Problem 43 - Intermediate Algebra

Question

Credit Cards. \(\quad\) A couple stopped using their VISA credit card charging \(1.5 \%\) per month interest and their Robinsons-May credit card charging \(1.75 \%\) per month interest because they had built up a combined debt of \(\$ 16,500\) on the two cards. For one month, they made no purchases or payments on the accounts. If the total amount of interest the credit cards accumulated during the month was \(\$ 259.25,\) what amount did they owe on each card when they stopped using them?

Step-by-Step Solution

Answer

The couple owed $11,800 on the VISA card and $4,700 on the Robinsons-May card.

Step 1: Define Variables

Let's define two variables to help set up the equations. Let \( x \) be the amount owed on the VISA card and \( y \) be the amount owed on the Robinsons-May card.

Step 2: Set Up the First Equation

The total debt on both cards is given as \( \$16,500 \). Therefore, we can write the equation as: \[ x + y = 16,500 \]

Step 3: Set Up the Second Equation

The total interest accumulated in the month is \( \$259.25 \). The interest on the VISA card is \( 1.5\% \), and for the Robinsons-May card, it is \( 1.75\% \). Thus, the second equation is: \[ 0.015x + 0.0175y = 259.25 \]

Step 4: Solve for One Variable

From the first equation, express \( x \) in terms of \( y \): \[ x = 16,500 - y \]

Step 5: Substitute in the Second Equation

Substitute \( x = 16,500 - y \) into the second equation: \[ 0.015(16,500 - y) + 0.0175y = 259.25 \] Expand and simplify the equation:\[ 247.5 - 0.015y + 0.0175y = 259.25 \] Combine like terms:\[ 247.5 + 0.0025y = 259.25 \]

Step 6: Solve for y

Subtract 247.5 from both sides:\[ 0.0025y = 11.75 \] Now, solve for \( y \) by dividing both sides by 0.0025:\[ y = \frac{11.75}{0.0025} = 4,700 \]

Step 7: Solve for x

Substitute \( y = 4,700 \) back into the equation from Step 4:\[ x = 16,500 - 4,700 \] Calculate \( x \):\[ x = 11,800 \]

Step 8: Verify the Solution

Ensure the values satisfy both equations.1. Total debt: \( 11,800 + 4,700 = 16,500 \) ✔️2. Total interest calculation: \[ 0.015 \times 11,800 + 0.0175 \times 4,700 = 177 + 82.25 = 259.25 \] ✔️

Key Concepts

Simple Interest Calculation

Interest is what you pay for borrowing money. In this exercise, interest is calculated monthly as a percentage of the money owed. This type of interest calculation is called "simple interest." For the VISA card, it’s charged at a rate of 1.5% per month and for the Robinsons-May card, it’s 1.75%. To calculate simple monthly interest, you use the formula: \[ \text{Interest} = \left( \frac{\text{interest rate}}{100} \right) \times \text{amount owed} \] Here's what you do:

• Multiply the amount on each card by its respective interest rate (convert that percentage to a decimal first).
• Add the two amounts you get to find the total interest.

In our case, both interests need to add up to $259.25 for one month without any purchases or payments made.

Linear Equations

A linear equation is one where the variable(s) are only raised to the power of one. In this exercise, we have two linear equations because we're dealing with two variables: the amounts owed on each credit card.The equations:

• First, we had \( x + y = 16,500 \), which tells us the total money owed on both cards is \(16,500.
• Second, \( 0.015x + 0.0175y = 259.25 \) tells us the total interest, calculated as explained before, adds up to \)259.25.

Both these equations help us form a system of equations necessary to figure out the specific debts on each card. It's like having two pieces of the same puzzle, each giving us unique information about the whole situation.

Problem Solving Steps

When you have a problem involving systems of equations, systematic problem-solving steps make it easier.These steps assist in breaking down complex problems:

• Step 1: Define Your Variables. Identify what each variable will represent in the context of the problem. Here, \( x \) and \( y \) were the amounts owed on the VISA and Robinsons-May credit cards.
• Step 2: Set Up Your Equations. Form equations using the given information, like total debt or interest, to reflect the relationships between your variables.
• Step 3: Solve One of the Equations. Simplify an equation to express one variable in terms of another. Use it as a tool for substitution in the other equation.
• Step 4: Substitute and Solve. Replace one variable in your second equation with the expression obtained previously and solve for the remaining variable.
• Step 5: Verification. It’s crucial to check your answers by plugging the values back into the original equations to ensure they satisfy all given conditions.

These steps streamline the problem-solving process and ensure you address all aspects of the problem effectively.