7th graders: click on the following link for practice problems for Thursday's test on Signed Numbers


Problem of the Week


Old Books 

A mathematics teacher collects old books. One day, a student asked him how many old math books he has. He replied, “If I divide the books into two unequal whole numbers, then 64 times the difference between the two numbers equals the difference between the squares of the two numbers.” How many old math books does the teacher have?


Mouse Race

Two mice are racing around the edges of a square whose sides are 2 feet in length. They start at the same vertex (corner) and both go in a clockwise direction. One mouse travels at a constant rate of 1 foot per second, and the second mouse travels at a constant rate of 2 feet per second. After 22 seconds, how far apart will the mice be from each other?


2 feet apart. Label a square with vertices A, B, C, and D in clockwise order. Suppose the mice start at vertex A. After 22 seconds, the first mouse will be on vertex D, while the second mouse will be on vertex C. Therefore, they will be two feet apart.


New Operation

Imagine that a new mathematical operation is being used. Its symbol is #. See the following equations:

1 # 1 = 2 

3 # 5 = 34 

6 # 9 = 117 

10 # 14 = 296

Find the value of 15 # 19, and explain your reasoning. 


586. a # b = a2 + b2. For example, 3 # 5 = 32 + 52 = 9 + 25 = 34. Therefore, 152 + 192 = 225 + 361 = 586.


Digging a Hole

If it takes two workers 4 hours and 35 minutes to dig a hole 3 meters long, 3 meters wide, and 3 meters deep, how long would it take five workers to dig a hole 6 meters long, 6 meters wide, and 6 meters deep if they worked at the same rate?


About 880 minutes or 14:40. The first hole has a volume of 3 × 3 × 3 = 27 cubic units. The new hole has a volume of 6 × 6 × 6 = 216 cubic units; 216/27 shows that 8 smaller holes would fit into the larger hole. In other words, the new hole is twice as wide, twice as long, and twice as deep as the original, so it would take 2 × 2 × 2 = 8 times as long. The smaller hole took two workers 4 hours and 35 minutes or 275 minutes, so it would take two workers 2,200 minutes to dig the larger hole.  With 5 workers, it would take 2/5 * 2200 minutes or 880 minutes to dig the larger hole.


There are 26 students in your class (including yourself). Every student shakes hands with exactly one-half of the students in the room. What is the minimum number of handshakes that occurred?


169. Each student shakes hands with 13 students; 26 × 13 = 338. Because each handshake is counted twice, you must divide 338 by 2 for the answer of 169. Examining a smaller problem helps. If there are four students, ABC, and D, then the handshakes are as follows. There are exactly 6 handshake combinations with 4 students: ABACADBCBD, and CD. Since each student only shakes hands with one-half the students in the class (2 students), one possible solution is ABADBC, and CD. Another possible solution is ACADBC, and BD. The resulting number pattern for other class sizes is the sequence of perfect squares as seen in the table where n is a counting number.


Your uncle buys a new car that comes with five new tires, one for each wheel and one spare tire. He decides to be economical by using the spare tire as much as the other four tires. If he drives 68,000 miles, what will be the wear in miles on each tire?


54,400 miles. 68,000/5×4 = 54,400. He must use four tires at a time, and he must use five tires equally. He has five tires: tires A, B, C, D, and E. Divide his total miles into 5 driving intervals: 68,000/5 = 13,600. 

Each tire is used for only 4 intervals; 4×13,600 = 54,400 miles.



A certain product is sold as either a liquid or a powder. Consumers were interviewed, and a survey revealed that—

1/5 do not use the product,

1/3 do not use the powder form,

427 use both the liquid and powder form, and

2/7 do not use the liquid form.


How many consumers were in the survey?


735 consumers. Divide the consumers into four sets:

A: They do not use the products. 
B: They do not use the powder form. 
C: They use both liquid and powder. 
D: They do not use the liquid form.

We know that— 

A + B = 1/3 of the total, 
A + D = 2/7 of the total, 
A = 1/5 is the total, and 
C = 427.

Then, B is 1/3-1/5 = 2/15 of the total. D is 2/7-1/5 = 3/35 of the total. A + B + D is 44/105 of the total. C is the remaining 61/105 of the total. The number of consumers interviewed is 427/(61/105) = 735.