \[ y = 18 - 5.4 = 12.6 \]

Thus, the amounts needed for the scrub are:

• Ounces of tropical oil blend: 5.4
• Ounces of sea salt: 12.6

Final results:

• Ounces of tropical oil blend: 5.4
• Ounces of sea salt: 12.6