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Difficulty: EasyRatios, Rates, and Proportions

A landscaping company uses a grass seed mixture containing perennial ryegrass and Kentucky bluegrass in a ratio of 5:35:3 by weight. If a bag of this mixture contains 1515 pounds of Kentucky bluegrass, how many pounds of perennial ryegrass does it contain?

  1. A
    9
  2. B
    24
  3. 25Answer
  4. D
    40
  5. E
    75

Answer

25
The ratio of perennial ryegrass to Kentucky bluegrass is 5:35:3, which means for every 55 pounds of ryegrass, there are 33 pounds of bluegrass. Setting up the proportion 53=x15\frac{5}{3} = \frac{x}{15} and solving for xx gives 3x=753x = 75, which simplifies to x=25x = 25 pounds of perennial ryegrass.

Step-by-Step Solution

1
Set up a proportion using the ratio of perennial ryegrass to Kentucky bluegrass.
53=x15\frac{5}{3} = \frac{x}{15}, where xx represents the weight of perennial ryegrass.
The ratio of perennial ryegrass to Kentucky bluegrass by weight is 5:35:3, and the actual weight of Kentucky bluegrass is 1515 pounds.
2
Solve the proportion for xx by cross-multiplying.
3x=5×15  ⟹  3x=753x = 5 \times 15 \implies 3x = 75
Cross-multiplication allows us to clear the denominators to solve for the unknown value.
3
Divide both sides of the equation by 33.
x=25x = 25
Isolating xx gives the weight of perennial ryegrass in pounds.

Key Concept

Solving proportions based on a given part-to-part ratio

Alternative Method

Identify the scaling factor: since the ratio of Kentucky bluegrass is 33 parts and the actual weight is 1515 pounds, each part of the ratio represents 15÷3=515 \div 3 = 5 pounds. Multiplying the 55 parts of perennial ryegrass by this factor of 55 gives 5×5=255 \times 5 = 25 pounds.
Estimated Time:45s
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Topics

Ratios, Rates, and Proportions

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