A 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? $\newline$ Write your answer as a whole number or as a decimal rounded to the nearest tenth. $\newline$ $\_\_\_\_$ tons

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Q. A 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?

\newline

Write your answer as a whole number or as a decimal rounded to the nearest tenth.

\newline

\_\_\_\_

tons

Denote compost needed as $x$ : Let's denote the number of tons of compost needed as $x$ . The cost of the soil on hand is $\$27.65$ per ton, and the company has $18$ tons of it. The cost of the compost is $\$42.65$ per ton.
Calculate cost of soil: The total cost of the soil on hand is $18$ tons multiplied by $\$27.65$ per ton. $\newline$ Total cost of soil = $18$ tons * $\$27.65/\text{ton}$
Calculate total cost of soil: Calculate the total cost of the soil. $\newline$ Total cost of soil = $$18$ \times \$$27$.$65$$\newline$Total cost of soil = \$$497$.$70$
Calculate total cost of final blend: The total cost of the final blend for the total weight $18 \text{ tons} + x \text{ tons}$ should be equal to the sum of the cost of the soil and the cost of the compost.$\newline$Total cost of final blend = $18 + x$ * $\$29.15$
Calculate cost of compost: The cost of the compost that needs to be added is $x$ tons multiplied by $\$42.65$ per ton.$\newline$Cost of compost = $x \times \$42.65$
Set up equation for total cost: Set up the equation based on the total cost of the final blend being the sum of the cost of the soil and the cost of the compost.$\$497.70 + (x \times \$42.65) = (18 + x) \times \$29.15$
Distribute $\$29.15$ on right side: Distribute the $\$29.15$ on the right side of the equation.$\newline$$\$497.70 + (x \times \$42.65) = \$525.70 + (x \times \$29.15)$
Subtract $\$497.70$ to isolate $x$: Subtract $\$497.70$ from both sides of the equation to isolate the terms with $x$ on one side.$\newline$$(x \times \$42.65) = \$525.70 + (x \times \$29.15) - \$497.70$
Simplify right side of equation: Simplify the right side of the equation. $x \times \$42.65$ = \$$28$.$00$ + $x \times \$29.15$
Subtract $\$29.15$ to isolate x: Subtract $(x \times \$29.15)$ from both sides to get x by itself.$\newline$$(x \times \$42.65) - (x \times \$29.15) = \$28.00$
Factor out x from left side: Factor out x from the left side of the equation.$\newline$$x \times (\$42.65 - \$29.15) = \$28.00$
Calculate difference in cost per ton: Calculate the difference in cost per ton between the compost and the blend.$\newline$$\$42.65 - \$29.15 = \$13.50$
Divide both sides by $\$13.50$: Divide both sides by $\$13.50$ to solve for $x$.\[x = \frac{\$28.00}{\$13.50}\]
Calculate value of x: Calculate the value of $x$.$x = 2.0741$ tons
Round $x$ to nearest tenth: Round $x$ to the nearest tenth.$x \approx 2.1$ tons