Problem 77 - Introductory and Intermediate Algebra for College Students 4th

Question

Make Sense? In Exercises \(75-78,\) determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I have \(\$ 100\) and my restaurant bill comes to \(\$ 80,\) which is not enough to leave a \(20 \%\) tip.

Step-by-Step Solution

Answer

The statement does make sense. After calculating, you'd find out that 20% of $80 is $16. After adding this to the original bill, the total comes to $96. Since $96 is less than $100, there is enough money to pay the bill and leave a 20% tip.

Step 1: Calculate the Tip Amount

To calculate the desired tip, we need to find the 20% of the total bill which is $80. To find the 20% of $80, we can multiply 80 by 20 and divide it by 100, i.e., \( (20 / 100) * 80\).

Step 2: Calculate Total Expenditure

Now add this tip amount to the original bill ($80) to get the total expenditure.

Step 3: Check against the Total Amount You Have

Check whether the total expenditure exceeds the total amount in hand, i.e., $100.

Key Concepts

Calculating Percentage Tips

Understanding how to calculate percentage tips is an essential everyday math skill, especially when dining out or engaging in services where tipping is customary. To calculate a tip based on a percentage, such as a commonly expected 20% tip at a restaurant, you need to know the basic formula for calculating percentages.

First, you need to identify the total cost of the service before tip, which in our given example is \(80. Next, you determine the percentage you want to give as a tip—in this case, 20%. To find out what 20% of \)80 is, you convert the percentage to a decimal by dividing it by 100, resulting in 0.20. Then, multiply this decimal by the total amount:

\[ \text{Tip} = 0.20 \times 80 \]
This multiplication gives you the tip amount, which is \[ \text{Tip} = 0.20 \times 80 = 16 \] dollars. To find out your total expenditure, simply add the tip to your original bill. For a smoother and faster calculation, remember this 20% tip trick: Divide the bill by 10 and then multiply by 2.

Mathematical Reasoning

Mathematical reasoning involves the ability to logically work through problems and arrive at a correct conclusion. It's a thought process that includes conceiving a plan to solve a problem, executing that plan, and then concluding and evaluating the results. In our example, the application of mathematical reasoning begins by interpreting the problem correctly. Firstly, you are asked if it 'makes sense' to claim that having \(100 is not enough to afford a 20% tip on an \)80 bill.

Through reasoning, we apply what we know about calculating percentages and addition to make sense of the situation. After calculating the 20% tip as \(16, we add it to the initial \)80 and find the total cost to be \(96. This does not exceed the \)100 in hand. Therefore, you conclude that the statement 'does not make sense' since you do in fact have enough to leave a 20% tip. This conclusion is based on executed arithmetical operations and understanding of the problem's context.

Algebra Problems

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols; it is a language of its own that represents numbers and relationships in equations and formulas. Working through algebra problems usually involves finding unknown values by applying these rules.

In the context of our tipping problem, we could let 'T' be the total expenditure, which comprises the original bill 'B' and the tip 'P', a percentage of the bill. Presented as an algebraic expression, it looks like:

\[ T = B + (P \times B) \]
We then insert our known values into the equation:

\[ T = 80 + (0.20 \times 80) \]
Solving the equation gives us the total expenditure. While this example does not present a traditional algebra problem with an unknown variable to solve for, it showcases how algebraic thinking can organize and simplify calculations and concepts, making it a valuable tool for solving a wide array of mathematical problems.