Problem 81 - Algebra 2

Question

Mackenzie has \(\$ 57\) in her bank account. She begins receiving a weekly allowance of \(\$ 15,\) of which she deposits 20\(\%\) in her bank account. Write an equation that represents how much money is in Mackenzie's account after \(x\) weeks. (lesson 2-4)

Step-by-Step Solution

Answer

The equation is \(57 + 3x\).

Step 1: Understanding the Problem

Mackenzie starts with $57 in her bank account and deposits 20% of her weekly $15 allowance. We need to represent the total amount in the account after x weeks.

Step 2: Calculate Weekly Deposit

Determine how much Mackenzie deposits each week: 20% of \(15 is calculated as \(0.20 \times 15 = 3\). She deposits \)3 each week.

Step 3: Write the Initial Amount Equation

Mackenzie starts with $57. This is a constant amount added to future deposits, so our equation needs to include this as the initial amount.

Step 4: Write the Equation for x Weeks

The money deposited over \(x\) weeks is \(3x\) dollars, where 3 is deposited weekly for \(x\) weeks. Add this to her initial amount. The equation becomes: \(57 + 3x\).

Step 5: Interpret the Equation

This equation \(57 + 3x\) represents the total money in Mackenzie's account after \(x\) weeks when depositing \(3 weekly from a \)15 allowance.

Key Concepts

Initial Value

The concept of the initial value in linear equations refers to the starting point of a sequence or series of numbers. In real-life situations, the initial value can often be understood as the starting amount or baseline from which changes occur. In Mackenzie's case, her initial value is the $57 she already has in her bank account. This amount is crucial as it serves as the foundation for all future calculations.

• It doesn't change over time unless withdrawals occur, but as given, it does not.
• This initial sum is represented in the linear equation as a constant term, which remains unchanged irrespective of what happens next (additional weeks or deposits).

Understanding the initial value is fundamental to interpreting the results of the linear equation, as it tells us what we start with before any new actions, like deposits, are taken into account.

Weekly Allowance

The weekly allowance in this context is the regular amount of money Mackenzie receives each week. She earns \(15 each week, but importantly, she only deposits 20% of this into her bank account. This weekly allowance influences the linear equation by determining the variable amount added to the account each week.

• Calculating the portion deposited requires understanding percentages, which converts the regular allowance into a consistent, weekly deposit.
• For Mackenzie, each week she deposits \)3 (20% of \(15) into the bank account.

This weekly formula is represented in the linear equation as a coefficient of the variable term. It shows how the bank account balance changes over time. In this case, an increment of \)3 for each week is efficiently demonstrated in the equation as \(3x\). Understanding this helps explain how regular increments add up over time.

Percentage Calculation

Percentage calculation is a mathematical technique used to determine the proportion of one amount related to another by expressing it as a part of 100. In Mackenzie's financial context, the percentage calculation is vital for deciding how much of her weekly allowance is to be saved. She decides on 20%, which requires the calculation of 20% of her \(15 allowance.

• To perform a percentage calculation, multiply the allowance by the percentage in decimal form: \(0.20 \times 15\).
• The result is the portion of the weekly allowance deposited in her account, which is \)3 every week.

By understanding how to convert percentages to decimal calculations, students can set up savings plans effectively. In broader applications, mastering percentage calculations can be useful for discounts, interests, or distributions, making it a universally important skill.