Question

landscaping company mixes its own proprietary blend of topsoil that costs $29.15 per ton to make. The blend contains soil that costs $27.65 per ton and organic compost that costs $42.65 per ton. If the company already has 18 tons of soil on hand, how many tons of compost do they need to mix in?
Write your answer as a whole number or as a decimal rounded to the nearest tenth.
tons

Answer 1

To determine how many tons of compost the landscaping company needs to mix in with the soil to create a proprietary blend of topsoil that costs $29.15 per ton, we can set up an equation based on the costs.

Let \( x \) be the number of tons of compost needed. The company has 18 tons of soil.

The total cost of the soil is:
\[
\text{Cost of soil} = 18 \text{ tons} \times 27.65 \text{ per ton} = 498.7
\]

The total cost of the compost is:
\[
\text{Cost of compost} = x \text{ tons} \times 42.65 \text{ per ton}
\]

The total weight of the mix is:
\[
\text{Total weight} = 18 \text{ tons of soil} + x \text{ tons of compost}
\]

The total cost of the mix is:
\[
\text{Total cost} = 498.7 + 42.65x
\]

To find the average cost per ton of the mix (which should equal $29.15), we can set up the following equation:
\[
\frac{498.7 + 42.65x}{18 + x} = 29.15
\]

Now, let's multiply both sides by \( 18 + x \) to eliminate the fraction:
\[
498.7 + 42.65x = 29.15(18 + x)
\]

Expanding the right side:
\[
498.7 + 42.65x = 524.7 + 29.15x
\]

Now, let's rearrange the equation to isolate \( x \):
\[
498.7 - 524.7 = 29.15x - 42.65x
\]
\[
-26 = -13.5x
\]

Now, divide both sides by -13.5:
\[
x = \frac{26}{13.5} \approx 1.9259
\]

Rounding to the nearest tenth, we find that:
\[
x \approx 1.9
\]

Thus, the company needs approximately **1.9 tons** of compost to mix in.

**Answer:** 1.9 tons