Problem 24 - Beginning and Intermediate Algebra

Question

Natural Fibers Clothing charges \(\$ 4\) for shipping orders of \(\$ 25\) or less, \(\$ 8\) for orders from \(\$ 25.01\) to \(\$ 75,\) and \(\$ 10\) for orders over \(\$ 75 .\) One week, shipping charges for 600 orders totaled \(\$ 4280 .\) Eighty more orders for \(\$ 25\) or less were shipped than orders for more than \(\$ 75 .\) Find the number of orders shipped at each rate.

Step-by-Step Solution

Answer

The number of orders shipped at each rate was: 140 orders for $25 or less, 400 orders for $25.01 to $75, and 60 orders for over $75.

Step 1: Set up the equations

We'll set up a system of linear equations using the information given in the problem: 1. Total number of orders: \(x + y + z = 600\) 2. Total shipping charges: \(4x + 8y + 10z = 4280\) 3. Relation between orders: \(x = z + 80\)

Step 2: Solve the equations

Since we have three unknowns (\(x, y, z\)) and three linear equations, we can solve these equations simultaneously to get the values of \(x, y, z\). Let's start by solving equation 3 for \(z\): \(z = x-80\) Now, substitute this equation in equations 1 and 2: 1. \(x+y+(x-80)=600\) 2. \(4x+8y+10(x-80)=4280\)

Step 3: Simplify the equations

Simplify the equations obtained in step 2: 1. \(2x+y=680\) (divide by 2) 2. \(14x+8y=8280\) (subtract \(10*80\) from the right side) Now, rearrange equation 1 to isolate y: \(y=680 - 2x\)

Step 4: Substitute the equation for y into the other equation

Next, substitute the equation for \(y\) in equation 2: \(14x+8(680-2x)=8280\) Now, solve for \(x\): \(14x+5440-16x=8280\) \(-2x=2840\) \(x=-1420\) However, since the number of orders can't be negative, we made an error somewhere in our calculations. Let's reevaluate our step 2.

Step 2 (correction): Substitute equation 3 into equations 1 and 2

Instead of substituting an equation for \(z\) in equations 1 and 2, substitute equation 3 (which is \(x = z + 80\)) directly: 1. \((z+80)+y+z=600\) 2. \(4(z+80)+8y+10z=4280\)

Step 3 (correction): Simplify the equations

Simplify the equations obtained in the corrected step 2: 1. \(2z+y=520\) (subtract \(80\) and divide by 2) 2. \(14z+8y=4040\) (subtract \(4*80\) from the right side) Now, rearrange equation 1 to isolate y: \(y=520 - 2z\)

Step 4 (correction): Substitute the equation for y into the other equation

Next, substitute the equation for \(y\) in equation 2: \(14z+8(520-2z)=4040\) Now, solve for \(z\): \(14z+4160-16z=4040\) \(-2z=-120\) \(z=60\) Now that we have found \(z\), go back to the equation for \(y\) and solve: \(y=520-2(60)\) \(y=520-120\) \(y=400\) Finally, use equation \(3\) to find \(x\): \(x=z+80\) \(x=60+80\) \(x=140\)

Step 5: State the answer

We've now found the number of orders shipped at each rate: 1. Orders of \(25 or less: x = 140\) 2. Orders from \(25.01 to \)75: y = 400$ 3. Orders over \(75: z = 60\) So, 140 orders were shipped for \(25 or less, 400 orders shipped for \)25.01 to \(75, and 60 orders shipped for over \)75 during that week.

Key Concepts

Linear Equations

Linear equations are fundamental in algebra, often expressed in the form \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants. They describe a straight line when plotted on a graph.
This exercise involves setting up a system of linear equations to solve a shipping cost problem.

• Each equation represents a relationship between the number of orders at different rates and the total shipping cost.
• The equations are essential as they simplify complex real-world problems into manageable calculations.

To better understand this problem, recognize the role of each variable:
- \(x\) denotes orders for \\(25 or less,- \(y\) for \\)25.01 to \\(75,- \(z\) for orders over \\)75.By setting up linear equations, you can transform a word problem into a mathematical model, making it easier to manage and solve.

Algebra Word Problems

Word problems in algebra can sometimes be tricky because they require translating language into mathematical expressions. The key is to carefully read the problem, identifying exactly what is being asked.
In this exercise, we see how the information provided needs to be systematically turned into algebraic equations.

• Recognize keywords that indicate operations, such as "more than," which translates to addition when forming equations.
• Clarify all relationships mentioned, for instance, "Eighty more orders for \\(25 or less were shipped than orders for more than \\)75," indicates \(x = z + 80\).

Algebra word problems like this prepare you for real-life situations where you derive and solve equations based on surrounding conditions.

Solving Equations

Solving equations involves finding the value of unknown variables that satisfy the given conditions. With linear equations, this often means simplifying the system to isolate each variable.To solve the system of linear equations provided:

• Substitute one equation into the others to reduce the number of variables, such as using \(x = z + 80\).
• Simplify the equations as much as possible to a single variable equation, then solve for that variable.

In our problem:- After substitution and simplification, solving \(14z + 8(520 - 2z) = 4040\) gives \(z = 60\).- Use this result in other simplified equations to find \(y\) and \(x\).The trick is to remain organized and methodically work through each equation, ensuring all relationships are preserved throughout the algebraic manipulations.