A restaurant has 19 tables that can sear a total of 84 people. Some of the tables seat 4 people and the others seat 5 people. How many tables seat 5 people?

Question

A restaurant has 19 tables that can sear a total of 84 people. Some of the tables seat 4 people and the others seat 5 people

Answer 1

Say x is the number of tables that seat 4 people and y is the number of tables that seat 5 people. You can set up a system of equations.

 

x + y = 19        <-- total number of tables must equal 19

4x + 5y = 84     <-- total number of people seated must equal 84

 

Using the first equation, x = 19 - y. Then substitute that equation in the second one.

 

4( 19-y ) + 5y = 84

76 - 4y + 5y = 84

y = 8

 

Since y is the number of tables that seat 5 people, 8 is your final answer

Answer 2

x=# of tables seating 5 people

19-x=# of tables seating 4 people

5x+4(19-x)=84

5x+76-4x=84

x+76=84

x=8 tables

 

Answer 3

Let's call 'n' the number of four-person tables and 'm' the number of five-person tables.  We just need to figure out some combination of n and m that (1) adds up to 19, and (2) seats a total of 84 people.  So, we have two requirements, and therefore, two equations:

Equation 1:  n + m = 19

Equation 2: 4*n + 5*m = 84

 

Two unknowns, two equations means you can solve for each of the unknowns.  Just follow this recipe:

 

Step 1:  Choose one of your two equations and move all but one variable to the right side of the equation, which we can call a new equation just for ease of discussion:

Equation 3 (based on Equation 1): n = 19 - m

(Equation 3 is really just a rearrangement of Equation 1, but it's easier to think of it as a new equation)

 

Step 2: The "new equation" defines one of the variables, so we can use that definition in the other original equation to create another new equation:

Equation 4 (based on Equations 2 & 3):

4*n + 5*m = 84  ⇒  4*(19 - m) + 5*m = 84  (since we know from Eq. 3 that n = 19 - m)

4*19 - 4*m + 5*m = 84  ⇒  76 + m = 84

 

Step 3: Solve Equation 4 for the one variable it contains.

m = 84 - 76 = 8

 

Step 4: Choose either of the original equations, substitute in the value of the variable we just computed, and solve for the other variable.

(Substituting m = 8 into Equation 1):

n + m = 19  ⇒  n + 8 = 19  ⇒ n = 19 - 8  ⇒  n = 11

 

Step 5: Check the math using the other original equation.

4*n + 5*m = 84  ?  4*11 + 5*8 = 44 + 40 = 84 ?

 

For more advanced questions involving more than two variables, simply repeat steps 1 and 2 until you can do Step 3.  (This will make sense when you run into an actual problem.  If it doesn't just send me a message!)

Answer 4

1. let "x" be the number of tables seating four people, and "y" be the number of tables seating five people.

Since the total number of tables is 19, then x+y=19.  That is one equation, with the two variables.  You need to equations to solve for two variables.

2.  the total number of guests is 84,  with four people at each "x" table and five people at each "y" table. Thus,

   4x + 5y = 84

3.  change the first equation  x+y=19  into x=19-y

4.  substitute the value of x into the second equation, i.e.,  4(19-y)+5y=84 and expand as 76-4y+5y=84 

5.  simplify as y=84-76=8

6.  substitute the value of y into the equation for x in # 3 above, i.e., x=19-8=11

7 check your work:  4 times 11 plus 8 time 5 does equal 44 + 11, or 84.   ###