7th and 8th graders, please click here for practice problems and an answer key before Tuesday's test:

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


Problem of the Week

12/11/17

Palindromic Speeding

The odometer of a family car shows 15,951 miles. The driver noticed that this number is palindromic: it reads the same backward as forward. Surprised, the driver saw his third palindromic odometer reading (not counting 15,951) exactly five hours later. How many miles per hour was the car traveling in those 5 hours (assuming speed was constant)?

 

12/4/17

Equal Row Sums 

Using the diagram, place the numbers 1 to 10 in the circles so that the sums in the rows of three circles are the same and the sums in the rows of four circles are the same.


Solution

One possible arrangement is shown. The sum of the rows of four is 23, and the sum of the rows of three is 16.


11/27/17

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?

Solution

0.999999. Let P(S) be the probability of a successful flight, P(S') the probability of an unsuccessful flight, and P(Fn) the probability of n engines failing. Since the flight is unsuccessful only when all three engines fail, then the probability of unsuccessful flight is

P(S') = P(F1 ∩ (F2 ∩ F3))
=(.01)(.01)(.01)
=(.01)3.

But

P(S) = 1 − P(S')
= 1 − (.01)3
= 1 − .000001
= 0.999999.

 

 

11/13/17

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 B represent the event that the customer buys bread, M the event that the customer buys milk. Then, according to the rule of addition, we have

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

 

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



11/6/17

Rectangular Solids

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

Solution

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).


10/30/17

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.

 

10/23/17

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


10/16/17

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 x and y.

x @ 0 = x

x @ y = y @ x

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

.

10/9/17

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.