How would you write a real-world problem that models the following equation:

Question

0.08x = 0.06x + 140

Answer 1

To create a real-world problem that models the equation \( 0.08x = 0.06x + 140 \), we can consider a scenario involving savings accounts with different interest rates.

Real-World Problem:

Scenario: Imagine you are deciding between two different savings accounts to put your money into.

• Account A offers an interest rate of 8% per year.
• Account B offers an interest rate of 6% per year.

You plan to invest some amount of money, denoted by \( x \), in these accounts.

After one year, you find yourself in a situation where the interest earned from Account A exceeds the interest earned from Account B by $140.

Question: How much money \( x \) do you need to invest in each account for the interest from Account A to be $140 more than the interest from Account B after one year?

Equation Explanation:

1. The interest from Account A after one year can be expressed as \( 0.08x \).
2. The interest from Account B after one year can be expressed as \( 0.06x \).
3. The problem states that the interest from Account A is $140 more than the interest from Account B, leading to the equation:

\[ 0.08x = 0.06x + 140 \]

This equation sets up the relationship between the investments in the two accounts based on their respective interest rates and the additional income from Account A.

Solution Steps:

To find the value of \( x \):

1. Subtract \( 0.06x \) from both sides:

\[ 0.08x - 0.06x = 140 \]

2. This simplifies to:

\[ 0.02x = 140 \]

3. To solve for \( x \), divide both sides by \( 0.02 \):

\[ x = \frac{140}{0.02} = 7000 \]

Thus, you would need to invest $7000 in each account to achieve the desired interest difference.

Answer 2

How would you write a real-world problem that models the following equation:
- x + 40 = x

Answer 3

To create a real-world problem that models the equation \(-x + 40 = x\), we can consider a scenario involving budgeting or savings.

Real-World Problem:

Scenario: You are managing a budget for a community event. You have a total of $40 allocated for decorations.

You decide to spend some amount of money, represented by \( x \), on decorations. However, due to unforeseen circumstances, you need to cancel a few items you planned to purchase for the decorations, causing you to save \(-x\) dollars instead (effectively meaning you're reducing your expenditure by that amount).

In this situation, you need to find out how much money \( x \) you originally planned to spend so that after accounting for the savings (canceled items), the total amount you would still have equals your original budget of $40.

Equation Explanation:

1. The left side of the equation, \(-x + 40\), represents the total amount you have after saving part of your budget by canceling some expenses.
2. The right side of the equation, \( x \), represents the amount you originally intended to spend.

The equation sets up the condition that the amount left after accounting for the savings should equal the amount originally planned for spending:

\[ -x + 40 = x \]

Solution Steps:

To solve for \( x \):

1. Add \( x \) to both sides of the equation:

\[ 40 = 2x \]

2. Divide both sides by 2 to find \( x \):

\[ x = 20 \]

This tells you that you initially planned to spend $20 on decorations before needing to cancel some items and save part of your budget.

Conclusion:

Thus, the real-world problem can be summarized as follows:

You want to allocate a budget of $40 for decorations for a community event. If you need to cut back on spending and save a portion of that budget equal to \( x \), how much did you initially plan to spend if you still want to end up with $40 after the adjustments? The solution reveals that the planned expenditure was $20.