A boy scout troop is selling Christmas trees at a local tree lot. In the morning, they sold 18 Douglas Fir trees and 5 Noble Fir trees, earning a total of $1,145. In the afternoon,... A boy scout troop is selling Christmas trees at a local tree lot. In the morning, they sold 18 Douglas Fir trees and 5 Noble Fir trees, earning a total of $1,145. In the afternoon, they sold 15 Douglas Fir trees and 13 Noble Fir trees, earning a total of $1,546. How much does each type of tree cost? A Douglas Fir costs $___ and a Noble Fir costs $___.

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Understand the Problem

The question is asking to set up and solve a system of equations based on the sales of two types of Christmas trees and their respective prices. We will define variables for the prices of the Douglas Fir and Noble Fir trees, create equations based on the sales data provided, and then solve for the values of these variables.

Answer

A Douglas Fir costs $45 and a Noble Fir costs $67.
Answer for screen readers

A Douglas Fir costs $45 and a Noble Fir costs $67.

Steps to Solve

  1. Define the Variables Let ( x ) be the price of one Douglas Fir tree, and ( y ) be the price of one Noble Fir tree.

  2. Create the Equations from the Sales Data From the morning sales: [ 18x + 5y = 1145 ] From the afternoon sales: [ 15x + 13y = 1546 ]

  3. Solve the System of Equations We can use the elimination or substitution method. Here, we'll use the elimination method.

First, multiply the first equation by 3 to align coefficients of ( y ): [ 54x + 15y = 3435 ]

We keep the second equation the same: [ 15x + 13y = 1546 ]

  1. Eliminate ( y ) Now, we can subtract the second equation from the modified first equation: [ (54x + 15y) - (15x + 13y) = 3435 - 1546 ] This simplifies to: [ 39x + 2y = 1889 ]

  2. Solve for ( y ) To isolate ( y ), we can solve for ( y ) in terms of ( x ): [ 2y = 1889 - 39x ] [ y = \frac{1889 - 39x}{2} ]

  3. Substitute back to find ( x ) We can substitute ( y ) back into one of the original equations to solve for ( x ). Let's use the first equation: [ 18x + 5\left(\frac{1889 - 39x}{2}\right) = 1145 ]

Multiply through by 2 to eliminate the fraction: [ 36x + 5(1889 - 39x) = 2290 ] [ 36x + 9445 - 195x = 2290 ] Combine like terms: [ -159x + 9445 = 2290 ] [ -159x = 2290 - 9445 ] [ -159x = -7155 ] Now divide by -159: [ x = \frac{7155}{159} = 45 ]

  1. Find ( y ) Now plug ( x ) back into the equation for ( y ): [ y = \frac{1889 - 39 \times 45}{2} ] [ y = \frac{1889 - 1755}{2} = \frac{134}{2} = 67 ]

A Douglas Fir costs $45 and a Noble Fir costs $67.

More Information

The system of equations allowed us to determine the prices based on sales data. This method is common in solving real-world problems involving multiple unknowns.

Tips

  • Forgetting to align coefficients correctly when using elimination.
  • Miscalculating in the substitution step, leading to wrong values of ( x ) or ( y ).

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