Question

Molly purchased Brand A binders at \(\$ 8.00\) apiece and Brand \(\mathrm{B}\) binders at \(\$ 5.60\) apiece. If she bought a total of 12 binders for \(\$ 84.00\), how many Brand A binders did she buy?

Short answer

Molly bought 7 Brand A binders.

Step-by-step solution

Identify the Variables

Let the number of Brand A binders be denoted by A. Let the number of Brand B binders be denoted by B.

Setup the Equations

Form two equations based on the information given:- The total number of binders is 12: A + B = 12.- The total cost of the binders is $84: 8A + 5.60B = 84.

Solve for One Variable

Solve the first equation for B: B = 12 - A.

Substitute and Solve

Substitute B in the second equation:8A + 5.60(12 - A) = 84. Simplify and solve for A:8A + 67.2 - 5.60A = 84 which simplifies to 2.4A = 16.8A = 7.

Verify the Solution

Substitute A back into the original equations to ensure both are satisfied:A + B = 12 7 + B = 12 B = 5And check the total cost:8(7) + 5.60(5) = 56 + 28 = 84.

Key concepts

Solving Linear Equations

Linear equations are equations of the first order and can be represented in the form of \(ax + by = c\). In the given problem, we start by forming linear equations based on definitions and given conditions. We know that Molly buys a total of 12 binders: \(A + B = 12\), and the total cost is $84: \(8A + 5.60B = 84\). The goal is to find the values of \(A\) and \(B\) that satisfy both equations. Solving linear equations involves techniques like isolation (solving for one variable) and substitution to arrive at the solution step-by-step.

Substitution Method

The substitution method is a common technique for solving a system of equations. To apply this method:

• First, solve one of the equations for one variable in terms of the other variable.
• Then, substitute this expression into the other equation.

In our problem, solve the first equation for \(B\): \(B = 12 - A\). Next, substitute \(B = 12 - A\) into the second equation: \(8A + 5.60(12 - A) = 84\). Simplifying this, results in \(2.4A = 16.8\). Solving further gives \(A = 7\). This substitution reduces the problem to one equation with one variable, simplifying the calculation.

System of Equations

A system of equations is a set of equations with multiple variables. The solution to the system is a set of values that satisfies all equations in the system simultaneously. In this problem, we have two equations:

• \(A + B = 12\)
• \(8A + 5.60B = 84\)

By solving these equations, we find the intersection point (A, B) that satisfies both conditions. Understanding how to manipulate and solve these systems is crucial for problems involving multiple equations and unknowns.

Mathematical Reasoning

Mathematical reasoning involves logical thinking and problem-solving skills. It requires identifying relationships, forming equations, and systematically solving them. In this example, the reasoning steps included:

• Defining variables for the quantities to be found (A for Brand A binders, B for Brand B binders).
• Setting up equations based on given total binders and total cost.
• Using mathematical techniques like substitution to find the solution.

Verifying the solution ensures accuracy: we checked that 7 binders of Brand A and 5 binders of Brand B satisfy both the total number and total cost conditions.