Problem 83 - 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. Write an equation that represents your friend's method of computing the tip.

Step-by-Step Solution

Answer

The equation that represents the friend's method of computing the tip is \(T=0.10C + 0.05C\).

Step 1: Identify the two components of the tip calculation

In your friend's method, there are two parts to the calculation: \n1. Calculating the 10% of the cost of the meal: This can be represented as \(0.10C\). \n2. Calculating half of 10% of the cost of the meal: Half of 10% is 5%, so this can be represented as \(0.05C\).

Step 2: Write the equation

The friend's method of calculating the tip can be written in equation form by adding together the two components identified in Step 1: \(T=0.10C + 0.05C\).

Key Concepts

Percentage Calculation

Calculating percentages is a useful skill in everyday life, especially when it comes to financial transactions like computing tips. When talking about percentages, they are simply a way of expressing a number as a fraction of 100. For example, a 15% tip means you're giving 15 out of every 100 units of currency spent on the meal. To find a percentage of a number, you multiply the number by the percentage expressed as a decimal.
For the original exercise, the computation of a 15% tip using the formula involves multiplying the cost of the meal, denoted as \( C \), by 0.15. This is mathematically expressed as \( T = 0.15C \).
Understanding how to calculate percentages is essential not only to compute tips but also in various scenarios like discounts, markups, and interest rates. Practice with real-life examples, such as determing discounts during shopping, will enhance your proficiency in percentage calculations.

Equation Formation

Creating equations from word problems or different methods of calculation harnesses the ability to translate verbal procedures into mathematical expressions. This is a vital skill in algebra. In the given task, your friend's mental math method for calculating the tip was transformed into an equation to make the calculation process clearer and more systematic.

The friend's method involves calculating 10% of the meal cost and then adding half of that 10% to find the total tip. The equation for this method is constructed as follows:

• First, find 10% of the cost of the meal, which can be expressed as \( 0.10C \).
• Then, add half of 10%, which equals 5%, expressed as \( 0.05C \).

Combining these parts, the equation becomes \( T = 0.10C + 0.05C \).
Knowing how to create an equation from a calculation process will help clarify your mathematical thinking and is beneficial for solving algebraic problems effectively.

Mental Math Techniques

Mental math is a powerful tool for simplifying calculations in your head, without any need for physical computation. This can be particularly useful in daily tasks like calculating expenses in a restaurant. The learning of mental math techniques involves understanding and using simple arithmetic shortcuts.

In the exercise, your friend's mental calculation for the tip demonstrates a clever method of breaking down the problem:

• Calculate 10% of the meal cost by moving the decimal point one place to the left, which is often easier than finding 15% directly.
• Then calculate 5% as half of the 10% already found, which is a simple division by 2.

This approach makes it easier to compute a 15% tip quickly and accurately.
By practicing mental math techniques, you not only increase your calculation speed but also sharpen your number sense, making mathematics more intuitive and less daunting.