Question

A local restaurant recently renovated its dining space, purchasing new tables and chairs to use in addition to the original tables and chairs. The new tables each seat six customers, while the original tables each seat four customers. Altogether, the restaurant now has 40 tables and is capable of seating 220 customers. How many more new tables than original tables does the restaurant have? A 10 B 20 C 30 D 34 E 36

Short answer

The restaurant has 20 more new tables than original tables.

Step-by-step solution

Define Variables

Let the number of new tables be represented by the variable \( n \) and the number of original tables be represented by the variable \( o \).

Create the System of Equations

Based on the problem, two equations can be formed:1. The total number of tables is 40. \( n + o = 40 \)2. The total seating capacity is 220 customers. \( 6n + 4o = 220 \)

Solve the First Equation for One Variable

Solve the first equation for \( o \):\( o = 40 - n \)

Substitute into the Second Equation

Substitute \( o = 40 - n \) into the second equation:\( 6n + 4(40 - n) = 220 \)

Simplify and Solve for \( n \)

Expand and simplify the equation:\( 6n + 160 - 4n = 220 \)Combine like terms:\( 2n + 160 = 220 \)Subtract 160 from both sides:\( 2n = 60 \)Divide by 2:\( n = 30 \)

Solve for \( o \)

Substitute \( n = 30 \) back into the first equation:\( o = 40 - 30 \)Therefore, \( o = 10 \).

Calculate the Difference

The number of new tables minus the number of original tables is:\( 30 - 10 = 20 \)

Verify Solution

Verify that the total seating capacity is correct:\( 6 \times 30 + 4 \times 10 = 180 + 40 = 220 \)It matches the given seating capacity. Therefore, the solution is verified.

Key concepts

Algebraic Equations

Algebraic equations are mathematical statements that show the equality of two expressions. In this GMAT math problem, we encounter two algebraic equations formed from the given conditions about tables and seating capacity.

The first equation derived is the total number of tables, given by: n + o = 40. (1)
Here, n represents the number of new tables, and o represents the number of original tables.
The second equation concerns the total seating capacity:
6n + 4o = 220. (2)
This equation reflects that new tables seat six customers each, and original tables seat four customers each. Understanding how to form and interpret these algebraic equations from word problems is critical for solving such problems efficiently.

System of Equations

A system of equations is a set of two or more equations with the same variables. To solve a word problem involving multiple conditions, we often need to set up and solve a system of equations.

In this case, our system is composed of equations (1) and (2): n + o = 40 6n + 4o = 220.

These equations represent the constraints given in the problem: the total number of tables and the total seating capacity. To find the values of n and o, we need to solve this system. Learning to work with systems of equations allows us to solve for multiple variables simultaneously and is a powerful technique in algebra.

Variable Substitution

Variable substitution is a method used to solve a system of equations, where we solve one equation for one variable and substitute that into another equation.

In our exercise, we solved the first equation for one of the variables: o = 40 - n.
This step isolates o in terms of n. Next, we substitute this expression into the second equation: 6n + 4(40 - n) = 220.
Expanding and combining like terms helps to simplify the equation, allowing us to solve for the variable n. This method reduces the number of variables in an equation, making it easier to solve complex problems.

Problem-Solving Steps

To tackle word problems effectively, it is essential to follow structured problem-solving steps:

• Step 1: Identify the variables and define them clearly. In our case, we used n for new tables and o for original tables.
• Step 2: Develop equations based on the problem's conditions. We derived n + o = 40 and 6n + 4o = 220.
• Step 3: Solve one of the equations for one variable, as we did with o = 40 - n.
• Step 4: Substitute the expression into the other equation to reduce the number of variables.
• Step 5: Simplify the resultant equation and solve for the remaining variable, which gave us n = 30.
• Step 6: Use the solution to find the value of the other variable, resulting in o = 10.
• Step 7: Compare the quantities to answer the original question.
• Step 8: Verify the solution to ensure it satisfies all given conditions.

Following these steps ensures a comprehensive and systematic approach, which is especially useful for complex problems.