Bike Savings: How Many Weeks To Reach $350?

Hey guys! Let's dive into a super relatable problem – saving up for something awesome! In this case, our friend Leon is dreaming of a shiny new bike, but he needs a bit of a financial boost to make that dream a reality. Leon needs to save more than $350 to finally call that bike his own. Right now, he's got a head start with $130 already stashed away. And here's the plan: he's going to save an additional $20 each week. Our mission, should we choose to accept it, is to figure out how many weeks Leon needs to keep up this savings strategy to surpass his $350 goal. We'll use an inequality to represent this situation, where $x$ stands for the number of weeks he needs to save.

Understanding the Inequality

Inequalities are mathematical statements that compare two values that are not necessarily equal. Instead of an equals sign (=), they use symbols like > (greater than), < (less than), ≥ (greater than or equal to), or ≤ (less than or equal to). In Leon's case, we need to ensure that his total savings exceeds $350, so we'll be using the "greater than" symbol (>).

Breaking Down the Components

Let's dissect the elements that make up our inequality:

• Initial Savings: Leon already has $130. This is our starting point.
• Weekly Savings: He adds $20 to his savings each week. If $x$ represents the number of weeks, then his total savings from weekly contributions will be $20x$.
• Total Savings: To find Leon's total savings after $x$ weeks, we add his initial savings to his weekly savings: $130 + 20x$.
• The Goal: Leon wants his total savings to be more than $350. So, we need to ensure that $130 + 20x$ is greater than $350.

Forming the Inequality

Putting it all together, the inequality that represents Leon's savings goal is:

$130 + 20x > 350$

This inequality is the key to unlocking the number of weeks Leon needs to save. Now, let's solve it!

Solving the Inequality: Step-by-Step

Alright, let's get our hands dirty and solve this inequality to figure out exactly how many weeks Leon needs to save. Here's a step-by-step breakdown:

Step 1: Isolate the Variable Term

Our goal is to get the term with the variable ($20x$) by itself on one side of the inequality. To do this, we need to get rid of the $130$ that's being added to it. We can achieve this by subtracting $130$ from both sides of the inequality. Remember, whatever we do to one side, we must do to the other to maintain the balance.

$130 + 20x > 350$

Subtract $130$ from both sides:

$130 + 20x - 130 > 350 - 130$

This simplifies to:

$20x > 220$

Step 2: Solve for the Variable

Now we have $20x > 220$. To isolate $x$, we need to get rid of the $20$ that's multiplying it. We can do this by dividing both sides of the inequality by $20$.

$20x > 220$

Divide both sides by $20$:

$\frac{20x}{20} > \frac{220}{20}$

This simplifies to:

$x > 11$

Step 3: Interpret the Solution

So, what does $x > 11$ actually mean? It means that $x$, the number of weeks Leon needs to save, must be greater than $11$. In other words, Leon needs to save for more than 11 weeks to have enough money to buy the bike.

Since Leon can't save for a fraction of a week, he needs to save for at least 12 full weeks to reach his goal.

Visualizing the Solution

Sometimes, it helps to visualize the solution to an inequality. We can do this using a number line.

Creating the Number Line

1. Draw a number line.
2. Mark the point $11$ on the number line. This is our critical value.
3. Since the inequality is $x > 11$ (greater than, not greater than or equal to), we use an open circle at $11$ to indicate that $11$ is not included in the solution.
4. Shade the region of the number line to the right of $11$. This represents all the values of $x$ that are greater than $11$.

Interpreting the Number Line

The shaded region on the number line visually shows us all the possible values for the number of weeks Leon needs to save. Any number in the shaded region (like 12, 13, 14, and so on) will satisfy the inequality and allow Leon to reach his savings goal.

Checking the Solution

It's always a good idea to check our solution to make sure it makes sense. Let's pick a number greater than 11, say 12, and plug it back into the original inequality:

$130 + 20x > 350$

Substitute $x = 12$:

$130 + 20(12) > 350$

$130 + 240 > 350$

$370 > 350$

This is true! So, saving for 12 weeks will indeed give Leon more than $350. Let's try a number less than or equal to 11, say 11:

$130 + 20x > 350$

Substitute $x = 11$:

$130 + 20(11) > 350$

$130 + 220 > 350$

$350 > 350$

This is false! Saving for only 11 weeks results in exactly $350, which is not more than $350. This confirms that our solution $x > 11$ is correct.

Real-World Implications and Considerations

Okay, so we've mathematically determined that Leon needs to save for more than 11 weeks. But let's bring this back to the real world. Here are some additional things Leon might consider:

• Bike Cost Fluctuations: The price of the bike might change! If the bike goes on sale, Leon might need to save for fewer weeks. Conversely, if the price increases, he might need to save for longer.
• Unexpected Expenses: Life happens! Leon might encounter unexpected expenses that dip into his bike savings. He'll need to adjust his savings plan accordingly.
• Earning More: Maybe Leon could find ways to earn extra money, like doing odd jobs or selling some of his old stuff. This would accelerate his savings and get him on that bike sooner.
• Interest: If Leon puts his savings in an account that earns interest, his money will grow faster, potentially shortening the saving time.

Conclusion: Leon's Path to Two Wheels

So, there you have it! By using an inequality, we've helped Leon figure out how many weeks he needs to save to achieve his dream of owning a new bike. Remember, the inequality $130 + 20x > 350$ tells us that Leon needs to save for more than 11 weeks. In practical terms, he'll need to save for at least 12 weeks.

Key Takeaways

• Inequalities are powerful tools for representing and solving real-world problems involving comparisons.
• Solving inequalities involves similar steps to solving equations, with a key difference: multiplying or dividing by a negative number reverses the direction of the inequality.
• Visualizing solutions on a number line can provide a clear understanding of the possible values.
• Real-world context is crucial for interpreting and applying mathematical solutions.

Now, let's all cheer Leon on as he diligently saves each week, bringing him closer to the joy of riding his brand-new bike! You got this, Leon! And remember guys, understanding math can help you achieve your own goals too, whether it's buying a bike, planning a vacation, or anything else you set your mind to!