Problem 85 - Algebra 1

Question

LEAVING A TIP In Exercises \(83-85\), use the following information. You and a friend decide to leave a \(15 \%\) tip for restaurant service. You compute the tip, \(T,\) as \(T=0.15 C,\) where \(C\) represents the cost of the meal. Your friend claims that an easier way to mentally compute the tip is to calculate \(10 \%\) of the cost of the meal plus one half of \(10 \%\) of the cost of the meal. Will both methods give the same results? Explain.

Step-by-Step Solution

Answer

Yes, both methods give the same results.

Step 1: Identifying the Two Methods

Method 1: The tip is calculated as 15% of the cost of the meal. This is represented mathematically as \(T = 0.15C\). \n Method 2: The tip is calculated by first determining 10% of the meal cost and then adding half of that result to the original 10%. Let's denote \(10\%C\) as \(A\). Then the tip is represented as \(T = A + 0.5A\).

Step 2: Comparing the Two Methods

We will equate the two methods to know if they result in the same tip. So, we need to compare \(0.15C\) with \(A + 0.5A\), where \(A = 0.10C\). Therefore, the comparison becomes \(0.15C = 0.10C + 0.5*0.10C\).

Step 3: Breakdown of the Second Method

For further comparison of the two methods, the method 2 should be simplified. The expression \(0.10C + 0.5*0.10C\) simplifies to \(0.10C + 0.05C = 0.15C\).

Step 4: Final Validation

With the breakdown of the second method, it can be concluded that \(0.15C = 0.15C\), practically showing both methods produce the same result. Therefore, two methods are indeed equivalent, giving the same tip amount.

Key Concepts

Calculating Tip

Leaving a tip at a restaurant is a way to show appreciation for good service. To figure out the tip, you can use different methods to reach the same result. For example, if you want to leave a 15% tip on a meal cost, you can calculate it directly using a simple formula:

• Method 1: Multiply the total cost of the meal (\(C\)) by 0.15. This can be represented as \(T = 0.15C\), where \(T\) is the tip.

This method is straightforward if you're comfortable with multiplying decimals. It gives you the exact percentage of the meal cost as a tip. If you prefer working with percentages in your head, you can also calculate the tip using another approach.

Percentage Calculation

Understanding how percentages work can make calculating tips and other similar tasks much easier. Here’s how you can break down a percentage calculation:

• Method 2 involves first finding 10% of the cost. If the meal costs \(C\), 10% of it is \(0.10C\).
• Next, calculate half of that 10%, which is \(0.05C\), and add it to the original 10%.

This gives you: \(0.10C + 0.05C = 0.15C\). This method is often easier for mental math, as calculating 10% of any amount is usually simple, and splitting that figure in half is much more manageable without a calculator. When doing everyday tasks like leaving a tip, grasping basic percentage calculations helps build confidence.

Mathematical Methods Comparison

When comparing different mathematical methods, it's essential to check if they yield the same result. In this case, both methods of calculating a 15% tip are equivalent, which can be shown by simple algebra:

• With Method 1, we have \(T = 0.15C\).
• With Method 2, we calculate \(T = 0.10C + 0.05C\).
• By simplifying Method 2, we also find \(T = 0.15C\).

This means both approaches lead to the same tip amount. Understanding how different methods work can assist in choosing the one that feels most intuitive for you. Both methods validate the same concept, confirming that math often offers multiple paths to the same solution.