Problem 29 - Algebra 2

Question

The Harvest Nut Company sells made-to-order trail mixes. Sam’s favorite mix contains peanuts, raisins, and carob-coated pretzels. Peanuts sell for \(\$ 3.20\) per pound, raisins are \(\$ 2.40\) per pound, and the carob-coated sretzels are \(\$ 4.00\) per pound. Sam bought a 5 -pound mixture for \(\$ 16.80\) that contained twice as many pounds of carob-coated pretzels as raisins. How many pounds of peanuts, raisins, and carob-coated pretzels did Sam buy?

Step-by-Step Solution

Answer

Sam bought 1 pound of raisins, 2 pounds of carob-coated pretzels, and 2 pounds of peanuts.

Step 1: Define Variables

Let \( x \) be the pounds of raisins, \( y \) be the pounds of carob-coated pretzels, and \( z \) be the pounds of peanuts in the mix. The total weight of the mix is given as 5 pounds, so we have \( x + y + z = 5 \).

Step 2: Relate Carob-coated Pretzels and Raisins

According to the problem, the weight of carob-coated pretzels is twice the weight of raisins, which gives us the equation \( y = 2x \).

Step 3: Write the Cost Equation

Using the cost per pound values, the total cost for each ingredient is \( 2.40x \) for raisins, \( 4.00y \) for carob-coated pretzels, and \( 3.20z \) for peanuts. Set the total cost equal to \( \$16.80 \), so we have the equation \( 2.40x + 4.00y + 3.20z = 16.80 \).

Step 4: Substitute and Simplify Equations

Substitute \( y = 2x \) into both the weight and cost equations: 1. \( x + 2x + z = 5 \) simplifies to \( 3x + z = 5 \). 2. \( 2.40x + 4.00(2x) + 3.20z = 16.80 \) simplifies to \( 2.40x + 8.00x + 3.20z = 16.80 \), which is \( 10.40x + 3.20z = 16.80 \).

Step 5: Solve the System of Linear Equations

We have two equations now: 1. \( 3x + z = 5 \) 2. \( 10.40x + 3.20z = 16.80 \).From the first equation, solve for \( z \): \( z = 5 - 3x \). Substitute \( z \) in the second equation to get:\[ 10.40x + 3.20(5 - 3x) = 16.80 \]. Expand this to \( 10.40x + 16 - 9.60x = 16.80 \), simplifying down to \( 0.80x = 0.80 \), which gives \( x = 1 \). With \( x = 1 \), substitute back to find \( z \): \( z = 5 - 3(1) = 2 \).

Step 6: Calculate the Pounds of Pretzels

Using \( x = 1 \) in the equation \( y = 2x \), calculate \( y \): \( y = 2(1) = 2 \).

Step 7: Verify the Solution

Verify that the solution \( x = 1, y = 2, z = 2 \) satisfies both the weight equation \( x + y + z = 5 \) and the cost equation \( 2.40x + 4.00y + 3.20z = 16.80 \). Substituting gives \( 2.40(1) + 4.00(2) + 3.20(2) = 2.40 + 8.00 + 6.40 = 16.80 \), which is correct.

Key Concepts

Substitution method

The substitution method is a way to solve a system of linear equations. It involves solving one equation for a variable and substituting that expression into another equation. This allows for the determination of one variable, which can then be used to find others. In the context of the problem about Sam's trail mix, we use the substitution method to solve the equations relating to the weights and costs of the ingredients.

Here's a quick overview:

• We begin by expressing one variable in terms of another. In this problem, we know that the weight of carob-coated pretzels, \( y \), is twice the weight of raisins, \( x \), making \( y = 2x \).
• The substitution method uses this relationship to eliminate one variable from the equations. This simplification steps the problem towards a solution by reducing the number of equations.
• The substitute values are then used in other equations in the system to find remaining variables. When you simplify and solve, you derive the values of all variables needed to solve the original equations.

After following these steps, you are usually left with a simple equation that is easy to solve. Once you have the solution, you should verify it by plugging the values into the original equations to ensure they hold true.

Word problems

Word problems are everyday scenarios described using language, requiring the translation of text into mathematical equations. These problems help students apply math in real-life situations. The trail mix problem describes Sam's purchase and prices and prompts finding the amount of each ingredient. Here's how you can approach solving a word problem:

• Read the problem thoroughly to understand what you're being asked to find.
• Identify the variables representing unknown quantities. For instance, in the problem, \( x \), \( y \), and \( z \) represented the pounds of raisins, pretzels, and peanuts.
• Break down the text into mathematical relationships and set up equations based on given information. Here, the total weight and cost, alongside proportional relationships between ingredients, provided these equations.
• Solve the equations using suitable methods such as substitution or elimination to find the values of the variables.

These steps bring you closer to solving word problems systematically and logically, allowing transitions from language to mathematics and back.

Cost equations

Cost equations are used to determine the total expense associated with certain quantities of objects or ingredients. They are a pivotal part of also solving real-world problems where cost is a factor, as in the trail mix problem.

• In the problem, each ingredient had a cost per pound: peanuts at \( \\(3.20 \), raisins at \( \\)2.40 \), and carob-coated pretzels at \( \$4.00 \).
• The total cost equation thus becomes the sum of the costs of all weighted amounts, \( 2.40x + 4.00y + 3.20z = 16.80 \).
• This equation enables us to substitute expressions from other equations, like from the weight or substitution relations, to eventually solve for each variable.

By understanding and utilizing cost equations, students see how mathematical operations have practical applications, predicting the total necessary expenses based on quantities and prices.