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.

Short answer

Yes, both methods give the same results.

Step-by-step solution

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 \(T = A + 0.5A\).

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 \(0.15C = 0.10C + 0.5*0.10C\).

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 \(0.10C + 0.05C = 0.15C\).

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.