Problem 28 - Algebra 1: Concepts and Skills

Question

You and your sister are saving money from your allowances. You have \(\$ 25\) and save \(\$ 3\) each week. Your sister has \(\$ 40\) and saves \(\$ 2\) each week. After how many weeks will you and your sister have the same amount of money?

Step-by-Step Solution

Answer

After 15 weeks, you and your sister will have the same amount of money.

Step 1: Setup the Equations

Let's assign x to represent the weeks and y to represent the amount of money saved. This leads to the following equations: \(y = 3x + 25\) for your savings and \(y = 2x + 40\) for your sister's savings.

Step 2: Equating the Two Expressions

As we need to find out the week count where you both will have the same amount saved, we will equate both expressions and solve for x; thus, \(3x + 25 = 2x + 40\).

Step 3: Solving for x

First we eliminate the smaller x on the right by subtracting \(2x\) from both sides, this gives: \(x = 15\). This tells us that after 15 weeks, you and your sister will have the same amount of money saved.

Key Concepts

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. In our exercise, we use algebra to represent and solve the problem of savings with variables. Here, the variable \(x\) represents the number of weeks. Understanding algebra involves learning how to work with equations, which are like balanced scales: what you do to one side, you must do to the other to keep things in balance.

When solving this problem, we set up two algebraic equations. These equations represent the money saved by you and your sister over several weeks. They are:

• \( y = 3x + 25 \) for your savings
• \( y = 2x + 40 \) for your sister's savings

These equations show how the total money \(y\) increases each week by a fixed amount: \( \\(3 \) for you, and \( \\)2 \) for your sister. Algebra helps us structure and solve problems with unknown quantities systematically.

Simultaneous Equations

Simultaneous equations involve two or more equations that are solved together because they share variables. They are very useful in real-life situations where multiple conditions must be satisfied at the same time, like in our savings problem.

To find out when you and your sister will have the same amount of money saved, we need to solve the simultaneous equations together. This means finding a common solution where both equations are true. From our problem, we have:

• Your equation: \( y = 3x + 25 \)
• Your sister's equation: \( y = 2x + 40 \)

To solve these, we equate the expressions for \(y\):

• \( 3x + 25 = 2x + 40 \)

This step is crucial as it allows us to find a single value for \(x\) that satisfies both conditions. This process is what makes solving simultaneous equations such a powerful tool in algebra.

Problem Solving

Problem-solving in mathematics involves interpreting the problem, setting up equations, and systematically finding the solution. It is about understanding the situation, translating it into mathematical language, and solving it logically.

In the example of saving money, the problem involves two people saving at different rates, and finding when they have equal savings. The key steps in problem solving include:

• Understanding the problem: Recognize what you know (starting amounts, saving rates) and what you need to find (the number of weeks).
• Setting up the problem: Use algebra to create equations that model the situation.
• Solving the problem: Utilize methods like solving simultaneous equations to find the solution.

For our problem, we calculated that after 15 weeks, both of you and your sister will have saved the same amount. This solution was reached by logically following the steps and correctly applying mathematical operations. Problem solving is not just about getting the answer, but also understanding the path to that answer.