Team A takes 10 days to complete a project. Team B takes 30 days to complete the same project. The two teams now work together on the same project. Meanwhile, Team A rest for 2 days and Team B rest for 8 days. None of the team rests in the same day. How many days are needed to complete the same project?
**Editorial note for this self-study exercise:** Assume constant daily work rates, and that all the stated non-overlapping rest days occur before the project is completed.
The product of 2860 and m(m ≠ 0) is a square number. Find the smallest possible value of m?
**Editorial note for this self-study exercise:** For this self-study exercise, take m to be a positive integer and the product to be an integer perfect square. The source omits the integer condition on m.
Mr. Nelson has 216 candies. He wanted to put them into packets so that he can sell it to his customer. List all possible numbers of candies that Mr. Nelson can put in each packet so that there are at least 4 and not more than 15 candies in each packet. Then, list the corresponding number of packets.
**Editorial note for this self-study exercise:** Use all 216 candies with no leftovers, and put the same number in every packet. Enter every pair as a:b, meaning candies per packet:number of packets, in increasing order of a; separate pairs by semicolons with no spaces.
A group of friends decided to divide the $\$800$ cost of a trip equally among themselves. When two of the friends decided not to go on the trip, those remaining still divided the $\$800$ cost equally, but each friend’s share of the cost increased by $\$20$. How many friends were in the group originally?
Mrs. Rama bought a box of paper clips. She shared the paper clips among her 4 children. Alice received 190 paper clips. Rita received 56 more paper clips than Jim. Jim received 25% of the paper clips while Tom received 20% of the paper clips. How many paper clips did Mrs. Rama buy altogether?
**Editorial note for this self-study exercise:** Assume all the paper clips are distributed among exactly these four children, with none left over.
An ATM password at Fred’s Bank is composed of four digits from 0 to 9, with repeated digits allowable. If no password may begin with the sequence 9, 1, 1, then how many passwords are possible?
**Editorial note for this self-study exercise:** Treat a password as a string of four digits, so leading zeroes are allowed. Only the first three positions forming 911 are forbidden.
Jack and Bill had $\$440$ altogether. Bill had $\$20$ more than Jack. When Jack spent $\frac13$ the amount of money spent by Bill, he had 3 times as much money as Bill. How much money did Bill spend?
The positive integers are arranged in rows and columns as shown below.
Row 1: 1
Row 2: 2 3
Row 3: 4 5 6
Row 4: 7 8 9 10
Row 5: 11 12 13 14 15
Row 6: 17 18 19 20 21 22
$\vdots$
More rows continue to list the positive integers in order, with each new row containing one more integer than the previous row. How many integers less than 2000 are in the column that contains the number 2000?
**Editorial note for this self-study exercise:** Follow the verbal rule of consecutive positive integers in left-aligned rows. The displayed sixth row has a source error: for this exercise replace it by 16, 17, 18, 19, 20, 21, and continue without omissions. Count only entries strictly less than 2000.
Maria is going to celebrate her birthday at home and invited 20 classmates. If we know that $\frac35$ of the invited classmates are girls and the rest are boys, how many girls did Maria invite to her party?
Driving from Boston to New York one day, Sasha covered the 250 miles in five hours. Because of heavy traffic, the 250-mile return took six hours and fifteen minutes. Calculate average speeds for the trip to New York, the trip from New York, and the round trip.
**Editorial note for this self-study exercise:** Give the three speeds in miles per hour in the order asked. Write the first two as integers and the last as an exact fraction a/b in lowest terms with a positive denominator; use semicolons with no spaces. Do not round the last speed.
In a list of seven integers, 13 is the smallest member, 37 is the largest member, the mean is 23, the median is 24, and the mode is 18. If the numbers 8 and 43 are then added to the list, which of the following will change: the mean, the mode, or the median?
