Problem 84 - 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. Simplify the equation. What property did you use to simplify the equation?

Step-by-Step Solution

Answer

The equation simplifies to \(T = 0.15C\) which is equivalent to the tip calculation used originally. The mathematical properties used to simplify the equation include the Distributive Property and Arithmetic operations to combine like terms.

Step 1: Setup the Equation

Start by expressing the 'Easier Method' for calculating the tip mentioned by your friend i.e., \(10 \%\) of the cost of the meal + \(\frac{1}{2}\) of \(10 \%\) of the cost of the meal. Take \(C\) to be the cost of the meal. Hence, the equation becomes: \(T = 0.10C + 0.5 \cdot 0.10C \)

Step 2: Simplify the Equation

Use the Distributive property of multiplication over addition to simplify the equation: \(T = 0.10C + 0.05C\)

Step 3: Combine Like Terms

The like terms - the terms with the same variable, to the same power can be combined to simplify the equation further. This gives us: \(T = 0.15C\)

Step 4: Identify the property

The properties which we applied in this problem are 'Distributive Property' to distribute 0.5 over 0.10C and then 'Combination of Like terms' which is part of Arithmetic operations to combine 0.10C and 0.05C.

Key Concepts

Distributive Property

The Distributive Property is a fundamental concept in algebra that allows you to multiply a term by a set of terms inside a parenthesis. It is useful when dealing with expressions such as \(a(b + c)\), where you need to multiply \(a\) by both \(b\) and \(c\).

This property can be expressed as: \\[aa(b + c) = ab + ac\]

In the context of the given exercise, you use the Distributive Property to simplify the term \(0.5 \times 0.10C\).

Here's how you apply it:

• First, recognize the multiplication \(0.5 \times 0.10C\) as if you're distributing the 0.5 over \(0.10C\).
• This results in \(0.05C\), effectively applying the property.

Thus, by deploying the Distributive Property, we transition from a more complex mental calculation into simpler arithmetic operations, fitting perfectly in practical scenarios like calculating tips quickly.

Simplifying Equations

Simplifying equations is an essential skill in algebra. It involves breaking down and reducing equations to their simplest form, making them easier to work with and understand. The primary goal when simplifying equations is to combine like terms and eliminate any unnecessary complexity.

In the exercise, the tip calculation starts as:

\[T = 0.10C + 0.5 \times 0.10C\]

After employing the Distributive Property, it becomes:

\[T = 0.10C + 0.05C\]

• To further simplify, we combine these like terms. "Like terms" are terms in an equation that have the same variable raised to the same power.
• Here, \(0.10C\) and \(0.05C\) are like terms because they both contain the variable \(C\).
• By combining them, we simplify the equation to \(T = 0.15C\).

Simplifying equations makes them more manageable, especially in day-to-day applications such as performing mental arithmetic or verifying calculations.

Calculating Percentages

Calculating percentages is a frequent necessity in various real-life situations, from determining discounts to calculating interest rates, and of course, adding tips to your restaurant bill! Understanding how to easily compute percentages helps you accurately handle these situations without much effort. In this exercise, you focused specifically on a 15% tip calculation.

The given formula \(T = 0.15C\) indicates that the tip \(T\) is 15% of the meal's cost \(C\). Here’s a breakdown of the process:

• To calculate 15% directly, multiply the cost by 0.15, as represented by the formula \(T = 0.15C\).
• Your friend’s approach simplifies the calculation by first figuring out 10% of \(C\) (i.e., \(0.10C\)), allowing an easy mental calculation.
• Then, you add half of this 10%—that is, \(0.05C\) - to reach 15% of the total cost.

This concept highlights a practical method of computing percentages without a calculator, aiding in quick, in-the-moment decisions like tipping in a restaurant.