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Question

Difficulty: MediumSolving Linear Equations

A landscaping company purchases bags of grass seed for 3030 each and bags of fertilizer for 1818 each. For a large project, the company purchased 22 fewer than 1.51.5 times as many bags of fertilizer as bags of grass seed. If the total cost of the grass seed and fertilizer was 420420, how many bags of fertilizer did the company purchase?

Answer: 10 bags

Answer

The company purchased 10 bags of fertilizer.
The correct answer is 10. By writing the number of fertilizer bags as f=1.5g−2f = 1.5g - 2 and substituting this expression into the total cost equation 30g+18f=42030g + 18f = 420, we get 30g+18(1.5g−2)=42030g + 18(1.5g - 2) = 420. Simplifying the expression leads to 57g−36=42057g - 36 = 420, which yields g=8g = 8. Substituting g=8g = 8 back into the relationship for ff gives f=1.5(8)−2=10f = 1.5(8) - 2 = 10.

Step-by-Step Solution

1
Define variables for the unknown quantities.
Let gg represent the number of grass seed bags and ff represent the number of fertilizer bags.
Setting up variables is the first step in translating the word problem into solvable algebraic equations.
2
Translate the relationship between the quantities of bags into an equation.
f=1.5g−2f = 1.5g - 2
The problem states that the number of fertilizer bags purchased is 2 fewer than 1.5 times the number of grass seed bags purchased.
3
Write the total cost equation using the prices and variables.
30g+18f=42030g + 18f = 420
The total cost of 420420 is the sum of the cost of the grass seed (3030 per bag) and the fertilizer (1818 per bag).
4
Substitute the equation from Step 2 into the cost equation from Step 3.
30g+18(1.5g−2)=42030g + 18(1.5g - 2) = 420
Substituting allows us to solve a single linear equation with one variable.
5
Distribute the 18 through the parentheses and combine like terms.
57g−36=42057g - 36 = 420
Distributing gives 18×1.5g=27g18 \times 1.5g = 27g and 18×−2=−3618 \times -2 = -36. Combining 30g30g and 27g27g yields 57g57g.
6
Isolate the variable term by adding 36 to both sides of the equation.
57g=45657g = 456
To solve for gg, we must isolate the term containing the variable.
7
Divide both sides by 57 to find the value of gg.
g=8g = 8
This division yields the number of grass seed bags purchased.
8
Substitute the value of gg back into the relation for ff to find the final answer.
f=1.5(8)−2=10f = 1.5(8) - 2 = 10
The question asks for the number of fertilizer bags (ff), not grass seed bags (gg).

Key Concept

Solving a linear equation by substitution and distributing across linear terms.
Estimated Time:1m 30s
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Topics

Solving Linear Equations

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