Question

I have some tables and chairs. If I place two chairs at each table, I have one extra chair. If I place three chairs at each table, I have one table with no chairs. What is the sum of the total number of tables and chairs?
A) 12
B) 7
C) 17
D) 13

Answer 1

Hint:
Here, we will assume the number of tables and chairs to be any variable. We will frame two linear equations with two variables using the given information. Then we will equate these linear equations and find the number of chairs and tables. We will then find the total sum by adding the number of tables and number of chairs.

Complete step by step solution:
Let \[c\] be the number of chairs and \[t\] be the number of tables.
If two chairs are placed at each table, then one extra chair is left without placed at the table. So, we get
 \[c = 2t + 1\] ……………………………………………\[\left( 1 \right)\]
If three chairs are placed at each table, then one table is left without a chair. So, we get
\[c = 3\left( {t - 1} \right)\] …………………………………………\[\left( 2 \right)\]
Now, by equating the equation \[\left( 1 \right)\]and equation\[\left( 2 \right)\], we get
\[2t + 1 = 3\left( {t - 1} \right)\]
By multiplying the terms, we get
\[ \Rightarrow 2t + 1 = 3t - 3\]
By rewriting the equation, we get
\[ \Rightarrow 3t - 2t = 3 + 1\]
By adding and subtracting the like terms, we get
\[ \Rightarrow t = 4\]
So, the Number of Tables is \[4\].

By substituting the number of tables in the equation \[\left( 1 \right)\] , we get
 \[c = 2\left( 4 \right) + 1\]
Multiplying the terms, we get
\[ \Rightarrow c = 8 + 1\]
Adding the terms, we get
\[ \Rightarrow c = 9\]
So, the Number of Chairs is \[9\]
Now, we will find the total number of chairs and tables by adding the number of chairs and the number of tables.
Total Number of Chairs and Tables \[ = \] Number of Chairs \[ + \] Number of Tables
Substituting the values in the above equation, we get
\[ \Rightarrow \] Total Number of Chairs and Tables \[ = 9 + 4\]
Adding the terms, we get
\[ \Rightarrow \] Total Number of Chairs and Tables \[ = 13\]
Therefore, the total number of chairs and tables is 13.

Thus, option (4) is the correct answer.

Note:
Linear equations are a combination of constants and variables. A linear equation is defined as an equation with the highest degree as 1 and it has only one solution. Here, we can make a mistake by just finding the numbers of chairs and tables and forget to find the total number of chairs and tables. We can make mistakes in framing the equation and thus end up getting the wrong answer.