A restaurant has 19 tables that can seat 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

SAT math problem. Possible answers are: 4, 5, 6, 7, 8.

Answer 1

Let x = table with 4 seats

 

Let y = table with 5 seats 

 

x + y =  total tables = 19

 

4x + 5y = total seats = 84

 

4x + 5y = 84

 

 x + y = 19

 

Multiply the bottom equation by 4

 

4x + 4y = 76

 

Subtract this equation from the first one you get

 

y = 8

 

There are 8 tables with 5 seats.

Answer 2

First, we know that the number of tables that seat 5 people must be even. We know this because 84 is an even number, and any multiple of 4 will be even, so 84-(#tables with 4 people) must also be even. This eliminates 5 and 7 as answer choices.

 

Second-- given that we now have three answer choices, pick the one in the middle and do the math:

 

6*5=30

(19-6)*4=13*4=52

52+30=82.

 

Since 82 is less than 84, we can assume that MORE than 6 tables seat five people. This leaves only one answer choice: 8. To check the logic, do the math:

 

8*5=40

(19-8)*4=11*4=44

40+44=84

 

The math works; correct answer is 8.

Answer 3

This is a classic system of equations problem often found on the SAT. Let's define our variables:

1. Let x be the number of tables that seat 4 people.
2. Let y be the number of tables that seat 5 people.

We can set up two equations based on the information given:

Equation 1 (Total Tables):

The restaurant has a total of 19 tables.

x+y=19

Equation 2 (Total Seating Capacity):

The tables can seat a total of 84 people.

4x+5y=84

Now we can solve this system of equations. One common method is substitution. From Equation 1, we can express x in terms of y:

x=19−y

Substitute this expression for x into Equation 2:

4(19−y)+5y=84

76−4y+5y=84

76+y=84

y=84−76

y=8

So, there are 8 tables that seat 5 people.

Let's check our answer by finding x:

x=19−y=19−8=11

So there are 11 tables that seat 4 people.

Now, check the total seating capacity:

(4 people/table×11 tables)+(5 people/table×8 tables)

44+40=84

This matches the given total seating capacity, so our answer is correct.

The question asks for "How many tables seat 5 people?", which is y.

The answer is 8.

Answer 4

For me, the easiest answer is to walk the 4 times multiplication table, looking for values where the remaining number of seats (out of 84 total) would be divisible by 5. These are small numbers we are dealing with.

The first such value is 4 (= 4x1), leaving 80/5 = 16 five-seaters. This is not our answer, as the total tables would be 17.

The next such value is 24 (= 4x6), leaving 60/5 = 12 five seaters. This is not our answer, as the total tables would be 18.

The next such value is 44 (= 4*11), leaving 40/5 = 8 five seaters. This gives us the 19 total tables we are looking for.

Thus 8 five-seaters solves our problem.

We could also have eliminated the first two trials immdiately, as they are not in the possible answer set. Not that it saves much time.

Answer 5

Sometimes you can do some algebra with equations capturing the numbers with unknown variables.

Or sometimes it may be easier to just plug in the possible solutions and see which one works.

Start with 19 tables seating 4 each, that's only 4 times 19 or 76 people total. You need 84 people

84 minus 76 = 8 more people, so make 8 of the tables for 5 each, with 19-8=11 for just 4 people each.

check the answer 4x11= 44, 8x5=40 44+40=84

x+y = 19

4x +5y = 84

4x+ 4y =4(19) = 76

y = 84-76 = 8 tables seating 5

x = 19-8 = 11 tables seating 4