Roger paid $915 rent for each of eight complete months - January through August - for a total of

He paid the rent for May two days late, so the penalty was

To break the lease, he paid two months' rent, or

Add these, add the prorated September rent, and subtract the returned deposit:

A number is divisible by 4 if and only if its last two digits form a number which itself is divisible by 4. The numbers 08, 28, 48, 68, and 88 are each divisible by 4, as seen here:

However, 18, 38, 58, 78, and 98 are not divisible by 4, as seen here:

Therefore, the ways to fill in the box to form a number divisible by 4 are 2, 4, 6, 8, and 0 - five ways.

An integer is divisible by 3 if and only if its digit sum is also divisible by 3. If we let  be the missing digit, then the digit sum is 

We can test each digit from 0 to 9 to see which sums are divisible by 3. We can see that this is the case if:

The other seven digits can be seen to form a digit sum that is not a mulitiple of 3 using the same method.

Therefore, there are three ways to fill in the box to form a multiple of 3.

For each of the four couples, there are two ways to select the table at which to seat each person - the man can be at the left table and the woman can be at the right, or vice versa. This makes  ways to make this decision total between the four couples.

Once these four choices are made, for each table, there are 

ways to seat the four chosen persons. 

By the multiplication principle. there will be 

 different arrangements that seat each husband and wife at separate tables.

First, it is necessary to choose the couples that are each table. There are

 ways to choose the two couples that will be at the left table; this is equal to

.

Once those two couples are chosen, the remaining two couples will be at right.

For each table, there are 

ways to seat the four chosen persons.

By the multiplication principle. there will be 

 different arrangements that seat all four couples together.

We can apply the multiplication principle here. First, there are two possible places Rachel can occupy. One this is decided, there are two ways to seat her parents - with Mr. Jones to her left and Mrs. Jones to her right, and vice versa. The number of ways to seat the remaining five guests in the remaining five seats is 

.

Therefore, the number of arrangements fulfilling the requirement is

.

For each candidate, add the two day totals:

Allen: 

Paul: 

Veronica: 

Wendy: 

Allen was the highest vote-getter. However, Allen got 157 votes, and his opponents got a total of  votes, so Allen did not get a majority. He will fac the second-highest vote-getter, Veronica, in a runoff.

Ahmad received 193 votes plus 21.7% of the previously uncounted 115 votes, for a total of

 votes.

Michiko received 598 votes plus 42.6% of the previously uncounted 115 votes, for a total of

 votes.

Quinn received 210 votes plus 18.3% of the previously uncounted 115 votes, for a total of

 votes.

Zane received 189 votes plus 17.4% of the previously uncounted 115 votes, for a total of

 votes.

Michiko got the most votes with 647. Her opponents won a total of

 votes, more than Michiko, so Michiko did not win a majority. She will face the second-highest vote-getter, Quinn, in a runoff.

A person in the set - the union of the sets  and  - would have to fit either or both of the following descriptions:

The person could be in the set , in which case (s)he speaks Arabic. Jim speaks Arabic, so he is in ; consequently, he is in . Julie does not qualify for inclusion in this set this way.

The person could be in set , which means (s)he is in both , the complement of , and , the complement of . (S)he would have to speak neither Dutch nor Spanish. Julie speaks Dutch, so she does not qualify for inclusion in this way either.

 is the set of dancers. , the complement of , is the set of persons not in  - that is, the set of persons who are not singers.  is the intersection of the two, or the set of all persons in both  and ; to be in , a person must be a dancer and must not be a singer. Mary is excluded from  because she is a singer; Florence is excluded because she is not a dancer.