91Ó°ÊÓ

A bank pays interest to its customers on the last day of the year. The interest paid to a customer is calculated as \(10 \%\) of the average monthly balance maintained by the customer. John is a customer at the bank. On the last day, when the interest was accumulated into his account, his bank balance doubled to $$\$ 5680$$. What is the average monthly balance maintained by John in his account during the year? (A) 2840 (B) 5680 (C) 6840 (D) 7540 (E) 28400 Use the table shown to answer the next two questions. $$ \begin{array}{|l|l|l|l|} \hline & \begin{array}{l} \% \text { (by weight) of } \\ \text { Protein } \end{array} & \begin{array}{l} \% \text { (by weight) of } \\ \text { Fat } \end{array} & \text { Cost of } 10 \text { grams of the food } \\ \hline \text { Food A } & 10 & 30 & \$ 2 \\ \text { Food B } & 20 & 10 & \$ 3 \\ \text { Food C } & 30 & 35 & \$ 7 \\ \hline \end{array} $$

Step by step solution

01

Understanding the Problem

We need to find the average monthly balance maintained by John, given that his final balance doubled to $5680 after interest was added. The interest is 10% of the average monthly balance.

02

Setting Up the Equation

Let the average monthly balance be represented by \( x \). The interest earned at the end of the year is 10% of this average balance, which is \( 0.1 \times x \). Since the balance doubles to \(5680 after the interest is added, we can say the balance before adding interest is \)2840 (half of $5680). So, we can set up the equation as: \[ 2840 + 0.1x = 5680 \].

03

Solving the Equation

Rearrange the equation \( 2840 + 0.1x = 5680 \) to solve for \( x \). First, subtract 2840 from both sides to isolate the interest term: \[ 0.1x = 5680 - 2840 \]. This simplifies to \[ 0.1x = 2840 \].

04

Calculating the Average Monthly Balance

Divide both sides of the equation \( 0.1x = 2840 \) by 0.1 to find \( x \): \[ x = \frac{2840}{0.1} \]. This gives \( x = 28400 \).

05

Verification

To verify, calculate 10% of \( 28400 \), which is \( 2840 \). Adding this to \( 2840 \) (the initial balance before doubling) results in $5680, confirming our solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Interest Calculation

Interest calculation in bank accounts often involves understanding how interest rates apply to average balances over a specific period. In John's case, the interest is 10% of his average monthly balance. This rate is applied at the end of the year. Knowing this, you realize that the interest calculation helps determine how much additional money John earns on top of his initial balance.

• Interest rate: 10%
• Interest amount: 10% of average monthly balance
• Applied: At the end of the year

To understand the connection, the average monthly balance is the amount on which the bank applies the percentage to calculate the interest awarded to the account.

Average Monthly Balance

The average monthly balance is a crucial element in interest-related problems. It represents the amount John had on average in his bank account each month over a year. For John, his average balance directly influenced the interest he received.
This balance is also half of the final balance before the interest doubles the total funds in his account.

• Key in determining interest earned
• Constant over each month of the year

In practical terms, you can think of it as the steady amount of money present in John's account each month if the balance never varied.

Problem-Solving Steps

Breaking down problems into smaller steps makes them easier to tackle. Begin by understanding the problem statement: John’s bank balance doubled after adding the interest at the year's end. Then, set up the equation based on provided information.

• Step 1: Understand problem requirements
• Step 2: Set up the equation with average balance and interest rate

Following these carefully crafted steps helps in systematically arriving at the solution while reinforcing the logic behind each movement in calculation.

Equation Solving

Equation solving involves rearranging and simplifying to find unknown variables. In this scenario, the goal was to find John's average monthly balance. Given that:\[ 2840 + 0.1x = 5680 \]
You break this down by isolating \( x \), the average monthly balance:

• Subtract initial balance: \( 0.1x = 5680 - 2840 \)
• Simplify further: \( 0.1x = 2840 \)
• Divide to solve for \( x \): \( x = \frac{2840}{0.1} \)
• Result: \( x = 28400 \)

These steps are crucial for effectively finding unknowns where multiple components come into play. By practicing equation solving, one learns to confidently tackle various algebraic challenges.