7th Graders - Here's a link to our online curriculum:


https://im.kendallhunt.com/MS/students/2/index.html



Here's a link to our online glossary:


https://im.kendallhunt.com/MS/students/2/glossary.html




Problem of the Week


Engine Failure

 

An aircraft is equipped with three engines that operate independently. The probability of an engine failure is .01. What is the probability of a successful flight if only one engine is needed for the successful operation of the aircraft?

 

Milk or Bread or Both?

 

A customer enters a supermarket. The probability that the customer buys bread is 0.6, the probability that he buys milk is 0.5, and that he buys both bread and milk is 0.3. What is the probability that the customer would buy either bread or milk or both?

 

Solution

.8, or 4/5. Let Brepresent the event that the customer buys bread, Mthe event that the customer buys milk. Then, according to the rule of addition, we have

P(BM) = P(B) + P(M) − P(B∩ M)
= .60 + .50 − .30
= .80

 

Alternate solution.A Venn diagram offers a visual approach. If we first show the probability that the customer buys mile andbread—P(M∩ B)—then we can complete the diagram by subtraction: P(MB) = .2 + .3 + .3 = .8


Rectangular Solids

 

How many rectangular solids are possible with a volume of 100 cubic meters and sides of only whole numbers?


Solution

8. The solutions by dimension are (1, 1, 100), (1, 2, 50), (1, 4, 25), (1, 5, 20), (1, 10, 10), (2, 2, 25), (2, 5, 10), and (4, 5, 5).


 

10 Digit Number

 

Write a ten-digit number so that the first digit indicates how many 0s are in the number, the second digit indicates how many 1s are in the number, the third digit indicates the number of 2s, etc.


Solution

6,210,001,000

 


Flat Tire

 

After a cyclist has gone 2/3 of his route, he gets a flat tire. Finishing on foot, he spends twice as long walking as he did riding. If his walking and riding rates are both constant, how much faster does he ride than walk?


Solution

4 times as fast. He walks one-third of the way, or half as far as he rides, but it takes him twice as long. Therefore, he rides four times as fast as he walks.

 


Pluto Math

 

Pluto's inhabitants use the same mathematical operators that we do (+, −, etc.). They also use an operator, @, that we do not know. The following are true for any real numbers and y.


x@ 0 = x

xyyx

x @ y = ((y–1) @ x) + (x+1) = (y-2) @ x + 2(x+1) = ((y–3) @ x) + 3(x+1)….

What is the value of 12 @ 5?


Solution

77. We have

12 @ 5 = 5 @ 12
= (4 @ 12) + 13
= (3 @ 12) + 13 + 13
= (3 @ 12) + 26.

Continuing,

(3 @ 12) + 26 = (2 @ 12 + 39)
= (1 @ 12) + 52
= (0 @ 12) + 65
= (12 @ 0) + 65
= 77




Path Distance

 

Jeremy walks along a spiral path (as shown). If the path is 2 meters wide, how far does he walk?




Solution

97 meters. By superimposing a grid of 2X2 squares, one can see that the path Jeremy travels can be broken into 13 sections of lengths 13, 12, 12, 10, 10, 8, 8, 6, 6, 4, 4, 2, and 2 meters.



Chicken Nuggets

 

Chicken nuggets come in packets of 6, 9, or 20. What is the largest number of nuggets that you cannot buy when combining various packets? 

 

Solution

43.

Any multiple of 3 that is greater than 3 can be obtained from packets of 6 and 9 nuggets. Since 36 = 9 + 9 + 9 + 9, 38 = 20 + 9 + 9, and 40 = 20 + 20, any even number ≥ 36 can be achieved by adding 6s to each of these. Similarly, by adding another 9, any odd number ≥ 45 can be achieved. But 43 is not yet guaranteed, so we need to examine the possibility of combinations that yield 43 nuggets. Since 43 is not a multiple of 3, likewise, 43 – 20 = 23 is not a multiple of 3, and 43 – 2 • 20 = 3 is too small to achieve. Consequently, 43 cannot be obtained.