An agricultural cooperative packages a premium seed mixture containing rye grass, fescue, … — ACT

An agricultural cooperative packages a premium seed mixture containing rye grass, fescue, and bluegrass. The weight of the fescue in the mixture is $10$ pounds less than twice the weight of the rye grass. The weight of the bluegrass is $15$ pounds more than half the weight of the fescue. If the total weight of the mixture is $120$ pounds, how many pounds of bluegrass are in the mixture?

Answer: 40 pounds

Answer

The weight of the bluegrass in the mixture is 40 pounds.

By translating the given relationships into equations (

f = 2r - 10

b = \frac{1}{2}f + 15

, and

r + f + b = 120

), we can express all variables in terms of

r

, yielding

r + (2r - 10) + (r + 10) = 120

. Solving this gives

r = 30

. Substituting this back gives the weight of bluegrass as

30 + 10 = 40

pounds.

Step-by-Step Solution

Define variables for each type of grass in the mixture.

Let

r

represent the weight of rye grass,

f

represent the weight of fescue, and

b

represent the weight of bluegrass.

Establishing variables is necessary to translate the verbal descriptions into algebraic terms.

Translate the given relationships into equations.

f = 2r - 10

and

b = \frac{1}{2}f + 15

The problem states the fescue is 10 pounds less than twice the rye grass, and the bluegrass is 15 pounds more than half the fescue.

Substitute the expression for f into the equation for b to express b solely in terms of r.

b = \frac{1}{2}(2r - 10) + 15 = r - 5 + 15 = r + 10

Reducing the number of variables simplifies the system of equations.

Set up the total weight equation and solve for r.

r + (2r - 10) + (r + 10) = 120 \implies 4r = 120 \implies r = 30

The sum of the three grass weights is given as 120 pounds.

Calculate the weight of the bluegrass using the value of r.

b = 30 + 10 = 40

The question asks specifically for the weight of the bluegrass.

Key Concept

Translating verbal relationships into linear equations and solving a system of equations

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