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Difficulty: MediumRatios, Rates, and Proportions

A landscaping company prepares a custom soil mixture by combining topsoil, compost, and coarse sand in the ratio 2:3:72 : 3 : 7 by weight. A project requires a total of 360 pounds360\text{ pounds} of this soil mixture. The cost per pound for each component is as follows:

- Topsoil: $0.40\$0.40 per pound
- Compost: $0.60\$0.60 per pound
- Coarse sand: $0.25\$0.25 per pound

What is the total cost of the components required to make the 360 pounds360\text{ pounds} of soil mixture?

  1. A
    $106.50
  2. B
    $124.50
  3. $130.50Answer
  4. D
    $150.00
  5. E
    $450.00

Answer

The total cost of the components required is $130.50.
To find the total cost, first determine the total number of parts in the ratio: 2+3+7=122 + 3 + 7 = 12 parts. Divide the total weight of 360 pounds360\text{ pounds} by 1212 to find that each ratio part corresponds to 30 pounds30\text{ pounds}. The weight of topsoil is 2×30=60 pounds2 \times 30 = 60\text{ pounds}, compost is 3×30=90 pounds3 \times 30 = 90\text{ pounds}, and coarse sand is 7×30=210 pounds7 \times 30 = 210\text{ pounds}. Multiplying each weight by its unit price yields: topsoil at 60×$0.40=$24.0060 \times \$0.40 = \$24.00, compost at 90×$0.60=$54.0090 \times \$0.60 = \$54.00, and sand at 210×$0.25=$52.50210 \times \$0.25 = \$52.50. Adding these amounts together gives a total cost of $130.50\$130.50.

Step-by-Step Solution

1
Find the total number of ratio parts.
2 + 3 + 7 = 12 parts
The total weight of the mixture is distributed across the sum of all parts in the ratio.
2
Determine the weight per single ratio part.
360 pounds / 12 parts = 30 pounds per part
Dividing the total weight by the total number of parts yields the scaling factor.
3
Calculate the weight of each component.
Topsoil: 2 x 30 = 60 pounds; Compost: 3 x 30 = 90 pounds; Coarse sand: 7 x 30 = 210 pounds
Multiply each component's ratio value by the weight per part.
4
Calculate the cost of each component and sum them up.
Topsoil: 60 x 0.40=0.40 = 24.00; Compost: 90 x 0.60=0.60 = 54.00; Sand: 210 x 0.25=0.25 = 52.50. Total cost = 24.00+24.00 + 54.00 + 52.50=52.50 = 130.50
Multiply the weight of each component by its price per pound and add the individual costs.

Key Concept

Solving multi-part ratio problems by converting part-to-part ratios into part-to-whole fractions to scale quantities and compute weighted totals.
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Topics

Ratios, Rates, and Proportions

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