Problem 19 - Algebra 1: Concepts and Skills

Question

You spend a total of $13.80 at a restaurant. This includes the price of dinner and a 15% tip. What is the price of dinner without the tip? Use a verbal model to solve the problem.

Step-by-Step Solution

Answer

The price of dinner without the tip is calculated to be \$12.00.

Step 1: Understand the Problem

To solve the problem, the fact that needs to be exploited is that the total payment made of $13.80 is essentially 115% of the cost of the dinner (100% for the cost of dinner and 15% as tip). Therefore \( x + 0.15x = \$13.80 \) where x represents the cost of dinner and 0.15x represents the tip.

Step 2: Simplify the Equation

Simplify the equation to have a standard form: \( 1.15x = \$13.80 \). The task is now to isolate x which is the price of the dinner.

Step 3: Solve for x

Solve for x by dividing both sides by 1.15 that gives \( x = \$13.80 / 1.15 \).

Step 4: Calculate the Final Answer

Perform the division to find the value of x.

Key Concepts

Verbal Model in Algebra

A verbal model in algebra is a way of representing a problem using words rather than mathematical symbols. It involves describing the relationships and components of a problem in a narrative form. For instance, when dealing with tips at a restaurant, one might say, 'The total bill is the sum of the meal cost and the tip, which is a percentage of the meal cost.' This approach helps to construct an algebraic equation in a more intuitive way. In the given exercise, the verbal model would articulate that the total amount paid is composed of the original price of the dinner plus an additional 15% of the dinner price as the tip.

Solving Percentage Problems

Percentage problems are a common occurrence in algebra, and they can be handled easily with the right approach. To solve percentage problems, it's essential to remember that percentages are essentially ratios out of 100. In this case, if you know the total amount and the percentage that this total represents, you can find the original amount before the percentage was added. In the provided exercise, we're looking for the price before the tip, which implies we are working backwards from a total that represents 115% (100% of the original price plus the 15% tip). The idea is to use this percentage to create an equation that will yield the original price.

Linear Equations in Algebra

Linear equations are fundamental in algebra and represent a direct relationship between two variables, often written in the form of \( ax + b = c \) with \( x \) being the variable we want to solve for. They're called 'linear' because their graph is a straight line. These equations can be manipulated by performing the same operation on both sides to isolate the variable. In the current example with the restaurant bill, the equation \( 1.15x = \$13.80 \) is a linear equation. To solve for \( x \)—the cost of the meal before the tip—you divide both sides by 1.15, maintaining the balance of the equation and isolating the \( x \) to find its value.

Algebraic Problem Solving

Algebraic problem solving is all about understanding the steps needed to isolate and solve for an unknown variable. It involves identifying given information, translating it into an algebraic equation, and then using algebraic techniques to find the solution. It's crucial to analyze the problem carefully, decide on a method to solve the equation, and be meticulous about the arithmetic involved. In this example, by recognizing the relationship between the bill and the meal cost plus tip, you are able to set up the right equation and then use basic algebraic operations to solve for the price before the tip. Throughout, it's essential to be systematic and check your work at each step.