7th and 8th graders, click on the following link for review problems for this week's tests:

https://sites.google.com/site/lassmath/my-forms

**Problem of the Week**

__9/26/16__

**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?

__9/19/16__

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?

**Solution**

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, *A*, *B*, *C*, and *D*, then the handshakes are as follows. There are exactly 6 handshake combinations with 4 students: *AB*, *AC*, *AD*, *BC*, *BD*, and *CD*. Since each student only shakes hands with one-half the students in the class (2 students), one possible solution is *AB*, *AD*, *BC*, and *CD*. Another possible solution is *AC*, *AD*, *BC*, 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.

__9/12/16__

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?

**Solution**

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.

__9/5/16__

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?

**Solution**

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.

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