Question

Susan is attending a talk at her son’s school. There are 8 rows of 10 chairs where 54 parents are sitting. Susan notices that every parent is either sitting on their own or next to just one other person. What is the largest possible number of adjacent empty chairs in a single row at that talk?

Answer 1

Answer:

4

Step-by-step explanation:

To have the largest number of empty chairs in a single row there would need to be the maximum possible number of parents sitting in the other seven rows.

Because no parent is sitting next to two other parents, there cannot be three consecutive chairs in which parents are sitting.

This means that there can be at most seven chairs that are occupied in each row of ten chairs. An example of how seven chairs in one row might be filled is shown in the diagram alongside.

Therefore the maximum number of parents that could be seated in seven rows is  

7

×

7

=

49

.

Since there are 54 seated parents the smallest number of parents that there could be in the remaining row is  

54

−

49

=

5

.

In seating five parents in one row there will be five empty chairs. The five parents must be seated in at least three groups.

So there will be at least two gaps in the row and five empty chairs making up these gaps.

Therefore the maximum gap that there can be is four chairs when there are two gaps,  with one empty chair in one gap, and four adjacent empty chairs in the other gap. An arrangement of this kind is shown alongside.

Therefore the largest possible number of adjacent empty chairs in a single row is four.

Answer 2

Given:

Total number of rows $=8$

Total number of chairs in each row $=10$

Total number of parents that are sitting $=54$

Every parent is either sitting on their own or next to just one other person.

To Find:

The largest possible number of adjacent empty chairs in a single row.

Solution:

• In order to have the largest number of adjacent empty chairs in a single row, we need to have the maximum possible number of parents sitting in the other seven rows.

• Since each parent is either sitting on their own or next to just one other person, therefore there cannot be three consecutive chairs in a row that are occupied.

• Therefore, according to this scheme, the maximum number of parents that can be seated in each row is $7$. An example of this arrangement can be: $\[{\rm{PP\_\,PP\_\,PP\_\,P}}\]$
• Hence, the maximum number of parents that can be seated in $7$ rows is $\[7 \times 7 = 49\]$.
• Since in total there are $54$ parents sitting, the number of parents that are seated in the remaining row is $54-49=5$. This is the smallest number of parents that can be seated in a single row.
• If $5$ parents are seated in a row, there will be $5$ empty chairs in that row.
• In order to have the largest possible number of adjacent empty chairs, the $5$ parents need to be seated in $3$ groups with two gaps in the row and $5$ empty chairs forming these gaps.
• An example of this arrangement is: $\[{\rm{PP\_\,PP\_\_\_\_\,P}}\]$
• Therefore the largest possible gap that there can be is $4$ chairs when there are two gaps, with one empty chair in one gap, and four adjacent empty chairs in the other gap.

Hence, the largest possible number of adjacent empty chairs in a single row is $4$.

#SPJ2