Question

Challenge Problem Sharing the cost of a Pizza Judy and Tom agree to share the cost of an 18 dollar pizza based on how much each ate. If Tom ate \(\frac{2}{3}\) the amount that Judy ate, how much should each pay? [Hint: Some pizza may be left.]

Short answer

Judy should pay \(10.80 and Tom should pay \)7.20.

Step-by-step solution

Define Variables

Let Judy's portion of pizza be represented by the variable \ (J). Tom ate \(\frac{2}{3}\) of the amount Judy ate, so Tom's portion can be represented as \(\frac{2}{3}J\).

Set Up the Total Cost Equation

The total cost of the pizza is \(18\) dollars. Since Judy and Tom agree to share the cost based on how much each ate, we need to find the fractions of the total cost each should pay. The total amount they ate together is \(J + \frac{2}{3}J = \frac{5}{3}J\).

Calculate Judy's Share

Judy’s share of the pizza is \(J\). The fraction of the total pizza she ate is \(\frac{J}{\frac{5}{3}J} = \frac{3}{5}\). Thus, Judy should pay \(\frac{3}{5} \times 18 = 10.80\) dollars.

Calculate Tom's Share

Tom's share of the pizza is \(\frac{2}{3}J\). The fraction of the total pizza he ate is \(\frac{\frac{2}{3}J}{\frac{5}{3}J} = \frac{2}{5}\). Thus, Tom should pay \(\frac{2}{5} \times 18 = 7.20\) dollars.

Key concepts

ratio and proportion

Ratio and proportion are essential concepts used in various mathematical problems, including cost-sharing scenarios. A ratio is a way to compare two or more quantities by showing the relative sizes of these quantities. For example, in the exercise, we are given that Tom ate \(\frac{2}{3}\) the amount Judy ate. This information allows us to set up a ratio between the quantities of pizza they each ate.
Proportion, on the other hand, is an equation that states that two ratios are equal. In this case, we use proportions to determine the fraction of the total cost each person should pay. By establishing that Judy's and Tom's shares are proportional to how much they ate, we can conclude how to divide the cost of the pizza based on their shares.
So in this problem, we set up the total amount of pizza eaten as the sum of Judy's and Tom's portions. The ratio of their portions helps us understand their respective shares in relation to the whole pizza and the cost.

fractional calculations

Fractional calculations are crucial when dealing with problems involving parts of a whole, such as the division of pizza in our exercise. Understanding fractions is key to solving such problems accurately. In the given problem, Tom ate \(\frac{2}{3}\) of the amount Judy ate. This gives us a fraction that we can manipulate to find different parts of the whole.
First, we define Judy's portion as \(J\) and Tom's portion as \(\frac{2}{3}J\). To find the total amount of pizza eaten, we add these: \(J + \frac{2}{3}J = \frac{5}{3}J\). This step shows us that the sum of these portions forms the whole, or the total pizza eaten.
Next, we use these fractions to determine the share each person should pay. Judy's share is determined by the fraction of the pizza she ate, computed as \(\frac{J}{\frac{5}{3}J} = \frac{3}{5}\). We then use this fraction to calculate her part of the total cost: \(\frac{3}{5} \times 18 = 10.80\) dollars. Similarly, we compute Tom's share: \(\frac{2}{3}J\) leads to \(\frac{\frac{2}{3}J}{\frac{5}{3}J} = \frac{2}{5}\) of the pizza, and therefore \(\frac{2}{5} \times 18 = 7.20\) dollars.
Working with fractions this way ensures we assign the costs accurately based on their portions of the pizza.

algebraic expressions

Algebraic expressions are combinations of variables, numbers, and operators that represent mathematical relationships. In this problem, we use algebraic expressions to represent the portions of pizza that Judy and Tom ate, and to find the total cost.
To solve the problem, we start by defining Judy's portion as \(J\). Tom's portion is then expressed as \(\frac{2}{3}J\). The combined amount they ate can be written as an algebraic expression: \(J + \frac{2}{3}J\). Combining these like terms gives \(\frac{5}{3}J\).
Next, we use this expression to calculate the cost each should pay. For Judy, we determine the fraction of the total pizza she ate: \(\frac{J}{\frac{5}{3}J} = \frac{3}{5}\). This fraction is then multiplied by the total cost of \(18\) dollars to find her share: \(\frac{3}{5} \times 18 = 10.80\) dollars. Similarly, Tom's portion is calculated as \(\frac{\frac{2}{3}J}{\frac{5}{3}J} = \frac{2}{5}\), leading to a cost of \(\frac{2}{5} \times 18 = 7.20\) dollars.
Using algebraic expressions in this way allows us to handle variables and relationships systematically, ensuring accurate and reliable solutions to mathematical problems like cost sharing.